Subgroups — C₃₁×#61843 #21518602

Order 7936.

1984 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3384	C₂	1	6	yes	maximal, complemented
4	Q₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3373	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3239	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3227	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3109	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3105	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₃₁	—	2	1	no	—
13	C₃₁×#559	C₂²	1	3	yes	complemented
14	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#554	C₂²	1	3	yes	complemented
16	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
17	D₄×Q₈×C₃₁	—	2	1	no	—
18	D₄×Q₈×C₃₁	—	2	3	no	—
19	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	D₄×Q₈×C₃₁	C₂²	1	6	yes	—
21	C₃₁×#548	—	2	1	no	—
22	C₃₁×#547	—	2	3	no	—
23	C₃₁×#542	—	2	3	no	—
24	C₃₁×#532	C₂²	1	3	yes	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
26	C₃₁×#529	C₂²	1	3	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	—	2	3	no	—
28	C₃₁×#520	—	2	1	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
31	C₃₁×#501	C₂²	1	6	yes	complemented
32	C₃₁×#498	C₂²	1	6	yes	complemented
33	C₃₁×#489	C₂²	1	6	yes	complemented
34	C₃₁×#486	C₂²	1	6	yes	complemented
35	C₃₁×#484	C₂²	1	3	yes	—
36	C₃₁×#481	C₂²	1	3	yes	—
37	C₃₁×#481	C₂²	1	6	yes	—
38	C₃₁×#478	C₂²	1	6	yes	—
39	C₃₁×#476	C₂²	1	6	yes	—
40	C₃₁×#476	C₂²	1	12	yes	complemented
41	C₃₁×#475	C₂²	1	6	yes	complemented
42	C₃₁×#474	C₂²	1	6	yes	complemented
43	C₃₁×#463	C₂²	1	6	yes	—
44	C₃₁×#459	C₂²	1	6	yes	—
45	C₃₁×#450	C₂²	1	1	yes	characteristic
46	C₃₁×#447	C₂²	1	3	yes	—
47	C₃₁×#446	C₂²	1	6	yes	complemented
48	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
49	C₃₁×#442	C₂²	1	12	yes	—
50	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	—
51	C₄×D₈×C₃₁	C₂²	1	6	yes	—
52	C₃₁×#432	C₂²	1	3	yes	—
53	C₃₁×#431	C₂²	1	3	yes	—
54	C₂×#41×C₃₁	—	2	1	no	—
55	C₂×#41×C₃₁	—	2	3	no	—
56	C₂²×Q₈×C₃₁	—	2	6	no	—
57	C₂²×Q₈×C₃₁	—	2	2	no	—
58	C₂²×Q₈×C₃₁	—	2	3	no	—
59	C₂²×Q₈×C₃₁	—	4	8	no	—
60	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	—	2	3	no	—
63	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×D₈×C₃₁	C₂³	1	6	yes	—
65	C₃₁×C₄⋊Q₈	—	2	3	no	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	—	2	3	no	—
68	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
71	C₃₁×#127	C₂³	1	3	yes	—
72	C₃₁×#126	—	2	3	no	—
73	C₃₁×#125	C₂³	1	3	yes	—
74	C₃₁×#124	—	2	3	no	—
75	C₃₁×#124	—	2	3	no	—
76	C₃₁×#124	C₂³	1	3	yes	—
77	C₃₁×#123	—	2	3	no	—
78	C₃₁×#123	—	2	6	no	—
79	C₃₁×C₂²⋊Q₈	—	2	3	no	—
80	C₃₁×C₂²⋊Q₈	—	2	6	no	—
81	C₃₁×C₂²⋊Q₈	—	2	6	no	—
82	C₃₁×C₂²⋊Q₈	—	2	12	no	—
83	C₃₁×C₂²⋊Q₈	—	2	6	no	—
84	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×#121	—	2	3	no	—
86	C₃₁×#121	—	2	3	no	—
87	C₃₁×#121	C₂³	1	3	yes	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₃₁	D₄	1	3	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
94	C₄×D₄×C₃₁	—	2	3	no	—
95	C₄×D₄×C₃₁	—	2	3	no	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	C₂³	1	3	yes	—
100	C₄×D₄×C₃₁	C₂³	1	6	yes	—
101	C₄×D₄×C₃₁	C₂³	1	6	yes	—
102	C₄×D₄×C₃₁	C₂³	1	6	yes	—
103	C₃₁×#117	—	2	3	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
109	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
110	C₃₁×#107	C₂³	1	6	yes	—
111	C₃₁×#106	C₂³	1	6	yes	—
112	C₃₁×#106	C₂³	1	12	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
115	C₃₁×#103	C₂³	1	6	yes	—
116	C₃₁×#102	C₂³	1	2	yes	—
117	C₃₁×#102	C₂³	1	6	yes	—
118	C₃₁×#102	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	12	yes	—
120	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
122	C₃₁×#97	C₂³	1	6	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	C₂⁴×C₃₁	—	2	1	no	—
125	#41×C₃₁	—	4	4	no	—
126	#41×C₃₁	—	8	6	no	—
127	C₂×Q₈×C₃₁	—	2	4	no	—
128	C₂×Q₈×C₃₁	—	2	6	no	—
129	C₂×Q₈×C₃₁	—	2	6	no	—
130	C₂×Q₈×C₃₁	—	2	3	no	—
131	C₂×Q₈×C₃₁	—	4	12	no	—
132	C₂×Q₈×C₃₁	—	4	24	no	—
133	C₂×Q₈×C₃₁	—	4	8	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
143	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₃₁	—	2	3	no	—
145	C₂²×C₄×C₃₁	—	2	3	no	—
146	C₂²×C₄×C₃₁	—	2	6	no	—
147	C₂²×C₄×C₃₁	—	2	6	no	—
148	C₂²×C₄×C₃₁	—	2	6	no	—
149	C₂²×C₄×C₃₁	—	4	12	no	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	SD₁₆×C₃₁	—	2	4	no	—
154	D₈×C₃₁	—	2	12	no	—
155	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
156	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
157	C₄⋊C₄×C₃₁	—	2	3	no	—
158	C₄⋊C₄×C₃₁	—	2	3	no	—
159	C₄⋊C₄×C₃₁	—	2	6	no	—
160	C₄⋊C₄×C₃₁	—	2	6	no	—
161	C₄⋊C₄×C₃₁	—	2	3	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	3	no	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
168	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
169	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²⋊C₄×C₃₁	—	2	1	no	—
172	C₂²⋊C₄×C₃₁	—	2	6	no	—
173	C₂²⋊C₄×C₃₁	—	2	6	no	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	3	no	—
176	C₂²⋊C₄×C₃₁	—	2	3	no	—
177	C₂²⋊C₄×C₃₁	—	2	3	no	—
178	C₂²⋊C₄×C₃₁	—	2	6	no	—
179	C₄²×C₃₁	—	2	3	no	—
180	C₄²×C₃₁	C₂×D₄	1	3	yes	—
181	C₄²×C₃₁	C₂×D₄	1	3	yes	—
182	C₄²×C₃₁	C₂⁴	1	3	yes	—
183	C₄²×C₃₁	C₂⁴	1	3	yes	—
184	#61843	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	4	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	3	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₃₁	—	2	6	no	—
192	Q₈×C₃₁	—	2	2	no	—
193	Q₈×C₃₁	—	4	12	no	—
194	Q₈×C₃₁	—	4	12	no	—
195	Q₈×C₃₁	—	8	3	no	—
196	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
197	D₄×C₃₁	—	2	4	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	6	no	—
200	D₄×C₃₁	—	4	6	no	—
201	D₄×C₃₁	—	4	6	no	—
202	D₄×C₃₁	—	8	3	no	—
203	D₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
204	C₂×C₄×C₃₁	—	2	6	no	—
205	C₂×C₄×C₃₁	—	2	6	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	3	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	6	no	—
211	C₂×C₄×C₃₁	—	4	12	no	—
212	C₂×C₄×C₃₁	—	4	6	no	—
213	C₂×C₄×C₃₁	—	4	12	no	—
214	C₂×C₄×C₃₁	—	4	6	no	—
215	C₂×C₄×C₃₁	—	8	6	no	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
222	C₈×C₃₁	—	2	6	no	—
223	C₈×C₃₁	—	2	2	no	—
224	#3522	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#3384	C₂×C₃₁	1	6	yes	complemented
227	Q₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
228	#3373	C₂×C₃₁	1	6	yes	complemented
229	#3358	C₂×C₃₁	1	2	yes	complemented
230	#3239	C₂×C₃₁	1	3	yes	complemented
231	#3227	C₂×C₃₁	1	3	yes	complemented
232	#3109	C₂×C₃₁	1	3	yes	complemented
233	#3105	C₂×C₃₁	1	3	yes	complemented
234	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
235	C₂²×C₃₁	—	2	1	no	—
236	C₂²×C₃₁	—	2	1	no	—
237	C₂²×C₃₁	—	2	4	no	—
238	C₂²×C₃₁	—	4	8	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	3	no	—
242	C₂²×C₃₁	—	4	3	no	—
243	C₂²×C₃₁	—	4	3	no	—
244	C₂²×C₃₁	#577	1	2	yes	—
245	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	2	3	no	—
248	C₄×C₃₁	—	2	3	no	—
249	C₄×C₃₁	—	2	3	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	4	6	no	—
253	C₄×C₃₁	—	4	3	no	—
254	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
255	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
256	C₂³×Q₈	—	2	1	no	—
257	#559	C₂²×C₃₁	1	3	yes	complemented
258	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
259	#554	C₂²×C₃₁	1	3	yes	complemented
260	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
261	D₄×Q₈	—	2	1	no	—
262	D₄×Q₈	—	2	3	no	—
263	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
264	D₄×Q₈	C₂²×C₃₁	1	6	yes	—
265	#548	—	2	1	no	—
266	#547	—	2	3	no	—
267	#542	—	2	3	no	—
268	#532	C₂²×C₃₁	1	3	yes	—
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
270	#529	C₂²×C₃₁	1	3	yes	—
271	C₂×C₂²⋊Q₈	—	2	3	no	—
272	#520	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
275	#501	C₂²×C₃₁	1	6	yes	complemented
276	#498	C₂²×C₃₁	1	6	yes	complemented
277	#489	C₂²×C₃₁	1	6	yes	complemented
278	#486	C₂²×C₃₁	1	6	yes	complemented
279	#484	C₂²×C₃₁	1	3	yes	—
280	#481	C₂²×C₃₁	1	3	yes	—
281	#481	C₂²×C₃₁	1	6	yes	—
282	#478	C₂²×C₃₁	1	6	yes	—
283	#476	C₂²×C₃₁	1	6	yes	—
284	#476	C₂²×C₃₁	1	12	yes	complemented
285	#475	C₂²×C₃₁	1	6	yes	complemented
286	#474	C₂²×C₃₁	1	6	yes	complemented
287	#463	C₂²×C₃₁	1	6	yes	—
288	#459	C₂²×C₃₁	1	6	yes	—
289	#450	C₂²×C₃₁	1	1	yes	characteristic
290	#447	C₂²×C₃₁	1	3	yes	—
291	#446	C₂²×C₃₁	1	6	yes	complemented
292	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
293	#442	C₂²×C₃₁	1	12	yes	—
294	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	—
295	C₄×D₈	C₂²×C₃₁	1	6	yes	—
296	#432	C₂²×C₃₁	1	3	yes	—
297	#431	C₂²×C₃₁	1	3	yes	—
298	C₂×C₃₁	—	2	2	no	—
299	C₂×C₃₁	—	4	4	no	—
300	C₂×C₃₁	—	8	3	no	—
301	C₂×C₃₁	#3486	1	1	yes	characteristic
302	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
304	C₂×#41	—	2	1	no	—
305	C₂×#41	—	2	3	no	—
306	C₂²×Q₈	—	2	6	no	—
307	C₂²×Q₈	—	2	2	no	—
308	C₂²×Q₈	—	2	3	no	—
309	C₂²×Q₈	—	4	8	no	—
310	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	3	no	—
313	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
314	C₂×D₈	C₂³×C₃₁	1	6	yes	—
315	C₄⋊Q₈	—	2	3	no	—
316	C₄⋊Q₈	—	2	3	no	—
317	C₄⋊Q₈	—	2	3	no	—
318	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
319	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
320	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
321	#127	C₂³×C₃₁	1	3	yes	—
322	#126	—	2	3	no	—
323	#125	C₂³×C₃₁	1	3	yes	—
324	#124	—	2	3	no	—
325	#124	—	2	3	no	—
326	#124	C₂³×C₃₁	1	3	yes	—
327	#123	—	2	3	no	—
328	#123	—	2	6	no	—
329	C₂²⋊Q₈	—	2	3	no	—
330	C₂²⋊Q₈	—	2	6	no	—
331	C₂²⋊Q₈	—	2	6	no	—
332	C₂²⋊Q₈	—	2	12	no	—
333	C₂²⋊Q₈	—	2	6	no	—
334	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	#121	—	2	3	no	—
336	#121	—	2	3	no	—
337	#121	C₂³×C₃₁	1	3	yes	—
338	C₄×Q₈	—	2	1	no	—
339	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
340	C₄×Q₈	D₄×C₃₁	1	3	yes	—
341	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
342	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	—	2	3	no	—
348	C₄×D₄	—	2	3	no	—
349	C₄×D₄	C₂³×C₃₁	1	6	yes	—
350	C₄×D₄	C₂³×C₃₁	1	6	yes	—
351	C₄×D₄	C₂³×C₃₁	1	3	yes	—
352	C₄×D₄	C₂³×C₃₁	1	6	yes	—
353	#117	—	2	3	no	—
354	C₂×C₄⋊C₄	—	2	3	no	—
355	C₂×C₄⋊C₄	—	2	3	no	—
356	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
358	C₂×C₂²⋊C₄	—	2	3	no	—
359	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
360	#107	C₂³×C₃₁	1	6	yes	—
361	#106	C₂³×C₃₁	1	6	yes	—
362	#106	C₂³×C₃₁	1	12	yes	—
363	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
364	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
365	#103	C₂³×C₃₁	1	6	yes	—
366	#102	C₂³×C₃₁	1	2	yes	—
367	#102	C₂³×C₃₁	1	6	yes	—
368	#102	C₂³×C₃₁	1	6	yes	—
369	#102	C₂³×C₃₁	1	12	yes	—
370	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
372	#97	C₂³×C₃₁	1	6	yes	—
373	C₄×C₈	C₂³×C₃₁	1	6	yes	—
374	C₃₁	#61843	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁴	—	2	1	no	—
376	#41	—	4	4	no	—
377	#41	—	8	6	no	—
378	C₂×Q₈	—	2	4	no	—
379	C₂×Q₈	—	2	6	no	—
380	C₂×Q₈	—	2	6	no	—
381	C₂×Q₈	—	2	3	no	—
382	C₂×Q₈	—	4	12	no	—
383	C₂×Q₈	—	4	24	no	—
384	C₂×Q₈	—	4	8	no	—
385	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
386	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	3	no	—
393	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
394	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
395	C₂²×C₄	—	2	3	no	—
396	C₂²×C₄	—	2	3	no	—
397	C₂²×C₄	—	2	6	no	—
398	C₂²×C₄	—	2	6	no	—
399	C₂²×C₄	—	2	6	no	—
400	C₂²×C₄	—	4	12	no	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	SD₁₆	—	2	4	no	—
405	D₈	—	2	12	no	—
406	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
407	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	3	no	—
410	C₄⋊C₄	—	2	6	no	—
411	C₄⋊C₄	—	2	6	no	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	—	2	3	no	—
416	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
418	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
419	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
420	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
421	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	6	no	—
424	C₂²⋊C₄	—	2	6	no	—
425	C₂²⋊C₄	—	2	3	no	—
426	C₂²⋊C₄	—	2	3	no	—
427	C₂²⋊C₄	—	2	3	no	—
428	C₂²⋊C₄	—	2	3	no	—
429	C₂²⋊C₄	—	2	6	no	—
430	C₄²	—	2	3	no	—
431	C₄²	C₂×D₄×C₃₁	1	3	yes	—
432	C₄²	C₂×D₄×C₃₁	1	3	yes	—
433	C₄²	C₂⁴×C₃₁	1	3	yes	—
434	C₄²	C₂⁴×C₃₁	1	3	yes	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	3	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	Q₈	—	2	6	no	—
442	Q₈	—	2	2	no	—
443	Q₈	—	4	12	no	—
444	Q₈	—	4	12	no	—
445	Q₈	—	8	3	no	—
446	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
447	D₄	—	2	4	no	—
448	D₄	—	2	2	no	—
449	D₄	—	2	6	no	—
450	D₄	—	4	6	no	—
451	D₄	—	4	6	no	—
452	D₄	—	8	3	no	—
453	D₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
454	C₂×C₄	—	2	6	no	—
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	3	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	6	no	—
461	C₂×C₄	—	4	12	no	—
462	C₂×C₄	—	4	6	no	—
463	C₂×C₄	—	4	12	no	—
464	C₂×C₄	—	4	6	no	—
465	C₂×C₄	—	8	6	no	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
471	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
472	C₈	—	2	6	no	—
473	C₈	—	2	2	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	4	8	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	3	no	—
481	C₂²	—	4	3	no	—
482	C₂²	—	4	3	no	—
483	C₂²	C₃₁×#577	1	2	yes	—
484	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	3	no	—
487	C₄	—	2	3	no	—
488	C₄	—	2	3	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	6	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	3	no	—
493	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
494	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
495	C₂	—	2	2	no	—
496	C₂	—	4	4	no	—
497	C₂	—	8	3	no	—
498	C₂	C₃₁×#3486	1	1	yes	characteristic
499	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
500	C₂	C₂×C₃₁×#577	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 ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3522	C₂	semidirect
2. C₂×D₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#3384	C₂	semidirect
4. Q₈×SD₁₆×C₃₁	C₂	semidirect
5. C₃₁×#3373	C₂	semidirect
6. C₃₁×#3358	C₂	semidirect
7. C₃₁×#3239	C₂	semidirect
8. C₃₁×#3227	C₂	semidirect
9. C₃₁×#3109	C₂	semidirect
10. C₃₁×#3105	C₂	semidirect
11. C₃₁×#2984	C₂	semidirect
13. C₃₁×#559	C₂²	semidirect
14. Q₈²×C₃₁	C₂²	semidirect
15. C₃₁×#554	C₂²	semidirect
16. C₃₁×#550	C₂²	semidirect
31. C₃₁×#501	C₂²	semidirect
32. C₃₁×#498	C₂²	semidirect
33. C₃₁×#489	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
47. C₃₁×#446	C₂²	semidirect
48. C₈×Q₈×C₃₁	C₂²	semidirect
184. #61843	C₃₁	direct (N × H)
224. #3522	C₂×C₃₁	semidirect
225. C₂×D₄×Q₈	C₂×C₃₁	semidirect
226. #3384	C₂×C₃₁	semidirect
227. Q₈×SD₁₆	C₂×C₃₁	semidirect
228. #3373	C₂×C₃₁	semidirect
229. #3358	C₂×C₃₁	semidirect
230. #3239	C₂×C₃₁	semidirect
231. #3227	C₂×C₃₁	semidirect
232. #3109	C₂×C₃₁	semidirect
233. #3105	C₂×C₃₁	semidirect
234. #2984	C₂×C₃₁	semidirect
257. #559	C₂²×C₃₁	semidirect
258. Q₈²	C₂²×C₃₁	semidirect
259. #554	C₂²×C₃₁	semidirect
260. #550	C₂²×C₃₁	semidirect
275. #501	C₂²×C₃₁	semidirect
276. #498	C₂²×C₃₁	semidirect
277. #489	C₂²×C₃₁	semidirect
278. #486	C₂²×C₃₁	semidirect
284. #476	C₂²×C₃₁	semidirect
285. #475	C₂²×C₃₁	semidirect
286. #474	C₂²×C₃₁	semidirect
291. #446	C₂²×C₃₁	semidirect
292. C₈×Q₈	C₂²×C₃₁	semidirect
374. C₃₁	#61843	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×#3522	2
C₂×D₄×Q₈×C₃₁	2
C₃₁×#3384	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3373	2
C₃₁×#3358	2
C₃₁×#3239	2
C₃₁×#3227	2
C₃₁×#3109	2
C₃₁×#3105	2
C₃₁×#2984	2
#61843	31

Subgroups — C31×#61843 #21518602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61843 #21518602