Subgroups — C₃₁×#61015 #21517774

Order 7936.

1768 conjugacy classes of subgroups.

The subgroup lattice has 564 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	2	yes	maximal, complemented
2	C₃₁×#3263	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3229	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3227	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3210	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3207	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3204	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3167	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3125	C₂	1	8	yes	maximal, complemented
12	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#556	—	2	2	no	—
14	C₃₁×#555	—	2	2	no	—
15	C₃₁×#554	—	2	2	no	—
16	C₃₁×#553	—	2	2	no	—
17	C₃₁×#553	—	2	2	no	—
18	C₃₁×#548	—	2	2	no	—
19	C₃₁×#548	—	2	2	no	—
20	C₃₁×#547	—	2	2	no	—
21	C₃₁×#542	C₂²	1	4	yes	—
22	C₃₁×#542	C₂²	1	4	yes	—
23	C₃₁×#537	C₂²	1	4	yes	—
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#533	C₂²	1	2	yes	—
26	C₃₁×#532	C₂²	1	2	yes	—
27	C₃₁×#532	C₂²	1	2	yes	—
28	C₃₁×#532	C₂²	1	2	yes	—
29	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
30	C₃₁×#520	C₂²	1	4	yes	—
31	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#498	C₂²	1	2	yes	—
33	C₃₁×#498	C₂²	1	1	yes	characteristic
34	C₃₁×#498	C₂²	1	1	yes	characteristic
35	C₃₁×#494	C₂²	1	1	yes	characteristic
36	C₃₁×#493	C₂²	1	1	yes	characteristic
37	C₃₁×#492	C₂²	1	2	yes	—
38	C₃₁×#487	C₂²	1	2	yes	—
39	C₃₁×#487	C₂²	1	2	yes	—
40	C₃₁×#486	C₂²	1	2	yes	—
41	C₃₁×#486	C₂²	1	2	yes	—
42	C₃₁×#483	C₂²	1	8	yes	—
43	C₃₁×#481	C₂²	1	8	yes	—
44	C₃₁×#476	C₂²	1	8	yes	—
45	C₃₁×#475	C₂²	1	8	yes	—
46	C₃₁×#470	C₂²	1	4	yes	—
47	C₃₁×#469	C₂²	1	4	yes	—
48	C₃₁×#465	C₂²	1	8	yes	—
49	C₃₁×#464	C₂²	1	8	yes	—
50	C₃₁×#463	C₂²	1	8	yes	—
51	C₃₁×#452	C₂²	1	8	yes	—
52	C₃₁×#449	C₂²	1	8	yes	—
53	C₃₁×#442	C₂²	1	4	yes	—
54	C₃₁×#441	C₂²	1	4	yes	—
55	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
56	C₃₁×#433	C₂²	1	4	yes	—
57	C₃₁×#431	C₂²	1	2	yes	—
58	C₃₁×#431	C₂²	1	2	yes	—
59	C₃₁×#424	C₂²	1	2	yes	—
60	C₃₁×#408	C₂²	1	2	yes	—
61	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	—	2	2	no	—
64	C₂×#41×C₃₁	—	2	2	no	—
65	C₂×#41×C₃₁	C₂³	1	4	yes	—
66	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	—	2	2	no	—
68	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
70	C₂×D₈×C₃₁	C₂³	1	4	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
73	C₃₁×C₄⋊Q₈	—	2	2	no	—
74	C₃₁×C₄⋊Q₈	—	2	2	no	—
75	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
77	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×#127	C₂³	1	2	yes	—
80	C₃₁×#126	—	2	4	no	—
81	C₃₁×#126	—	2	4	no	—
82	C₃₁×#125	—	2	2	no	—
83	C₃₁×#125	—	2	2	no	—
84	C₃₁×#124	—	2	4	no	—
85	C₃₁×#124	—	2	4	no	—
86	C₃₁×#124	—	2	2	no	—
87	C₃₁×#124	—	2	2	no	—
88	C₃₁×#124	C₂³	1	4	yes	—
89	C₃₁×#123	—	2	4	no	—
90	C₃₁×#123	—	2	4	no	—
91	C₃₁×C₂²⋊Q₈	—	2	4	no	—
92	C₃₁×C₂²⋊Q₈	—	2	2	no	—
93	C₃₁×C₂²⋊Q₈	—	2	2	no	—
94	C₃₁×C₂²⋊Q₈	—	2	4	no	—
95	C₃₁×C₂²⋊Q₈	—	2	4	no	—
96	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
97	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	—	2	4	no	—
101	C₃₁×#121	—	2	2	no	—
102	C₃₁×#121	—	2	2	no	—
103	C₃₁×#121	—	2	2	no	—
104	C₃₁×#121	—	2	2	no	—
105	C₃₁×#121	C₂³	1	4	yes	—
106	C₃₁×#121	C₂³	1	4	yes	—
107	C₄×Q₈×C₃₁	—	2	2	no	—
108	C₄×Q₈×C₃₁	—	2	2	no	—
109	C₄×Q₈×C₃₁	D₄	1	2	yes	—
110	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
111	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
112	C₄×D₄×C₃₁	—	2	4	no	—
113	C₄×D₄×C₃₁	—	2	4	no	—
114	C₄×D₄×C₃₁	—	2	2	no	—
115	C₄×D₄×C₃₁	—	2	2	no	—
116	C₄×D₄×C₃₁	—	2	2	no	—
117	C₄×D₄×C₃₁	—	2	2	no	—
118	C₄×D₄×C₃₁	D₄	1	2	yes	—
119	C₄×D₄×C₃₁	D₄	1	2	yes	—
120	C₄×D₄×C₃₁	D₄	1	2	yes	—
121	C₄×D₄×C₃₁	C₂³	1	4	yes	—
122	C₄×D₄×C₃₁	C₂³	1	4	yes	—
123	C₃₁×#117	—	2	2	no	—
124	C₃₁×#117	—	2	2	no	—
125	C₃₁×#117	—	2	4	no	—
126	C₃₁×#117	C₂³	1	1	yes	characteristic
127	C₃₁×#117	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#106	C₂³	1	8	yes	—
134	C₃₁×#106	C₂³	1	4	yes	—
135	C₃₁×#106	C₂³	1	4	yes	—
136	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
138	C₃₁×#103	C₂³	1	8	yes	—
139	C₃₁×#103	C₂³	1	8	yes	—
140	C₃₁×#102	C₂³	1	8	yes	—
141	C₃₁×#102	C₂³	1	8	yes	—
142	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
143	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
144	C₃₁×#97	C₂³	1	2	yes	—
145	C₄×C₈×C₃₁	C₂³	1	2	yes	—
146	C₄×C₈×C₃₁	C₂³	1	2	yes	—
147	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
149	#41×C₃₁	—	4	2	no	—
150	#41×C₃₁	—	4	8	no	—
151	#41×C₃₁	—	8	4	no	—
152	#41×C₃₁	—	8	4	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	2	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	8	4	no	—
157	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
159	C₂×D₄×C₃₁	—	2	4	no	—
160	C₂×D₄×C₃₁	—	2	4	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	2	no	—
163	C₂×D₄×C₃₁	—	2	2	no	—
164	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
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	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₃₁	—	4	4	no	—
177	SD₁₆×C₃₁	—	4	8	no	—
178	D₈×C₃₁	—	4	4	no	—
179	M₁₆×C₃₁	—	4	2	no	—
180	C₂×C₈×C₃₁	—	4	2	no	—
181	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
182	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
183	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	2	4	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₂²⋊C₄×C₃₁	—	2	4	no	—
207	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
208	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
209	C₂²⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	—	2	4	no	—
212	C₄²×C₃₁	—	2	4	no	—
213	C₄²×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	C₂×D₄	1	2	yes	—
215	C₄²×C₃₁	C₂×D₄	1	2	yes	—
216	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄²×C₃₁	C₂⁴	1	2	yes	—
219	#61015	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₃₁	—	2	4	no	—
221	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
223	Q₈×C₃₁	—	2	8	no	—
224	Q₈×C₃₁	—	4	1	no	—
225	Q₈×C₃₁	—	4	4	no	—
226	Q₈×C₃₁	—	4	4	no	—
227	Q₈×C₃₁	—	8	4	no	—
228	D₄×C₃₁	—	2	8	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	1	no	—
231	D₄×C₃₁	—	4	8	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	8	4	no	—
235	D₄×C₃₁	—	8	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	4	8	no	—
245	C₂×C₄×C₃₁	—	8	4	no	—
246	C₂×C₄×C₃₁	—	8	4	no	—
247	C₂×C₄×C₃₁	—	8	4	no	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
255	C₈×C₃₁	—	2	4	no	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	—	2	2	no	—
258	#3511	C₂×C₃₁	1	2	yes	complemented
259	#3263	C₂×C₃₁	1	4	yes	complemented
260	#3229	C₂×C₃₁	1	2	yes	complemented
261	#3227	C₂×C₃₁	1	2	yes	complemented
262	#3210	C₂×C₃₁	1	2	yes	complemented
263	#3207	C₂×C₃₁	1	2	yes	complemented
264	#3204	C₂×C₃₁	1	4	yes	complemented
265	#3185	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#3184	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#3167	C₂×C₃₁	1	2	yes	complemented
268	#3125	C₂×C₃₁	1	8	yes	complemented
269	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂²×C₃₁	—	2	2	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	4	4	no	—
276	C₂²×C₃₁	—	4	4	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	4	no	—
287	C₄×C₃₁	#577	1	4	yes	—
288	C₄×C₃₁	#585	1	2	yes	—
289	#556	—	2	2	no	—
290	#555	—	2	2	no	—
291	#554	—	2	2	no	—
292	#553	—	2	2	no	—
293	#553	—	2	2	no	—
294	#548	—	2	2	no	—
295	#548	—	2	2	no	—
296	#547	—	2	2	no	—
297	#542	C₂²×C₃₁	1	4	yes	—
298	#542	C₂²×C₃₁	1	4	yes	—
299	#537	C₂²×C₃₁	1	4	yes	—
300	#533	C₂²×C₃₁	1	2	yes	—
301	#533	C₂²×C₃₁	1	2	yes	—
302	#532	C₂²×C₃₁	1	2	yes	—
303	#532	C₂²×C₃₁	1	2	yes	—
304	#532	C₂²×C₃₁	1	2	yes	—
305	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
306	#520	C₂²×C₃₁	1	4	yes	—
307	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
308	#498	C₂²×C₃₁	1	2	yes	—
309	#498	C₂²×C₃₁	1	1	yes	characteristic
310	#498	C₂²×C₃₁	1	1	yes	characteristic
311	#494	C₂²×C₃₁	1	1	yes	characteristic
312	#493	C₂²×C₃₁	1	1	yes	characteristic
313	#492	C₂²×C₃₁	1	2	yes	—
314	#487	C₂²×C₃₁	1	2	yes	—
315	#487	C₂²×C₃₁	1	2	yes	—
316	#486	C₂²×C₃₁	1	2	yes	—
317	#486	C₂²×C₃₁	1	2	yes	—
318	#483	C₂²×C₃₁	1	8	yes	—
319	#481	C₂²×C₃₁	1	8	yes	—
320	#476	C₂²×C₃₁	1	8	yes	—
321	#475	C₂²×C₃₁	1	8	yes	—
322	#470	C₂²×C₃₁	1	4	yes	—
323	#469	C₂²×C₃₁	1	4	yes	—
324	#465	C₂²×C₃₁	1	8	yes	—
325	#464	C₂²×C₃₁	1	8	yes	—
326	#463	C₂²×C₃₁	1	8	yes	—
327	#452	C₂²×C₃₁	1	8	yes	—
328	#449	C₂²×C₃₁	1	8	yes	—
329	#442	C₂²×C₃₁	1	4	yes	—
330	#441	C₂²×C₃₁	1	4	yes	—
331	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
332	#433	C₂²×C₃₁	1	4	yes	—
333	#431	C₂²×C₃₁	1	2	yes	—
334	#431	C₂²×C₃₁	1	2	yes	—
335	#424	C₂²×C₃₁	1	2	yes	—
336	#408	C₂²×C₃₁	1	2	yes	—
337	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₃₁	—	4	2	no	—
340	C₂×C₃₁	—	4	1	no	—
341	C₂×C₃₁	—	8	4	no	—
342	C₂×C₃₁	#3449	1	1	yes	characteristic
343	C₂×C₃₁	C₂×#577	1	2	yes	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	C₂³×C₃₁	1	4	yes	—
347	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
348	C₂²×Q₈	—	2	2	no	—
349	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
350	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
351	C₂×D₈	C₂³×C₃₁	1	4	yes	—
352	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
353	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
357	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
359	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
360	#127	C₂³×C₃₁	1	2	yes	—
361	#126	—	2	4	no	—
362	#126	—	2	4	no	—
363	#125	—	2	2	no	—
364	#125	—	2	2	no	—
365	#124	—	2	4	no	—
366	#124	—	2	4	no	—
367	#124	—	2	2	no	—
368	#124	—	2	2	no	—
369	#124	C₂³×C₃₁	1	4	yes	—
370	#123	—	2	4	no	—
371	#123	—	2	4	no	—
372	C₂²⋊Q₈	—	2	4	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	4	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
378	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
379	#121	—	2	4	no	—
380	#121	—	2	4	no	—
381	#121	—	2	4	no	—
382	#121	—	2	2	no	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	C₂³×C₃₁	1	4	yes	—
387	#121	C₂³×C₃₁	1	4	yes	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	D₄×C₃₁	1	2	yes	—
391	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
392	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	D₄×C₃₁	1	2	yes	—
400	C₄×D₄	D₄×C₃₁	1	2	yes	—
401	C₄×D₄	D₄×C₃₁	1	2	yes	—
402	C₄×D₄	C₂³×C₃₁	1	4	yes	—
403	C₄×D₄	C₂³×C₃₁	1	4	yes	—
404	#117	—	2	2	no	—
405	#117	—	2	2	no	—
406	#117	—	2	4	no	—
407	#117	C₂³×C₃₁	1	2	yes	—
408	#117	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	—	2	2	no	—
410	C₂×C₄⋊C₄	—	2	4	no	—
411	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
412	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
414	#106	C₂³×C₃₁	1	8	yes	—
415	#106	C₂³×C₃₁	1	4	yes	—
416	#106	C₂³×C₃₁	1	4	yes	—
417	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
418	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
419	#103	C₂³×C₃₁	1	8	yes	—
420	#103	C₂³×C₃₁	1	8	yes	—
421	#102	C₂³×C₃₁	1	8	yes	—
422	#102	C₂³×C₃₁	1	8	yes	—
423	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
424	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
425	#97	C₂³×C₃₁	1	2	yes	—
426	C₄×C₈	C₂³×C₃₁	1	2	yes	—
427	C₄×C₈	C₂³×C₃₁	1	2	yes	—
428	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
430	C₃₁	#61015	1	1	yes	characteristic, direct product factor, Sylow
431	#41	—	4	2	no	—
432	#41	—	4	8	no	—
433	#41	—	8	4	no	—
434	#41	—	8	4	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	8	4	no	—
439	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
448	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
458	Q₁₆	—	4	4	no	—
459	SD₁₆	—	4	8	no	—
460	D₈	—	4	4	no	—
461	M₁₆	—	4	2	no	—
462	C₂×C₈	—	4	2	no	—
463	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
464	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
465	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
478	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
480	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
490	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
491	C₂²⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
492	C₂²⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₄²	—	2	4	no	—
494	C₄²	—	2	4	no	—
495	C₄²	C₂×D₄×C₃₁	1	2	yes	—
496	C₄²	C₂×D₄×C₃₁	1	2	yes	—
497	C₄²	C₂×D₄×C₃₁	1	2	yes	—
498	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
499	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
500	C₄²	C₂⁴×C₃₁	1	2	yes	—
501	C₂³	—	2	4	no	—
502	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
504	Q₈	—	2	8	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	4	no	—
507	Q₈	—	4	4	no	—
508	Q₈	—	8	4	no	—
509	D₄	—	2	8	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	8	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	4	no	—
515	D₄	—	8	4	no	—
516	D₄	—	8	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	8	4	no	—
527	C₂×C₄	—	8	4	no	—
528	C₂×C₄	—	8	4	no	—
529	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
535	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
536	C₈	—	2	4	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	4	no	—
556	C₄	C₃₁×#577	1	4	yes	—
557	C₄	C₃₁×#585	1	2	yes	—
558	C₂	—	4	2	no	—
559	C₂	—	4	1	no	—
560	C₂	—	8	4	no	—
561	C₂	C₃₁×#3449	1	1	yes	characteristic
562	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3263	C₂	semidirect
3. C₃₁×#3229	C₂	semidirect
4. C₃₁×#3227	C₂	semidirect
5. C₃₁×#3210	C₂	semidirect
6. C₃₁×#3207	C₂	semidirect
7. C₃₁×#3204	C₂	semidirect
8. C₃₁×#3185	C₂	semidirect
9. C₃₁×#3184	C₂	semidirect
10. C₃₁×#3167	C₂	semidirect
11. C₃₁×#3125	C₂	semidirect
12. C₃₁×#2922	C₂	semidirect
219. #61015	C₃₁	direct (N × H)
258. #3511	C₂×C₃₁	semidirect
259. #3263	C₂×C₃₁	semidirect
260. #3229	C₂×C₃₁	semidirect
261. #3227	C₂×C₃₁	semidirect
262. #3210	C₂×C₃₁	semidirect
263. #3207	C₂×C₃₁	semidirect
264. #3204	C₂×C₃₁	semidirect
265. #3185	C₂×C₃₁	semidirect
266. #3184	C₂×C₃₁	semidirect
267. #3167	C₂×C₃₁	semidirect
268. #3125	C₂×C₃₁	semidirect
269. #2922	C₂×C₃₁	semidirect
430. C₃₁	#61015	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×#3263	2
C₃₁×#3229	2
C₃₁×#3227	2
C₃₁×#3210	2
C₃₁×#3207	2
C₃₁×#3204	2
C₃₁×#3185	2
C₃₁×#3184	2
C₃₁×#3167	2
C₃₁×#3125	2
C₃₁×#2922	2
#61015	31

Subgroups — C31×#61015 #21517774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61015 #21517774