Subgroups — C₃₁×#61014 #21517773

Order 7936.

1768 conjugacy classes of subgroups.

The subgroup lattice has 586 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×#3262	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3230	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3226	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×#3203	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3124	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×#500	C₂²	1	1	yes	characteristic
33	C₃₁×#500	C₂²	1	1	yes	characteristic
34	C₃₁×#500	C₂²	1	2	yes	—
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×#484	C₂²	1	8	yes	—
43	C₃₁×#480	C₂²	1	8	yes	—
44	C₃₁×#477	C₂²	1	8	yes	—
45	C₃₁×#474	C₂²	1	8	yes	—
46	C₃₁×#468	C₂²	1	8	yes	—
47	C₃₁×#467	C₂²	1	4	yes	—
48	C₃₁×#466	C₂²	1	4	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×#440	C₂²	1	8	yes	—
54	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
55	C₄×D₈×C₃₁	C₂²	1	4	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	1	yes	characteristic
110	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
111	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
112	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
113	C₄×D₄×C₃₁	—	2	4	no	—
114	C₄×D₄×C₃₁	—	2	4	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	—	2	2	no	—
119	C₄×D₄×C₃₁	D₄	1	2	yes	—
120	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
121	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
122	C₄×D₄×C₃₁	D₄	1	2	yes	—
123	C₄×D₄×C₃₁	C₂³	1	4	yes	—
124	C₄×D₄×C₃₁	C₂³	1	4	yes	—
125	C₃₁×#117	—	2	2	no	—
126	C₃₁×#117	—	2	2	no	—
127	C₃₁×#117	—	2	4	no	—
128	C₃₁×#117	C₂³	1	1	yes	characteristic
129	C₃₁×#117	C₂³	1	2	yes	—
130	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
132	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
135	C₃₁×#107	C₂³	1	4	yes	—
136	C₃₁×#107	C₂³	1	4	yes	—
137	C₃₁×#107	C₂³	1	8	yes	—
138	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
139	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
140	C₃₁×#103	C₂³	1	8	yes	—
141	C₃₁×#103	C₂³	1	8	yes	—
142	C₃₁×#102	C₂³	1	8	yes	—
143	C₃₁×#102	C₂³	1	8	yes	—
144	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
146	C₃₁×#97	C₂³	1	2	yes	—
147	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
149	C₄×C₈×C₃₁	C₂³	1	2	yes	—
150	C₄×C₈×C₃₁	C₂³	1	2	yes	—
151	#41×C₃₁	—	4	2	no	—
152	#41×C₃₁	—	4	8	no	—
153	#41×C₃₁	—	8	4	no	—
154	#41×C₃₁	—	8	4	no	—
155	C₂×Q₈×C₃₁	—	2	2	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	—	2	4	no	—
158	C₂×Q₈×C₃₁	—	8	4	no	—
159	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	4	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	2	4	no	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₃₁	—	4	4	no	—
180	SD₁₆×C₃₁	—	4	8	no	—
181	D₈×C₃₁	—	4	4	no	—
182	M₁₆×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	—	4	2	no	—
184	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
185	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
198	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
202	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
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	—	2	4	no	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	—	2	4	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
214	C₄²×C₃₁	—	2	4	no	—
215	C₄²×C₃₁	—	2	4	no	—
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	2	yes	—
218	C₄²×C₃₁	C₂×D₄	1	2	yes	—
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₄²×C₃₁	C₂⁴	1	2	yes	—
223	#61014	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₃₁	—	2	4	no	—
225	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
227	Q₈×C₃₁	—	2	8	no	—
228	Q₈×C₃₁	—	4	1	no	—
229	Q₈×C₃₁	—	4	4	no	—
230	Q₈×C₃₁	—	4	4	no	—
231	Q₈×C₃₁	—	8	4	no	—
232	D₄×C₃₁	—	2	8	no	—
233	D₄×C₃₁	—	4	2	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	8	no	—
236	D₄×C₃₁	—	4	4	no	—
237	D₄×C₃₁	—	4	4	no	—
238	D₄×C₃₁	—	8	4	no	—
239	D₄×C₃₁	—	8	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	8	no	—
250	C₂×C₄×C₃₁	—	8	4	no	—
251	C₂×C₄×C₃₁	—	8	4	no	—
252	C₂×C₄×C₃₁	—	8	4	no	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
262	C₈×C₃₁	—	2	2	no	—
263	C₈×C₃₁	—	2	2	no	—
264	C₈×C₃₁	—	2	4	no	—
265	#3511	C₂×C₃₁	1	2	yes	complemented
266	#3262	C₂×C₃₁	1	4	yes	complemented
267	#3230	C₂×C₃₁	1	2	yes	complemented
268	#3226	C₂×C₃₁	1	2	yes	complemented
269	#3210	C₂×C₃₁	1	2	yes	complemented
270	#3207	C₂×C₃₁	1	2	yes	complemented
271	#3203	C₂×C₃₁	1	4	yes	complemented
272	#3182	C₂×C₃₁	1	2	yes	complemented
273	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#3124	C₂×C₃₁	1	8	yes	complemented
276	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	4	no	—
286	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	4	4	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	—	4	4	no	—
296	C₄×C₃₁	#576	1	2	yes	—
297	C₄×C₃₁	#578	1	2	yes	—
298	C₄×C₃₁	#585	1	2	yes	—
299	#556	—	2	2	no	—
300	#555	—	2	2	no	—
301	#554	—	2	2	no	—
302	#553	—	2	2	no	—
303	#553	—	2	2	no	—
304	#548	—	2	2	no	—
305	#548	—	2	2	no	—
306	#547	—	2	2	no	—
307	#542	C₂²×C₃₁	1	4	yes	—
308	#542	C₂²×C₃₁	1	4	yes	—
309	#537	C₂²×C₃₁	1	4	yes	—
310	#533	C₂²×C₃₁	1	2	yes	—
311	#533	C₂²×C₃₁	1	2	yes	—
312	#532	C₂²×C₃₁	1	2	yes	—
313	#532	C₂²×C₃₁	1	2	yes	—
314	#532	C₂²×C₃₁	1	2	yes	—
315	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
316	#520	C₂²×C₃₁	1	4	yes	—
317	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
318	#500	C₂²×C₃₁	1	1	yes	characteristic
319	#500	C₂²×C₃₁	1	1	yes	characteristic
320	#500	C₂²×C₃₁	1	2	yes	—
321	#494	C₂²×C₃₁	1	1	yes	characteristic
322	#493	C₂²×C₃₁	1	1	yes	characteristic
323	#492	C₂²×C₃₁	1	2	yes	—
324	#487	C₂²×C₃₁	1	2	yes	—
325	#487	C₂²×C₃₁	1	2	yes	—
326	#486	C₂²×C₃₁	1	2	yes	—
327	#486	C₂²×C₃₁	1	2	yes	—
328	#484	C₂²×C₃₁	1	8	yes	—
329	#480	C₂²×C₃₁	1	8	yes	—
330	#477	C₂²×C₃₁	1	8	yes	—
331	#474	C₂²×C₃₁	1	8	yes	—
332	#468	C₂²×C₃₁	1	8	yes	—
333	#467	C₂²×C₃₁	1	4	yes	—
334	#466	C₂²×C₃₁	1	4	yes	—
335	#464	C₂²×C₃₁	1	8	yes	—
336	#463	C₂²×C₃₁	1	8	yes	—
337	#452	C₂²×C₃₁	1	8	yes	—
338	#449	C₂²×C₃₁	1	8	yes	—
339	#440	C₂²×C₃₁	1	8	yes	—
340	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
341	C₄×D₈	C₂²×C₃₁	1	4	yes	—
342	#433	C₂²×C₃₁	1	4	yes	—
343	#431	C₂²×C₃₁	1	2	yes	—
344	#431	C₂²×C₃₁	1	2	yes	—
345	#424	C₂²×C₃₁	1	2	yes	—
346	#408	C₂²×C₃₁	1	2	yes	—
347	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
349	C₂×C₃₁	—	4	2	no	—
350	C₂×C₃₁	—	4	1	no	—
351	C₂×C₃₁	—	8	4	no	—
352	C₂×C₃₁	#3449	1	1	yes	characteristic
353	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
354	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	C₂³×C₃₁	1	4	yes	—
358	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
359	C₂²×Q₈	—	2	2	no	—
360	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
361	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
362	C₂×D₈	C₂³×C₃₁	1	4	yes	—
363	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
364	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
365	C₄⋊Q₈	—	2	2	no	—
366	C₄⋊Q₈	—	2	2	no	—
367	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
368	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
371	#127	C₂³×C₃₁	1	2	yes	—
372	#126	—	2	4	no	—
373	#126	—	2	4	no	—
374	#125	—	2	2	no	—
375	#125	—	2	2	no	—
376	#124	—	2	4	no	—
377	#124	—	2	4	no	—
378	#124	—	2	2	no	—
379	#124	—	2	2	no	—
380	#124	C₂³×C₃₁	1	4	yes	—
381	#123	—	2	4	no	—
382	#123	—	2	4	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	—	2	4	no	—
388	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
389	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
390	#121	—	2	4	no	—
391	#121	—	2	4	no	—
392	#121	—	2	4	no	—
393	#121	—	2	2	no	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	2	no	—
397	#121	C₂³×C₃₁	1	4	yes	—
398	#121	C₂³×C₃₁	1	4	yes	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
402	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
403	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
404	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
405	C₄×D₄	—	2	4	no	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	D₄×C₃₁	1	2	yes	—
412	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
413	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
414	C₄×D₄	D₄×C₃₁	1	2	yes	—
415	C₄×D₄	C₂³×C₃₁	1	4	yes	—
416	C₄×D₄	C₂³×C₃₁	1	4	yes	—
417	#117	—	2	2	no	—
418	#117	—	2	2	no	—
419	#117	—	2	4	no	—
420	#117	C₂³×C₃₁	1	2	yes	—
421	#117	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	4	no	—
424	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
425	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
426	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
427	#107	C₂³×C₃₁	1	4	yes	—
428	#107	C₂³×C₃₁	1	4	yes	—
429	#107	C₂³×C₃₁	1	8	yes	—
430	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
431	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
432	#103	C₂³×C₃₁	1	8	yes	—
433	#103	C₂³×C₃₁	1	8	yes	—
434	#102	C₂³×C₃₁	1	8	yes	—
435	#102	C₂³×C₃₁	1	8	yes	—
436	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
437	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
438	#97	C₂³×C₃₁	1	2	yes	—
439	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
440	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
441	C₄×C₈	C₂³×C₃₁	1	2	yes	—
442	C₄×C₈	C₂³×C₃₁	1	2	yes	—
443	C₃₁	#61014	1	1	yes	characteristic, direct product factor, Sylow
444	#41	—	4	2	no	—
445	#41	—	4	8	no	—
446	#41	—	8	4	no	—
447	#41	—	8	4	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	8	4	no	—
452	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
461	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
472	Q₁₆	—	4	4	no	—
473	SD₁₆	—	4	8	no	—
474	D₈	—	4	4	no	—
475	M₁₆	—	4	2	no	—
476	C₂×C₈	—	4	2	no	—
477	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
478	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
479	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
491	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
492	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
495	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
505	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
506	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂×D₄×C₃₁	1	2	yes	—
511	C₄²	C₂×D₄×C₃₁	1	2	yes	—
512	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
514	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
515	C₄²	C₂⁴×C₃₁	1	2	yes	—
516	C₂³	—	2	4	no	—
517	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
519	Q₈	—	2	8	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	4	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	8	4	no	—
524	D₄	—	2	8	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	1	no	—
527	D₄	—	4	8	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	8	4	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	—	8	4	no	—
544	C₂×C₄	—	8	4	no	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
553	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	2	4	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	2	1	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	1	no	—
572	C₄	—	2	2	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	C₃₁×#576	1	2	yes	—
577	C₄	C₃₁×#578	1	2	yes	—
578	C₄	C₃₁×#585	1	2	yes	—
579	C₂	—	4	2	no	—
580	C₂	—	4	1	no	—
581	C₂	—	8	4	no	—
582	C₂	C₃₁×#3449	1	1	yes	characteristic
583	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
584	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	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×#3262	C₂	semidirect
3. C₃₁×#3230	C₂	semidirect
4. C₃₁×#3226	C₂	semidirect
5. C₃₁×#3210	C₂	semidirect
6. C₃₁×#3207	C₂	semidirect
7. C₃₁×#3203	C₂	semidirect
8. C₃₁×#3182	C₂	semidirect
9. C₃₁×#3170	C₂	semidirect
10. C₃₁×#3169	C₂	semidirect
11. C₃₁×#3124	C₂	semidirect
12. C₃₁×#2922	C₂	semidirect
223. #61014	C₃₁	direct (N × H)
265. #3511	C₂×C₃₁	semidirect
266. #3262	C₂×C₃₁	semidirect
267. #3230	C₂×C₃₁	semidirect
268. #3226	C₂×C₃₁	semidirect
269. #3210	C₂×C₃₁	semidirect
270. #3207	C₂×C₃₁	semidirect
271. #3203	C₂×C₃₁	semidirect
272. #3182	C₂×C₃₁	semidirect
273. #3170	C₂×C₃₁	semidirect
274. #3169	C₂×C₃₁	semidirect
275. #3124	C₂×C₃₁	semidirect
276. #2922	C₂×C₃₁	semidirect
443. C₃₁	#61014	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×#3262	2
C₃₁×#3230	2
C₃₁×#3226	2
C₃₁×#3210	2
C₃₁×#3207	2
C₃₁×#3203	2
C₃₁×#3182	2
C₃₁×#3170	2
C₃₁×#3169	2
C₃₁×#3124	2
C₃₁×#2922	2
#61014	31

Subgroups — C31×#61014 #21517773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61014 #21517773