Subgroups — C₃₁×#17726 #21474485

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1554	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1543	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1542	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#556	C₄	1	2	yes	—
13	C₃₁×#555	—	2	1	no	—
14	C₃₁×#554	—	2	1	no	—
15	C₃₁×#553	—	2	2	no	—
16	C₃₁×#548	C₄	1	4	yes	—
17	C₃₁×#547	C₄	1	2	yes	—
18	C₃₁×#542	—	2	2	no	—
19	C₃₁×#537	—	2	1	no	—
20	C₃₁×#533	C₂²	1	2	yes	—
21	C₃₁×#532	C₂²	1	2	yes	—
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×#520	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#431	C₂²	1	2	yes	—
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
38	C₃₁×#408	—	2	1	no	—
39	C₃₁×#405	—	2	1	no	—
40	C₃₁×#333	C₂²	1	2	yes	—
41	C₃₁×#333	C₂²	1	2	yes	—
42	C₃₁×#333	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#332	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	1	yes	characteristic
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	1	yes	characteristic
48	C₃₁×#331	C₂²	1	1	yes	characteristic
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	—	2	1	no	—
51	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#127	C₂³	1	1	yes	characteristic
64	C₃₁×#126	—	2	4	no	—
65	C₃₁×#125	—	2	2	no	—
66	C₃₁×#124	—	2	4	no	—
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#124	C₂×C₄	1	4	yes	—
69	C₃₁×#123	—	2	4	no	—
70	C₃₁×C₂²⋊Q₈	—	2	2	no	—
71	C₃₁×C₂²⋊Q₈	—	2	2	no	—
72	C₃₁×C₂²⋊Q₈	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	—	2	4	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₃₁×#121	—	2	2	no	—
79	C₃₁×#121	C₂×C₄	1	2	yes	—
80	C₄×Q₈×C₃₁	—	2	2	no	—
81	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
82	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
84	C₄×D₄×C₃₁	—	2	4	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	1	no	—
88	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₄×D₄×C₃₁	D₄	1	2	yes	—
92	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
93	C₃₁×#117	—	2	2	no	—
94	C₃₁×#117	—	2	2	no	—
95	C₃₁×#117	D₄	1	1	yes	characteristic
96	C₃₁×#117	D₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	2	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
107	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
109	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	1	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	2	no	—
119	C₄×C₈×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	4	2	no	—
123	#41×C₃₁	—	4	2	no	—
124	#41×C₃₁	—	8	4	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	8	2	no	—
128	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	4	no	—
132	C₂×D₄×C₃₁	—	2	1	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
135	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
136	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	M₁₆×C₃₁	—	4	2	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	4	2	no	—
154	C₂×C₈×C₃₁	—	8	1	no	—
155	C₂×C₈×C₃₁	—	8	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
166	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	2	no	—
172	C₂²⋊C₄×C₃₁	—	2	2	no	—
173	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
174	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
175	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
176	C₄²×C₃₁	—	2	4	no	—
177	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
179	C₄²×C₃₁	C₂×D₄	1	2	yes	—
180	C₄²×C₃₁	C₂×D₄	1	2	yes	—
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
184	#17726	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	4	4	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	4	1	no	—
192	Q₈×C₃₁	—	8	2	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	4	2	no	—
196	D₄×C₃₁	—	4	4	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	1	no	—
199	D₄×C₃₁	—	4	1	no	—
200	D₄×C₃₁	—	4	1	no	—
201	D₄×C₃₁	—	8	4	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	8	4	no	—
214	C₂×C₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	#135	1	2	yes	—
222	C₂×C₄×C₃₁	#135	1	2	yes	—
223	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₃₁	—	4	1	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	—	4	1	no	—
232	#3511	C₂×C₃₁	1	1	yes	characteristic
233	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1554	C₂×C₃₁	1	2	yes	complemented
236	#1551	C₂×C₃₁	1	2	yes	complemented
237	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1543	C₂×C₃₁	1	2	yes	complemented
240	#1542	C₂×C₃₁	1	2	yes	complemented
241	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	2	no	—
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	#416	1	2	yes	—
252	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	1	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	1	no	—
259	C₄×C₃₁	—	4	4	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	#416	1	2	yes	—
262	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
263	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
264	C₄×C₃₁	#577	1	1	yes	characteristic
265	C₄×C₃₁	#578	1	1	yes	characteristic
266	#556	C₄×C₃₁	1	2	yes	—
267	#555	—	2	1	no	—
268	#554	—	2	1	no	—
269	#553	—	2	2	no	—
270	#548	C₄×C₃₁	1	4	yes	—
271	#547	C₄×C₃₁	1	2	yes	—
272	#542	—	2	2	no	—
273	#537	—	2	1	no	—
274	#533	C₂²×C₃₁	1	2	yes	—
275	#532	C₂²×C₃₁	1	2	yes	—
276	#532	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	#520	—	2	1	no	—
279	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
280	C₈×Q₈	—	2	1	no	—
281	C₈×D₄	—	2	2	no	—
282	C₈×D₄	—	2	1	no	—
283	#433	—	2	2	no	—
284	#432	C₂²×C₃₁	1	2	yes	—
285	#431	C₂²×C₃₁	1	1	yes	characteristic
286	#431	C₂²×C₃₁	1	2	yes	—
287	#431	C₂²×C₃₁	1	2	yes	—
288	#423	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
292	#408	—	2	1	no	—
293	#405	—	2	1	no	—
294	#333	C₂²×C₃₁	1	2	yes	—
295	#333	C₂²×C₃₁	1	2	yes	—
296	#333	C₂²×C₃₁	1	2	yes	—
297	#333	C₂²×C₃₁	1	2	yes	—
298	#332	C₂²×C₃₁	1	1	yes	characteristic
299	#332	C₂²×C₃₁	1	1	yes	characteristic
300	#332	C₂²×C₃₁	1	2	yes	—
301	#331	C₂²×C₃₁	1	1	yes	characteristic
302	#331	C₂²×C₃₁	1	1	yes	characteristic
303	#331	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	2	2	no	—
305	C₂×C₃₁	—	4	2	no	—
306	C₂×C₃₁	—	8	2	no	—
307	C₂×C₃₁	#2888	1	1	yes	characteristic
308	C₂×C₃₁	C₂×#416	1	1	yes	characteristic
309	C₂×C₃₁	#2898	1	1	yes	characteristic
310	C₂×#41	—	2	1	no	—
311	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
312	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
313	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
314	C₂×M₁₆	—	2	1	no	—
315	C₂²×C₈	—	2	1	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	—	2	1	no	—
320	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
322	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
323	#127	C₂³×C₃₁	1	1	yes	characteristic
324	#126	—	2	4	no	—
325	#125	—	2	2	no	—
326	#124	—	2	4	no	—
327	#124	—	2	1	no	—
328	#124	C₂×C₄×C₃₁	1	4	yes	—
329	#123	—	2	4	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
334	#121	—	2	2	no	—
335	#121	—	2	4	no	—
336	#121	—	2	2	no	—
337	#121	—	2	1	no	—
338	#121	—	2	2	no	—
339	#121	C₂×C₄×C₃₁	1	2	yes	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
342	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
343	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
349	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
351	C₄×D₄	D₄×C₃₁	1	2	yes	—
352	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
353	#117	—	2	2	no	—
354	#117	—	2	2	no	—
355	#117	D₄×C₃₁	1	1	yes	characteristic
356	#117	D₄×C₃₁	1	2	yes	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
362	C₄⋊C₈	—	2	1	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	2	1	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
368	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
371	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	—	2	1	no	—
375	C₂²⋊C₈	—	2	2	no	—
376	#97	—	2	2	no	—
377	C₄×C₈	—	2	2	no	—
378	C₄×C₈	—	2	2	no	—
379	C₄×C₈	—	2	1	no	—
380	C₃₁	#17726	1	1	yes	characteristic, direct product factor, Sylow
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	—	4	2	no	—
384	#41	—	4	2	no	—
385	#41	—	8	4	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	8	2	no	—
389	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
390	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
396	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
397	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	4	2	no	—
415	C₂×C₈	—	8	1	no	—
416	C₂×C₈	—	8	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
425	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
426	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
427	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
428	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
436	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
437	C₄²	—	2	4	no	—
438	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
439	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
440	C₄²	C₂×D₄×C₃₁	1	2	yes	—
441	C₄²	C₂×D₄×C₃₁	1	2	yes	—
442	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
444	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂³	—	2	2	no	—
446	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
447	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	Q₈	—	2	2	no	—
449	Q₈	—	4	4	no	—
450	Q₈	—	4	1	no	—
451	Q₈	—	4	1	no	—
452	Q₈	—	8	2	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	4	no	—
457	D₄	—	4	1	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	8	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	8	4	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
476	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
480	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₃₁×#135	1	2	yes	—
482	C₂×C₄	C₃₁×#135	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₈	—	4	1	no	—
487	C₈	—	4	1	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	C₃₁×#416	1	2	yes	—
501	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
502	C₄	—	2	1	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	2	no	—
510	C₄	C₃₁×#416	1	2	yes	—
511	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
512	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
513	C₄	C₃₁×#577	1	1	yes	characteristic
514	C₄	C₃₁×#578	1	1	yes	characteristic
515	C₂	—	2	2	no	—
516	C₂	—	4	2	no	—
517	C₂	—	8	2	no	—
518	C₂	C₃₁×#2888	1	1	yes	characteristic
519	C₂	C₂×C₃₁×#416	1	1	yes	characteristic
520	C₂	C₃₁×#2898	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×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2988	C₂	semidirect
3. C₃₁×#2925	C₂	semidirect
4. C₃₁×#1554	C₂	semidirect
5. C₃₁×#1551	C₂	semidirect
6. C₃₁×#1548	C₂	semidirect
7. C₃₁×#1547	C₂	semidirect
8. C₃₁×#1543	C₂	semidirect
9. C₃₁×#1542	C₂	semidirect
10. C₂×C₃₁×#333	C₂	semidirect
11. C₃₁×#1540	C₂	semidirect
184. #17726	C₃₁	direct (N × H)
233. #2988	C₂×C₃₁	semidirect
234. #2925	C₂×C₃₁	semidirect
235. #1554	C₂×C₃₁	semidirect
236. #1551	C₂×C₃₁	semidirect
237. #1548	C₂×C₃₁	semidirect
238. #1547	C₂×C₃₁	semidirect
239. #1543	C₂×C₃₁	semidirect
240. #1542	C₂×C₃₁	semidirect
241. C₂×#333	C₂×C₃₁	semidirect
242. #1540	C₂×C₃₁	semidirect
380. C₃₁	#17726	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×#3511	2
C₃₁×#2988	2
C₃₁×#2925	2
C₃₁×#1554	2
C₃₁×#1551	2
C₃₁×#1548	2
C₃₁×#1547	2
C₃₁×#1543	2
C₃₁×#1542	2
C₂×C₃₁×#333	2
C₃₁×#1540	2
#17726	31

Subgroups — C31×#17726 #21474485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17726 #21474485