Subgroups — C₃₁×#17774 #21474533

Order 7936.

904 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1556	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1544	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#332	C₂	1	2	yes	maximal, complemented
12	Q₈²×C₃₁	—	2	1	no	—
13	C₃₁×#557	—	2	1	no	—
14	C₃₁×#554	—	2	1	no	—
15	C₃₁×#554	—	2	2	no	—
16	D₄×Q₈×C₃₁	—	2	1	no	—
17	D₄×Q₈×C₃₁	—	2	2	no	—
18	C₃₁×#544	C₄	1	4	yes	—
19	C₃₁×#544	C₄	1	2	yes	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×#520	C₄	1	2	yes	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#446	—	2	1	no	—
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#435	—	2	2	no	—
29	C₃₁×#435	—	2	2	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₃₁×#432	C₂²	1	2	yes	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	1	yes	characteristic
41	C₃₁×#332	C₂²	1	2	yes	—
42	C₃₁×#332	C₂²	1	2	yes	—
43	C₃₁×#332	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	1	yes	characteristic
47	C₃₁×#332	C₂²	1	2	yes	—
48	C₃₁×#332	C₂²	1	2	yes	—
49	C₃₁×#332	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	—	2	2	no	—
52	C₂²×Q₈×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	4	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#125	—	2	4	no	—
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	C₂×C₄	1	2	yes	—
71	C₃₁×#125	C₂×C₄	1	4	yes	—
72	C₃₁×C₂²⋊Q₈	—	2	2	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₄×Q₈×C₃₁	—	2	2	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
86	C₄×D₄×C₃₁	—	2	4	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
89	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
90	C₄×D₄×C₃₁	D₄	1	2	yes	—
91	C₄×D₄×C₃₁	D₄	1	2	yes	—
92	C₃₁×#117	C₂×C₄	1	2	yes	—
93	C₃₁×#117	C₂×C₄	1	4	yes	—
94	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
95	C₃₁×#117	C₂×C₄	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×C₄⋊C₈	—	2	2	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	1	yes	characteristic
107	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
109	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
110	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₂²⋊C₈	—	2	1	no	—
113	C₃₁×C₂²⋊C₈	—	2	1	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₃₁×#97	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₄×C₈×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	4	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	4	no	—
123	C₂×Q₈×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	8	2	no	—
127	C₂×Q₈×C₃₁	—	8	1	no	—
128	C₂×Q₈×C₃₁	—	8	2	no	—
129	C₂×Q₈×C₃₁	—	8	1	no	—
130	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
132	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	M₁₆×C₃₁	—	8	1	no	—
142	M₁₆×C₃₁	—	8	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	8	2	no	—
150	C₂×C₈×C₃₁	—	8	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
163	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
169	C₂²⋊C₄×C₃₁	—	2	2	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
172	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
173	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
174	C₄²×C₃₁	—	2	4	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
177	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂×D₄	1	2	yes	—
181	C₄²×C₃₁	C₂×D₄	1	2	yes	—
182	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
183	#17774	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	2	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	4	2	no	—
192	Q₈×C₃₁	—	4	1	no	—
193	Q₈×C₃₁	—	8	4	no	—
194	Q₈×C₃₁	—	8	2	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	#101	1	4	yes	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	8	4	no	—
211	C₂×C₄×C₃₁	—	8	2	no	—
212	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₈×C₃₁	—	4	1	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	—	4	2	no	—
226	C₈×C₃₁	—	4	1	no	—
227	#3503	C₂×C₃₁	1	1	yes	characteristic
228	#2990	C₂×C₃₁	1	1	yes	characteristic
229	#2990	C₂×C₃₁	1	1	yes	characteristic
230	#1561	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1556	C₂×C₃₁	1	2	yes	complemented
232	#1552	C₂×C₃₁	1	2	yes	complemented
233	#1552	C₂×C₃₁	1	2	yes	complemented
234	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1544	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂×#332	C₂×C₃₁	1	2	yes	complemented
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	C₂×#101	1	4	yes	—
244	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₄×C₃₁	—	2	1	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	1	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	4	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	C₂×#101	1	2	yes	—
255	C₄×C₃₁	#576	1	1	yes	characteristic
256	C₄×C₃₁	#577	1	1	yes	characteristic
257	C₄×C₃₁	#577	1	1	yes	characteristic
258	C₄×C₃₁	#578	1	1	yes	characteristic
259	Q₈²	—	2	1	no	—
260	#557	—	2	1	no	—
261	#554	—	2	1	no	—
262	#554	—	2	2	no	—
263	D₄×Q₈	—	2	1	no	—
264	D₄×Q₈	—	2	2	no	—
265	#544	C₄×C₃₁	1	4	yes	—
266	#544	C₄×C₃₁	1	2	yes	—
267	#533	C₂²×C₃₁	1	1	yes	characteristic
268	#533	C₂²×C₃₁	1	2	yes	—
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	#520	C₄×C₃₁	1	2	yes	—
272	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
273	#446	—	2	1	no	—
274	#446	—	2	1	no	—
275	#435	—	2	2	no	—
276	#435	—	2	2	no	—
277	C₈×D₄	—	2	1	no	—
278	C₈×D₄	—	2	1	no	—
279	#432	C₂²×C₃₁	1	2	yes	—
280	#432	C₂²×C₃₁	1	2	yes	—
281	#431	C₂²×C₃₁	1	1	yes	characteristic
282	#431	C₂²×C₃₁	1	1	yes	characteristic
283	#423	C₂²×C₃₁	1	1	yes	characteristic
284	#423	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
287	#332	C₂²×C₃₁	1	1	yes	characteristic
288	#332	C₂²×C₃₁	1	2	yes	—
289	#332	C₂²×C₃₁	1	2	yes	—
290	#332	C₂²×C₃₁	1	2	yes	—
291	#332	C₂²×C₃₁	1	1	yes	characteristic
292	#332	C₂²×C₃₁	1	1	yes	characteristic
293	#332	C₂²×C₃₁	1	1	yes	characteristic
294	#332	C₂²×C₃₁	1	2	yes	—
295	#332	C₂²×C₃₁	1	2	yes	—
296	#332	C₂²×C₃₁	1	2	yes	—
297	C₂×C₃₁	—	2	4	no	—
298	C₂×C₃₁	—	4	1	no	—
299	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
300	C₂×C₃₁	#2897	1	1	yes	characteristic
301	C₂×C₃₁	#2898	1	1	yes	characteristic
302	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂²×Q₈	—	2	2	no	—
304	C₂²×Q₈	—	2	1	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	1	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₂²×C₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	4	no	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
316	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
317	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
319	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	#125	—	2	4	no	—
321	#125	—	2	2	no	—
322	#125	C₂×C₄×C₃₁	1	2	yes	—
323	#125	C₂×C₄×C₃₁	1	4	yes	—
324	C₂²⋊Q₈	—	2	2	no	—
325	C₂²⋊Q₈	—	2	1	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	1	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	4	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	1	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
337	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
341	C₄×D₄	D₄×C₃₁	1	2	yes	—
342	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
343	C₄×D₄	D₄×C₃₁	1	2	yes	—
344	#117	C₂×C₄×C₃₁	1	2	yes	—
345	#117	C₂×C₄×C₃₁	1	4	yes	—
346	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
347	#117	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
351	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
352	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	2	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
360	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
361	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
362	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²⋊C₈	—	2	1	no	—
365	C₂²⋊C₈	—	2	1	no	—
366	C₂²⋊C₈	—	2	2	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	#97	—	2	2	no	—
369	#97	—	2	2	no	—
370	C₄×C₈	—	2	1	no	—
371	C₄×C₈	—	2	1	no	—
372	C₃₁	#17774	1	1	yes	characteristic, direct product factor, Sylow
373	#41	—	2	4	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	8	2	no	—
380	C₂×Q₈	—	8	1	no	—
381	C₂×Q₈	—	8	2	no	—
382	C₂×Q₈	—	8	1	no	—
383	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
384	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
385	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	M₁₆	—	8	1	no	—
395	M₁₆	—	8	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	8	2	no	—
403	C₂×C₈	—	8	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
416	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
417	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
418	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
419	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
420	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
421	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
422	C₂²⋊C₄	—	2	2	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
425	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
426	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
427	C₄²	—	2	4	no	—
428	C₄²	—	2	2	no	—
429	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
430	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₃₁	1	2	yes	—
434	C₄²	C₂×D₄×C₃₁	1	2	yes	—
435	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
437	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
438	Q₈	—	2	2	no	—
439	Q₈	—	4	1	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	4	1	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	8	4	no	—
446	Q₈	—	8	2	no	—
447	D₄	—	2	4	no	—
448	D₄	C₃₁×#101	1	4	yes	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	8	4	no	—
463	C₂×C₄	—	8	2	no	—
464	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
465	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
467	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
473	C₈	—	4	1	no	—
474	C₈	—	4	2	no	—
475	C₈	—	4	1	no	—
476	C₈	—	4	1	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	C₂×C₃₁×#101	1	4	yes	—
485	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
486	C₄	—	2	1	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	1	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	2	no	—
495	C₄	C₂×C₃₁×#101	1	2	yes	—
496	C₄	C₃₁×#576	1	1	yes	characteristic
497	C₄	C₃₁×#577	1	1	yes	characteristic
498	C₄	C₃₁×#577	1	1	yes	characteristic
499	C₄	C₃₁×#578	1	1	yes	characteristic
500	C₂	—	2	4	no	—
501	C₂	—	4	1	no	—
502	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
503	C₂	C₃₁×#2897	1	1	yes	characteristic
504	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	C₂×#41×C₃₁	C₂×C₄
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
4. C₃₁×#1561	C₂	semidirect
5. C₃₁×#1556	C₂	semidirect
6. C₃₁×#1552	C₂	semidirect
7. C₃₁×#1552	C₂	semidirect
8. C₃₁×#1549	C₂	semidirect
9. C₃₁×#1546	C₂	semidirect
10. C₃₁×#1544	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
183. #17774	C₃₁	direct (N × H)
230. #1561	C₂×C₃₁	semidirect
231. #1556	C₂×C₃₁	semidirect
232. #1552	C₂×C₃₁	semidirect
233. #1552	C₂×C₃₁	semidirect
234. #1549	C₂×C₃₁	semidirect
235. #1546	C₂×C₃₁	semidirect
236. #1544	C₂×C₃₁	semidirect
237. C₂×#332	C₂×C₃₁	semidirect
372. C₃₁	#17774	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×#3503	2
C₃₁×#2990	2
C₃₁×#1561	2
C₃₁×#1556	2
C₃₁×#1552	2
C₃₁×#1549	2
C₃₁×#1546	2
C₃₁×#1544	2
C₂×C₃₁×#332	2
#17774	31

Subgroups — C31×#17774 #21474533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17774 #21474533