Subgroups — C₃₁×#17607 #21474366

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3494	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1521	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1519	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1514	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1513	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1509	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1507	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1502	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1501	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#555	—	2	1	no	—
13	C₃₁×#553	—	2	1	no	—
14	C₃₁×#553	—	2	1	no	—
15	C₃₁×#551	—	2	2	no	—
16	C₃₁×#547	—	2	1	no	—
17	C₃₁×#546	—	2	2	no	—
18	C₃₁×#544	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#542	C₄	1	2	yes	complemented
20	C₃₁×#540	C₄	1	2	yes	complemented
21	C₃₁×#535	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₃₁×#529	C₂²	1	2	yes	—
24	C₃₁×#525	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₃₁×#520	C₄	1	1	yes	characteristic, complemented
27	C₃₁×#518	C₄	1	1	yes	characteristic, complemented
28	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	complemented
31	C₃₁×#431	C₂²	1	2	yes	complemented
32	C₃₁×#424	—	2	1	no	—
33	C₃₁×#407	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#405	—	2	1	no	—
36	C₃₁×#389	—	2	1	no	—
37	C₃₁×#388	—	2	2	no	—
38	C₃₁×#383	C₂²	1	2	yes	complemented
39	C₃₁×#380	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
42	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#328	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#328	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#328	C₂²	1	2	yes	complemented
46	C₃₁×#327	C₂²	1	2	yes	—
47	C₃₁×#327	C₂²	1	2	yes	—
48	C₃₁×#327	C₂²	1	2	yes	—
49	C₃₁×#327	C₂²	1	2	yes	—
50	C₃₁×#327	C₂²	1	1	yes	characteristic, complemented
51	C₃₁×#327	C₂²	1	1	yes	characteristic, complemented
52	C₃₁×#327	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
62	C₃₁×#127	C₂×C₄	1	2	yes	—
63	C₃₁×#126	—	2	2	no	—
64	C₃₁×#126	—	2	2	no	—
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#125	C₂×C₄	1	2	yes	—
67	C₃₁×#124	—	2	2	no	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	C₂×C₄	1	2	yes	—
70	C₃₁×#123	—	2	1	no	—
71	C₃₁×#123	—	2	2	no	—
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×#123	—	2	2	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
78	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×#121	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	2	no	—
86	C₃₁×#121	C₂×C₄	1	2	yes	—
87	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
88	C₃₁×#121	C₂³	1	2	yes	—
89	C₃₁×#121	C₂³	1	1	yes	characteristic
90	C₃₁×#120	—	2	1	no	—
91	C₃₁×#120	—	2	2	no	—
92	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	2	no	—
100	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
101	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
102	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
106	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
107	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
108	C₃₁×#117	C₂×C₄	1	2	yes	—
109	C₃₁×#117	D₄	1	1	yes	characteristic
110	C₃₁×#117	D₄	1	1	yes	characteristic
111	C₃₁×#117	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	—	2	1	no	—
124	C₂×C₄²×C₃₁	—	2	2	no	—
125	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₄⋊C₈	—	2	2	no	—
128	C₃₁×C₂²⋊C₈	—	2	1	no	—
129	C₃₁×C₂²⋊C₈	—	2	1	no	—
130	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
131	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
132	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
133	C₃₁×#97	—	2	2	no	—
134	C₄×C₈×C₃₁	—	2	2	no	—
135	C₃₁×#95	—	2	2	no	—
136	C₃₁×#95	—	2	2	no	—
137	C₃₁×#95	C₂³	1	1	yes	characteristic
138	C₃₁×#95	C₂³	1	1	yes	characteristic
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₂⁴×C₃₁	—	2	1	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
142	#41×C₃₁	—	4	2	no	—
143	#41×C₃₁	—	8	2	no	—
144	#41×C₃₁	—	8	2	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	1	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	4	2	no	—
156	C₂×D₄×C₃₁	—	8	1	no	—
157	C₂×D₄×C₃₁	—	8	1	no	—
158	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
159	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	—	2	1	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	4	2	no	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	M₁₆×C₃₁	—	4	2	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	2	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	4	2	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	4	2	no	—
190	C₄⋊C₄×C₃₁	—	4	4	no	—
191	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
196	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
198	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
201	C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₂²⋊C₄×C₃₁	—	2	1	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	1	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	1	no	—
212	C₂²⋊C₄×C₃₁	—	4	2	no	—
213	C₂²⋊C₄×C₃₁	—	8	1	no	—
214	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
215	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
216	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
217	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
218	C₄²×C₃₁	—	2	2	no	—
219	C₄²×C₃₁	—	2	1	no	—
220	C₄²×C₃₁	—	4	2	no	—
221	C₄²×C₃₁	—	8	2	no	—
222	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
223	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
224	C₄²×C₃₁	C₂×D₄	1	2	yes	—
225	C₄²×C₃₁	C₂×D₄	1	2	yes	—
226	#17607	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂³×C₃₁	—	2	1	no	—
228	C₂³×C₃₁	—	2	1	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	1	no	—
231	C₂³×C₃₁	—	4	1	no	—
232	C₂³×C₃₁	—	4	1	no	—
233	C₂³×C₃₁	—	4	2	no	—
234	C₂³×C₃₁	—	8	1	no	—
235	C₂³×C₃₁	—	8	1	no	—
236	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₃₁	—	4	1	no	—
240	Q₈×C₃₁	—	8	2	no	—
241	D₄×C₃₁	—	4	2	no	—
242	D₄×C₃₁	—	4	1	no	—
243	D₄×C₃₁	—	4	2	no	—
244	D₄×C₃₁	—	4	2	no	—
245	D₄×C₃₁	—	4	1	no	—
246	D₄×C₃₁	—	4	2	no	—
247	D₄×C₃₁	—	4	1	no	—
248	D₄×C₃₁	—	4	2	no	—
249	D₄×C₃₁	—	8	2	no	—
250	D₄×C₃₁	—	8	2	no	—
251	D₄×C₃₁	—	8	1	no	—
252	D₄×C₃₁	—	8	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	4	1	no	—
267	C₂×C₄×C₃₁	—	4	1	no	—
268	C₂×C₄×C₃₁	—	4	1	no	—
269	C₂×C₄×C₃₁	—	4	2	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	4	2	no	—
274	C₂×C₄×C₃₁	—	4	2	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	8	2	no	—
277	C₂×C₄×C₃₁	—	8	2	no	—
278	C₂×C₄×C₃₁	—	8	2	no	—
279	C₂×C₄×C₃₁	—	8	1	no	—
280	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
283	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
288	C₈×C₃₁	—	4	1	no	—
289	C₈×C₃₁	—	4	1	no	—
290	C₈×C₃₁	—	4	2	no	—
291	#3494	C₂×C₃₁	1	1	yes	characteristic
292	#2924	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#1521	C₂×C₃₁	1	2	yes	complemented
295	#1519	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#1514	C₂×C₃₁	1	2	yes	complemented
297	#1513	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#1509	C₂×C₃₁	1	2	yes	complemented
299	#1507	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#1502	C₂×C₃₁	1	2	yes	complemented
301	#1501	C₂×C₃₁	1	1	yes	characteristic, complemented
302	C₂²×C₃₁	—	2	1	no	—
303	C₂²×C₃₁	—	2	1	no	—
304	C₂²×C₃₁	—	2	1	no	—
305	C₂²×C₃₁	—	2	1	no	—
306	C₂²×C₃₁	—	2	1	no	—
307	C₂²×C₃₁	—	2	1	no	—
308	C₂²×C₃₁	—	2	1	no	—
309	C₂²×C₃₁	—	2	1	no	—
310	C₂²×C₃₁	—	4	1	no	—
311	C₂²×C₃₁	—	4	2	no	—
312	C₂²×C₃₁	—	4	1	no	—
313	C₂²×C₃₁	—	4	2	no	—
314	C₂²×C₃₁	—	4	2	no	—
315	C₂²×C₃₁	—	4	1	no	—
316	C₂²×C₃₁	—	8	1	no	—
317	C₂²×C₃₁	—	8	1	no	—
318	C₂²×C₃₁	—	16	2	no	—
319	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
320	C₂²×C₃₁	#576	1	1	yes	characteristic
321	C₂²×C₃₁	#577	1	1	yes	characteristic
322	C₄×C₃₁	—	2	1	no	—
323	C₄×C₃₁	—	2	1	no	—
324	C₄×C₃₁	—	2	1	no	—
325	C₄×C₃₁	—	2	2	no	—
326	C₄×C₃₁	—	2	2	no	—
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	4	2	no	—
330	C₄×C₃₁	—	4	1	no	—
331	C₄×C₃₁	—	4	2	no	—
332	C₄×C₃₁	—	4	1	no	—
333	C₄×C₃₁	—	4	1	no	—
334	C₄×C₃₁	—	8	2	no	—
335	C₄×C₃₁	—	8	1	no	—
336	C₄×C₃₁	#410	1	2	yes	—
337	#555	—	2	1	no	—
338	#553	—	2	1	no	—
339	#553	—	2	1	no	—
340	#551	—	2	2	no	—
341	#547	—	2	1	no	—
342	#546	—	2	2	no	—
343	#544	C₄×C₃₁	1	1	yes	characteristic, complemented
344	#542	C₄×C₃₁	1	2	yes	complemented
345	#540	C₄×C₃₁	1	2	yes	complemented
346	#535	C₄×C₃₁	1	1	yes	characteristic, complemented
347	#532	C₂²×C₃₁	1	2	yes	—
348	#529	C₂²×C₃₁	1	2	yes	—
349	#525	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
351	#520	C₄×C₃₁	1	1	yes	characteristic, complemented
352	#518	C₄×C₃₁	1	1	yes	characteristic, complemented
353	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
354	#433	—	2	2	no	—
355	#432	C₂²×C₃₁	1	2	yes	complemented
356	#431	C₂²×C₃₁	1	2	yes	complemented
357	#424	—	2	1	no	—
358	#407	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
360	#405	—	2	1	no	—
361	#389	—	2	1	no	—
362	#388	—	2	2	no	—
363	#383	C₂²×C₃₁	1	2	yes	complemented
364	#380	C₂²×C₃₁	1	1	yes	characteristic
365	C₄×C₄⋊C₄	—	2	1	no	—
366	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
367	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
368	#328	C₂²×C₃₁	1	1	yes	characteristic, complemented
369	#328	C₂²×C₃₁	1	1	yes	characteristic, complemented
370	#328	C₂²×C₃₁	1	2	yes	complemented
371	#327	C₂²×C₃₁	1	2	yes	—
372	#327	C₂²×C₃₁	1	2	yes	—
373	#327	C₂²×C₃₁	1	2	yes	—
374	#327	C₂²×C₃₁	1	2	yes	—
375	#327	C₂²×C₃₁	1	1	yes	characteristic, complemented
376	#327	C₂²×C₃₁	1	1	yes	characteristic, complemented
377	#327	C₂²×C₃₁	1	2	yes	complemented
378	C₂×C₃₁	—	2	1	no	—
379	C₂×C₃₁	—	2	1	no	—
380	C₂×C₃₁	—	4	1	no	—
381	C₂×C₃₁	—	4	1	no	—
382	C₂×C₃₁	—	8	1	no	—
383	C₂×C₃₁	—	8	2	no	—
384	C₂×C₃₁	C₂×#410	1	1	yes	characteristic
385	C₂×C₃₁	#2897	1	1	yes	characteristic
386	C₂×C₃₁	#2901	1	1	yes	characteristic
387	C₂×#41	—	2	2	no	—
388	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
389	C₂²×D₄	—	2	1	no	—
390	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×M₁₆	—	2	1	no	—
393	C₂²×C₈	—	2	1	no	—
394	C₄⋊Q₈	—	2	1	no	—
395	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
396	#127	C₂×C₄×C₃₁	1	2	yes	—
397	#126	—	2	2	no	—
398	#126	—	2	2	no	—
399	#125	—	2	1	no	—
400	#125	C₂×C₄×C₃₁	1	2	yes	—
401	#124	—	2	2	no	—
402	#124	—	2	2	no	—
403	#124	C₂×C₄×C₃₁	1	2	yes	—
404	#123	—	2	1	no	—
405	#123	—	2	2	no	—
406	#123	—	2	2	no	—
407	#123	—	2	2	no	—
408	#123	—	2	2	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
412	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
413	#121	—	2	1	no	—
414	#121	—	2	1	no	—
415	#121	—	2	2	no	—
416	#121	—	2	1	no	—
417	#121	—	2	1	no	—
418	#121	—	2	2	no	—
419	#121	—	2	2	no	—
420	#121	C₂×C₄×C₃₁	1	2	yes	—
421	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
422	#121	C₂³×C₃₁	1	2	yes	—
423	#121	C₂³×C₃₁	1	1	yes	characteristic
424	#120	—	2	1	no	—
425	#120	—	2	2	no	—
426	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
427	C₄×D₄	—	2	1	no	—
428	C₄×D₄	—	2	1	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	—	2	1	no	—
432	C₄×D₄	—	2	2	no	—
433	C₄×D₄	—	2	2	no	—
434	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
435	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
436	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
437	#117	—	2	1	no	—
438	#117	—	2	2	no	—
439	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
440	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
441	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
442	#117	C₂×C₄×C₃₁	1	2	yes	—
443	#117	D₄×C₃₁	1	2	yes	—
444	#117	D₄×C₃₁	1	1	yes	characteristic
445	#117	D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄⋊C₄	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
450	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
451	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₂²⋊C₄	—	2	1	no	—
456	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
457	C₂×C₄²	—	2	1	no	—
458	C₂×C₄²	—	2	2	no	—
459	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
460	C₄⋊C₈	—	2	2	no	—
461	C₄⋊C₈	—	2	2	no	—
462	C₂²⋊C₈	—	2	1	no	—
463	C₂²⋊C₈	—	2	1	no	—
464	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
465	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
467	#97	—	2	2	no	—
468	C₄×C₈	—	2	2	no	—
469	#95	—	2	2	no	—
470	#95	—	2	2	no	—
471	#95	C₂³×C₃₁	1	1	yes	characteristic
472	#95	C₂³×C₃₁	1	1	yes	characteristic
473	#95	C₂³×C₃₁	1	2	yes	—
474	C₃₁	#17607	1	1	yes	characteristic, direct product factor, Sylow
475	C₂⁴	—	2	1	no	—
476	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
477	#41	—	4	2	no	—
478	#41	—	8	2	no	—
479	#41	—	8	2	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	2	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	2	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	4	2	no	—
491	C₂×D₄	—	8	1	no	—
492	C₂×D₄	—	8	1	no	—
493	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
494	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
505	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
510	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
512	M₁₆	—	4	2	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	4	2	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	4	no	—
526	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
530	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
531	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
532	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
533	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
534	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	1	no	—
547	C₂²⋊C₄	—	4	2	no	—
548	C₂²⋊C₄	—	8	1	no	—
549	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
550	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
551	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
552	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
553	C₄²	—	2	2	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	4	2	no	—
556	C₄²	—	8	2	no	—
557	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
558	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
559	C₄²	C₂×D₄×C₃₁	1	2	yes	—
560	C₄²	C₂×D₄×C₃₁	1	2	yes	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	1	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	2	no	—
568	C₂³	—	8	1	no	—
569	C₂³	—	8	1	no	—
570	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
571	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
572	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
573	Q₈	—	4	1	no	—
574	Q₈	—	8	2	no	—
575	D₄	—	4	2	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	2	no	—
578	D₄	—	4	2	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	2	no	—
581	D₄	—	4	1	no	—
582	D₄	—	4	2	no	—
583	D₄	—	8	2	no	—
584	D₄	—	8	2	no	—
585	D₄	—	8	1	no	—
586	D₄	—	8	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	8	2	no	—
611	C₂×C₄	—	8	2	no	—
612	C₂×C₄	—	8	2	no	—
613	C₂×C₄	—	8	1	no	—
614	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
615	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
616	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
617	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
618	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
619	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
620	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
621	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
622	C₈	—	4	1	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	2	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	2	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	8	1	no	—
640	C₂²	—	8	1	no	—
641	C₂²	—	16	2	no	—
642	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
643	C₂²	C₃₁×#576	1	1	yes	characteristic
644	C₂²	C₃₁×#577	1	1	yes	characteristic
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	1	no	—
657	C₄	—	8	2	no	—
658	C₄	—	8	1	no	—
659	C₄	C₃₁×#410	1	2	yes	—
660	C₂	—	2	1	no	—
661	C₂	—	2	1	no	—
662	C₂	—	4	1	no	—
663	C₂	—	4	1	no	—
664	C₂	—	8	2	no	—
665	C₂	—	8	1	no	—
666	C₂	C₂×C₃₁×#410	1	1	yes	characteristic
667	C₂	C₃₁×#2897	1	1	yes	characteristic
668	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3494	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2924	C₂	semidirect
3. C₃₁×#2347	C₂	semidirect
4. C₃₁×#1521	C₂	semidirect
5. C₃₁×#1519	C₂	semidirect
6. C₃₁×#1514	C₂	semidirect
7. C₃₁×#1513	C₂	semidirect
8. C₃₁×#1509	C₂	semidirect
9. C₃₁×#1507	C₂	semidirect
10. C₃₁×#1502	C₂	semidirect
11. C₃₁×#1501	C₂	semidirect
18. C₃₁×#544	C₄	semidirect
19. C₃₁×#542	C₄	semidirect
20. C₃₁×#540	C₄	semidirect
21. C₃₁×#535	C₄	semidirect
26. C₃₁×#520	C₄	semidirect
27. C₃₁×#518	C₄	semidirect
30. C₃₁×#432	C₂²	semidirect
31. C₃₁×#431	C₂²	semidirect
38. C₃₁×#383	C₂²	semidirect
41. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
43. C₃₁×#328	C₂²	semidirect
44. C₃₁×#328	C₂²	semidirect
45. C₃₁×#328	C₂²	semidirect
50. C₃₁×#327	C₂²	semidirect
51. C₃₁×#327	C₂²	semidirect
52. C₃₁×#327	C₂²	semidirect
226. #17607	C₃₁	direct (N × H)
292. #2924	C₂×C₃₁	semidirect
293. #2347	C₂×C₃₁	semidirect
294. #1521	C₂×C₃₁	semidirect
295. #1519	C₂×C₃₁	semidirect
296. #1514	C₂×C₃₁	semidirect
297. #1513	C₂×C₃₁	semidirect
298. #1509	C₂×C₃₁	semidirect
299. #1507	C₂×C₃₁	semidirect
300. #1502	C₂×C₃₁	semidirect
301. #1501	C₂×C₃₁	semidirect
343. #544	C₄×C₃₁	semidirect
344. #542	C₄×C₃₁	semidirect
345. #540	C₄×C₃₁	semidirect
346. #535	C₄×C₃₁	semidirect
351. #520	C₄×C₃₁	semidirect
352. #518	C₄×C₃₁	semidirect
355. #432	C₂²×C₃₁	semidirect
356. #431	C₂²×C₃₁	semidirect
363. #383	C₂²×C₃₁	semidirect
366. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
368. #328	C₂²×C₃₁	semidirect
369. #328	C₂²×C₃₁	semidirect
370. #328	C₂²×C₃₁	semidirect
375. #327	C₂²×C₃₁	semidirect
376. #327	C₂²×C₃₁	semidirect
377. #327	C₂²×C₃₁	semidirect
474. C₃₁	#17607	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3494	2
C₃₁×#2924	2
C₃₁×#2347	2
C₃₁×#1521	2
C₃₁×#1519	2
C₃₁×#1514	2
C₃₁×#1513	2
C₃₁×#1509	2
C₃₁×#1507	2
C₃₁×#1502	2
C₃₁×#1501	2
#17607	31

Subgroups — C31×#17607 #21474366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17607 #21474366