Subgroups — C₃₁×#17677 #21474436

Order 7936.

976 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1535	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1532	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1531	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1530	C₂	1	2	yes	maximal, complemented
8	C₃₁×#533	C₄	1	1	yes	characteristic
9	C₃₁×#533	C₄	1	1	yes	characteristic
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₄	1	1	yes	characteristic
12	C₃₁×#532	C₄	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	C₂²	1	2	yes	—
19	C₃₁×#529	C₂²	1	2	yes	—
20	C₃₁×#525	—	2	1	no	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	C₄	1	2	yes	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₄	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#432	—	2	1	no	—
28	C₃₁×#432	—	2	1	no	—
29	C₃₁×#432	—	2	1	no	—
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#432	C₂²	1	2	yes	—
32	C₃₁×#432	C₂²	1	2	yes	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#432	C₂²	1	2	yes	—
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
37	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
38	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
39	C₂×C₃₁×#97	C₂²	1	2	yes	—
40	C₂×C₃₁×#97	C₂²	1	2	yes	—
41	C₃₁×#330	C₂²	1	8	yes	complemented
42	C₃₁×#329	C₂²	1	8	yes	complemented
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#126	—	2	2	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#125	C₂³	1	2	yes	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#124	C₂³	1	2	yes	—
63	C₃₁×#123	—	2	2	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂×C₄	1	2	yes	—
78	C₃₁×#120	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
96	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₃₁	—	2	2	no	—
104	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	—	2	2	no	—
117	C₃₁×C₄⋊C₈	—	2	2	no	—
118	C₃₁×C₄⋊C₈	—	2	2	no	—
119	C₃₁×C₄⋊C₈	—	2	2	no	—
120	C₃₁×C₂²⋊C₈	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	2	no	—
122	C₃₁×C₂²⋊C₈	—	2	2	no	—
123	C₃₁×C₂²⋊C₈	—	2	2	no	—
124	C₃₁×#97	—	2	2	no	—
125	C₃₁×#97	—	2	2	no	—
126	C₃₁×#97	C₂³	1	4	yes	—
127	C₃₁×#97	C₂³	1	4	yes	—
128	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
129	#41×C₃₁	—	4	4	no	—
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	4	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	2	no	—
146	C₂²×C₄×C₃₁	—	4	2	no	—
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₃₁	—	2	4	no	—
166	C₂×C₈×C₃₁	—	2	4	no	—
167	C₂×C₈×C₃₁	—	4	4	no	—
168	C₂×C₈×C₃₁	—	4	4	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	2	2	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	2	2	no	—
192	C₄²×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
202	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
209	#17677	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	4	1	no	—
214	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	Q₈×C₃₁	—	4	2	no	—
218	Q₈×C₃₁	—	4	2	no	—
219	Q₈×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	2	no	—
222	D₄×C₃₁	—	4	4	no	—
223	D₄×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	4	4	no	—
225	D₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	4	no	—
248	C₂×C₄×C₃₁	—	4	4	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
262	C₈×C₃₁	—	4	4	no	—
263	C₈×C₃₁	—	4	4	no	—
264	#3444	C₂×C₃₁	1	1	yes	characteristic
265	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1535	C₂×C₃₁	1	4	yes	complemented
268	#1532	C₂×C₃₁	1	4	yes	complemented
269	#1531	C₂×C₃₁	1	2	yes	complemented
270	#1530	C₂×C₃₁	1	2	yes	complemented
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	1	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	4	4	no	—
279	C₂²×C₃₁	—	4	2	no	—
280	C₂²×C₃₁	—	4	2	no	—
281	C₂²×C₃₁	—	4	2	no	—
282	C₂²×C₃₁	#413	1	2	yes	—
283	C₂²×C₃₁	#421	1	2	yes	—
284	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	1	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	—	4	2	no	—
298	C₄×C₃₁	—	4	2	no	—
299	C₄×C₃₁	#421	1	2	yes	—
300	#533	C₄×C₃₁	1	1	yes	characteristic
301	#533	C₄×C₃₁	1	1	yes	characteristic
302	#533	C₂²×C₃₁	1	1	yes	characteristic
303	#532	C₄×C₃₁	1	1	yes	characteristic
304	#532	C₄×C₃₁	1	1	yes	characteristic
305	#532	C₂²×C₃₁	1	1	yes	characteristic
306	#529	—	2	1	no	—
307	#529	—	2	1	no	—
308	#529	—	2	1	no	—
309	#529	—	2	1	no	—
310	#529	C₂²×C₃₁	1	2	yes	—
311	#529	C₂²×C₃₁	1	2	yes	—
312	#525	—	2	1	no	—
313	#525	—	2	1	no	—
314	#525	C₄×C₃₁	1	2	yes	—
315	C₄×#41	—	2	1	no	—
316	C₄×#41	—	2	1	no	—
317	C₄×#41	C₄×C₃₁	1	2	yes	—
318	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
319	#432	—	2	1	no	—
320	#432	—	2	1	no	—
321	#432	—	2	1	no	—
322	#432	—	2	1	no	—
323	#432	C₂²×C₃₁	1	2	yes	—
324	#432	C₂²×C₃₁	1	2	yes	—
325	#432	C₂²×C₃₁	1	2	yes	—
326	#432	C₂²×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₈	—	2	1	no	—
328	C₂×C₄⋊C₈	—	2	1	no	—
329	C₂×C₂²⋊C₈	—	2	1	no	—
330	C₂×C₂²⋊C₈	—	2	1	no	—
331	C₂×#97	C₂²×C₃₁	1	2	yes	—
332	C₂×#97	C₂²×C₃₁	1	2	yes	—
333	#330	C₂²×C₃₁	1	8	yes	complemented
334	#329	C₂²×C₃₁	1	8	yes	complemented
335	C₂×C₃₁	—	2	2	no	—
336	C₂×C₃₁	—	2	2	no	—
337	C₂×C₃₁	—	4	1	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
341	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
342	C₂×#41	—	2	1	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
345	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₈	—	2	2	no	—
349	C₂²×C₈	—	2	2	no	—
350	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
353	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
354	#126	—	2	2	no	—
355	#126	—	2	2	no	—
356	#125	—	2	1	no	—
357	#125	—	2	1	no	—
358	#125	C₂³×C₃₁	1	2	yes	—
359	#124	—	2	1	no	—
360	#124	—	2	1	no	—
361	#124	C₂³×C₃₁	1	2	yes	—
362	#123	—	2	2	no	—
363	#123	—	2	2	no	—
364	#123	—	2	2	no	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	2	1	no	—
369	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
370	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
371	#121	—	2	1	no	—
372	#121	—	2	1	no	—
373	#121	—	2	1	no	—
374	#121	—	2	1	no	—
375	#121	C₂×C₄×C₃₁	1	2	yes	—
376	#121	C₂×C₄×C₃₁	1	2	yes	—
377	#120	—	2	2	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
383	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
395	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
396	#117	—	2	1	no	—
397	#117	—	2	1	no	—
398	#117	—	2	2	no	—
399	#117	—	2	2	no	—
400	#117	—	2	2	no	—
401	#117	C₂×C₄×C₃₁	1	2	yes	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
406	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
412	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
415	C₄⋊C₈	—	2	2	no	—
416	C₄⋊C₈	—	2	2	no	—
417	C₄⋊C₈	—	2	2	no	—
418	C₄⋊C₈	—	2	2	no	—
419	C₂²⋊C₈	—	2	2	no	—
420	C₂²⋊C₈	—	2	2	no	—
421	C₂²⋊C₈	—	2	2	no	—
422	C₂²⋊C₈	—	2	2	no	—
423	#97	—	2	2	no	—
424	#97	—	2	2	no	—
425	#97	C₂³×C₃₁	1	4	yes	—
426	#97	C₂³×C₃₁	1	4	yes	—
427	C₃₁	#17677	1	1	yes	characteristic, direct product factor, Sylow
428	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
429	#41	—	4	4	no	—
430	#41	—	4	4	no	—
431	#41	—	4	4	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₈	—	2	4	no	—
466	C₂×C₈	—	2	4	no	—
467	C₂×C₈	—	4	4	no	—
468	C₂×C₈	—	4	4	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
480	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	2	no	—
493	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
494	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
495	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
496	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
497	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
498	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
499	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
500	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
501	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
502	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
506	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
507	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	4	1	no	—
513	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
514	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
515	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
516	Q₈	—	4	2	no	—
517	Q₈	—	4	2	no	—
518	Q₈	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	4	no	—
522	D₄	—	4	4	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
550	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
552	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
553	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
561	C₈	—	4	4	no	—
562	C₈	—	4	4	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	C₃₁×#413	1	2	yes	—
575	C₂²	C₃₁×#421	1	2	yes	—
576	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	2	no	—
591	C₄	C₃₁×#421	1	2	yes	—
592	C₂	—	2	2	no	—
593	C₂	—	2	2	no	—
594	C₂	—	4	1	no	—
595	C₂	—	8	2	no	—
596	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
597	C₂	C₂×C₃₁×#421	1	1	yes	characteristic
598	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#432	C₂	semidirect
3. C₂×C₃₁×#432	C₂	semidirect
4. C₃₁×#1535	C₂	semidirect
5. C₃₁×#1532	C₂	semidirect
6. C₃₁×#1531	C₂	semidirect
7. C₃₁×#1530	C₂	semidirect
41. C₃₁×#330	C₂²	semidirect
42. C₃₁×#329	C₂²	semidirect
209. #17677	C₃₁	direct (N × H)
265. C₂×#432	C₂×C₃₁	semidirect
266. C₂×#432	C₂×C₃₁	semidirect
267. #1535	C₂×C₃₁	semidirect
268. #1532	C₂×C₃₁	semidirect
269. #1531	C₂×C₃₁	semidirect
270. #1530	C₂×C₃₁	semidirect
333. #330	C₂²×C₃₁	semidirect
334. #329	C₂²×C₃₁	semidirect
427. C₃₁	#17677	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×#3444	2
C₂×C₃₁×#432	2
C₃₁×#1535	2
C₃₁×#1532	2
C₃₁×#1531	2
C₃₁×#1530	2
#17677	31

Subgroups — C31×#17677 #21474436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17677 #21474436