Subgroups — C₃₁×#61374 #21518133

Order 7936.

1778 conjugacy classes of subgroups.

The subgroup lattice has 482 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	2	yes	maximal, complemented
2	C₃₁×#3387	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3384	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3381	C₂	1	4	yes	maximal, complemented
5	Q₈×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3163	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3084	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3079	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2952	C₂	1	2	yes	maximal
12	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	Q₈²×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#557	C₂²	1	4	yes	—
16	C₃₁×#554	—	2	4	no	—
17	C₃₁×#554	C₂²	1	4	yes	—
18	D₄×Q₈×C₃₁	—	2	4	no	—
19	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#544	—	2	4	no	—
21	C₃₁×#544	C₂²	1	4	yes	—
22	C₃₁×#533	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₃₁×#520	C₂²	1	4	yes	—
27	C₄×#41×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#501	C₂²	1	8	yes	—
29	C₃₁×#499	C₂²	1	4	yes	—
30	C₃₁×#498	C₂²	1	4	yes	—
31	C₃₁×#498	C₂²	1	4	yes	—
32	C₃₁×#498	C₂²	1	4	yes	—
33	C₃₁×#479	C₂²	1	8	yes	—
34	C₃₁×#478	C₂²	1	8	yes	—
35	C₃₁×#477	C₂²	1	4	yes	—
36	C₃₁×#476	C₂²	1	4	yes	—
37	C₃₁×#476	C₂²	1	8	yes	—
38	C₃₁×#475	C₂²	1	4	yes	—
39	C₃₁×#475	C₂²	1	8	yes	—
40	C₃₁×#474	C₂²	1	4	yes	—
41	C₃₁×#446	C₂²	1	4	yes	—
42	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
43	C₃₁×#442	C₂²	1	4	yes	—
44	C₃₁×#441	C₂²	1	4	yes	—
45	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
46	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
48	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
49	C₄×D₈×C₃₁	C₂²	1	2	yes	—
50	C₃₁×#428	C₂²	1	4	yes	—
51	C₃₁×#427	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#106	C₂²	1	2	yes	—
53	C₂×C₃₁×#106	C₂²	1	2	yes	—
54	C₃₁×#424	C₂²	1	2	yes	—
55	C₃₁×#423	C₂²	1	1	yes	characteristic
56	C₃₁×#417	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	2	yes	—
58	C₃₁×#405	C₂²	1	2	yes	—
59	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂²×Q₈×C₃₁	—	2	4	no	—
62	C₂²×Q₈×C₃₁	—	2	2	no	—
63	C₃₁×#135	Q₈	1	8	yes	complemented
64	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×C₄⋊Q₈	—	2	4	no	—
71	C₃₁×C₄⋊Q₈	—	2	4	no	—
72	C₃₁×C₄⋊Q₈	—	2	4	no	—
73	C₃₁×C₄⋊Q₈	—	2	8	no	—
74	C₃₁×C₄⋊Q₈	—	2	2	no	—
75	C₃₁×C₄⋊Q₈	—	2	2	no	—
76	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
77	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
80	C₃₁×#125	—	2	4	no	—
81	C₃₁×#125	—	2	8	no	—
82	C₃₁×#125	D₄	1	2	yes	—
83	C₃₁×#125	C₂³	1	8	yes	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	—	2	4	no	—
86	C₃₁×C₂²⋊Q₈	—	2	8	no	—
87	C₃₁×C₂²⋊Q₈	—	2	8	no	—
88	C₃₁×C₂²⋊Q₈	—	2	8	no	—
89	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
90	C₄×Q₈×C₃₁	—	2	4	no	—
91	C₄×Q₈×C₃₁	—	2	4	no	—
92	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	8	no	—
99	C₄×D₄×C₃₁	C₂³	1	2	yes	—
100	C₄×D₄×C₃₁	C₂³	1	2	yes	—
101	C₄×D₄×C₃₁	C₂³	1	8	yes	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	C₂³	1	2	yes	—
105	C₃₁×#117	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄²×C₃₁	—	2	2	no	—
111	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
112	C₃₁×#107	C₂³	1	2	yes	—
113	C₃₁×#107	C₂³	1	8	yes	—
114	C₃₁×#106	C₂³	1	4	yes	—
115	C₃₁×#106	C₂³	1	4	yes	—
116	C₃₁×#106	C₂³	1	8	yes	—
117	C₃₁×#106	C₂³	1	1	yes	characteristic
118	C₃₁×#106	C₂³	1	1	yes	characteristic
119	C₃₁×#106	C₂³	1	8	yes	—
120	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
121	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
122	C₃₁×#103	C₂³	1	2	yes	—
123	C₃₁×#103	C₂³	1	2	yes	—
124	C₃₁×#103	C₂³	1	8	yes	—
125	C₃₁×#102	C₂³	1	2	yes	—
126	C₃₁×#102	C₂³	1	2	yes	—
127	C₃₁×#102	C₂³	1	8	yes	—
128	C₃₁×#97	C₂³	1	4	yes	—
129	C₄×C₈×C₃₁	C₂³	1	2	yes	—
130	C₄×C₈×C₃₁	C₂³	1	2	yes	—
131	C₄×C₈×C₃₁	C₂³	1	4	yes	—
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	C₂×Q₈	1	8	yes	complemented
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	8	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	—	8	4	no	—
138	C₂×Q₈×C₃₁	—	8	8	no	—
139	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
140	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
151	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
152	D₈×C₃₁	C₂×Q₈	1	4	yes	complemented
153	M₁₆×C₃₁	—	2	4	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
156	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
157	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
158	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	8	no	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	8	no	—
169	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
170	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
171	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
172	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
173	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	2	8	no	—
179	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	2	no	—
183	C₄²×C₃₁	—	2	8	no	—
184	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂⁴	1	2	yes	—
187	C₄²×C₃₁	C₂⁴	1	4	yes	—
188	#61374	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₃₁	—	2	8	no	—
192	Q₈×C₃₁	—	2	2	no	—
193	Q₈×C₃₁	—	4	8	no	—
194	Q₈×C₃₁	—	8	4	no	—
195	Q₈×C₃₁	—	8	8	no	—
196	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
197	D₄×C₃₁	—	2	8	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	8	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	4	4	no	—
207	C₂×C₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	8	8	no	—
209	C₂×C₄×C₃₁	—	8	4	no	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
215	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
216	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	2	4	no	—
219	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
220	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
221	#3503	C₂×C₃₁	1	2	yes	complemented
222	#3387	C₂×C₃₁	1	4	yes	complemented
223	#3384	C₂×C₃₁	1	4	yes	complemented
224	#3381	C₂×C₃₁	1	4	yes	complemented
225	Q₈×SD₁₆	C₂×C₃₁	1	4	yes	complemented
226	#3163	C₂×C₃₁	1	2	yes	complemented
227	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#3084	C₂×C₃₁	1	4	yes	complemented
229	#3079	C₂×C₃₁	1	2	yes	complemented
230	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2952	C₂×C₃₁	1	2	yes	—
232	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
238	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	8	no	—
242	C₄×C₃₁	—	4	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	8	no	—
245	C₄×C₃₁	D₄×Q₈	1	2	yes	—
246	C₄×C₃₁	#577	1	4	yes	—
247	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
248	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
249	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
250	Q₈²	C₂²×C₃₁	1	4	yes	—
251	#557	C₂²×C₃₁	1	4	yes	—
252	#554	—	2	4	no	—
253	#554	C₂²×C₃₁	1	4	yes	—
254	D₄×Q₈	—	2	4	no	—
255	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
256	#544	—	2	4	no	—
257	#544	C₂²×C₃₁	1	4	yes	—
258	#533	—	2	2	no	—
259	#533	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	—	2	2	no	—
261	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
262	#520	C₂²×C₃₁	1	4	yes	—
263	C₄×#41	C₂²×C₃₁	1	2	yes	—
264	#501	C₂²×C₃₁	1	8	yes	—
265	#499	C₂²×C₃₁	1	4	yes	—
266	#498	C₂²×C₃₁	1	4	yes	—
267	#498	C₂²×C₃₁	1	4	yes	—
268	#498	C₂²×C₃₁	1	4	yes	—
269	#479	C₂²×C₃₁	1	8	yes	—
270	#478	C₂²×C₃₁	1	8	yes	—
271	#477	C₂²×C₃₁	1	4	yes	—
272	#476	C₂²×C₃₁	1	4	yes	—
273	#476	C₂²×C₃₁	1	8	yes	—
274	#475	C₂²×C₃₁	1	4	yes	—
275	#475	C₂²×C₃₁	1	8	yes	—
276	#474	C₂²×C₃₁	1	4	yes	—
277	#446	C₂²×C₃₁	1	4	yes	—
278	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
279	#442	C₂²×C₃₁	1	4	yes	—
280	#441	C₂²×C₃₁	1	4	yes	—
281	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
282	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
283	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
284	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
285	C₄×D₈	C₂²×C₃₁	1	2	yes	—
286	#428	C₂²×C₃₁	1	4	yes	—
287	#427	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#106	C₂²×C₃₁	1	2	yes	—
289	C₂×#106	C₂²×C₃₁	1	2	yes	—
290	#424	C₂²×C₃₁	1	2	yes	—
291	#423	C₂²×C₃₁	1	1	yes	characteristic
292	#417	C₂²×C₃₁	1	4	yes	—
293	#416	C₂²×C₃₁	1	2	yes	—
294	#405	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₃₁	—	2	2	no	—
297	C₂×C₃₁	—	4	4	no	—
298	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
299	C₂×C₃₁	C₂×#577	1	2	yes	—
300	C₂×#41	C₂³×C₃₁	1	2	yes	—
301	C₂²×Q₈	—	2	4	no	—
302	C₂²×Q₈	—	2	2	no	—
303	#135	Q₈×C₃₁	1	8	yes	complemented
304	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
306	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄⋊Q₈	—	2	4	no	—
311	C₄⋊Q₈	—	2	4	no	—
312	C₄⋊Q₈	—	2	4	no	—
313	C₄⋊Q₈	—	2	8	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
317	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
318	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
319	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
320	#125	—	2	4	no	—
321	#125	—	2	8	no	—
322	#125	D₄×C₃₁	1	2	yes	—
323	#125	C₂³×C₃₁	1	8	yes	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	—	2	8	no	—
327	C₂²⋊Q₈	—	2	8	no	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
330	C₄×Q₈	—	2	4	no	—
331	C₄×Q₈	—	2	4	no	—
332	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
333	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
334	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
335	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	8	no	—
339	C₄×D₄	C₂³×C₃₁	1	2	yes	—
340	C₄×D₄	C₂³×C₃₁	1	2	yes	—
341	C₄×D₄	C₂³×C₃₁	1	8	yes	—
342	#117	—	2	4	no	—
343	#117	—	2	2	no	—
344	#117	C₂³×C₃₁	1	2	yes	—
345	#117	C₂³×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	—	2	4	no	—
348	C₂×C₄⋊C₄	—	2	4	no	—
349	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#107	C₂³×C₃₁	1	2	yes	—
353	#107	C₂³×C₃₁	1	8	yes	—
354	#106	C₂³×C₃₁	1	4	yes	—
355	#106	C₂³×C₃₁	1	4	yes	—
356	#106	C₂³×C₃₁	1	8	yes	—
357	#106	C₂³×C₃₁	1	1	yes	characteristic
358	#106	C₂³×C₃₁	1	1	yes	characteristic
359	#106	C₂³×C₃₁	1	8	yes	—
360	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
361	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
362	#103	C₂³×C₃₁	1	2	yes	—
363	#103	C₂³×C₃₁	1	2	yes	—
364	#103	C₂³×C₃₁	1	8	yes	—
365	#102	C₂³×C₃₁	1	2	yes	—
366	#102	C₂³×C₃₁	1	2	yes	—
367	#102	C₂³×C₃₁	1	8	yes	—
368	#97	C₂³×C₃₁	1	4	yes	—
369	C₄×C₈	C₂³×C₃₁	1	2	yes	—
370	C₄×C₈	C₂³×C₃₁	1	2	yes	—
371	C₄×C₈	C₂³×C₃₁	1	4	yes	—
372	C₃₁	#61374	1	1	yes	characteristic, direct product factor, Sylow
373	#41	—	2	4	no	—
374	#41	C₂×Q₈×C₃₁	1	8	yes	complemented
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	8	no	—
377	C₂×Q₈	—	2	4	no	—
378	C₂×Q₈	—	8	4	no	—
379	C₂×Q₈	—	8	8	no	—
380	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
381	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
392	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
393	D₈	C₂×Q₈×C₃₁	1	4	yes	complemented
394	M₁₆	—	2	4	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
397	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
398	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
399	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
400	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
413	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
414	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	8	no	—
425	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₄²	C₂⁴×C₃₁	1	2	yes	—
428	C₄²	C₂⁴×C₃₁	1	4	yes	—
429	C₂³	—	2	2	no	—
430	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
431	Q₈	—	2	8	no	—
432	Q₈	—	2	2	no	—
433	Q₈	—	4	8	no	—
434	Q₈	—	8	4	no	—
435	Q₈	—	8	8	no	—
436	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
437	D₄	—	2	8	no	—
438	D₄	—	2	2	no	—
439	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	8	8	no	—
449	C₂×C₄	—	8	4	no	—
450	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
451	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
455	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
456	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
457	C₈	—	2	2	no	—
458	C₈	—	2	4	no	—
459	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
460	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
466	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
467	C₄	—	2	4	no	—
468	C₄	—	2	2	no	—
469	C₄	—	2	8	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	8	no	—
473	C₄	D₄×Q₈×C₃₁	1	2	yes	—
474	C₄	C₃₁×#577	1	4	yes	—
475	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
476	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
477	C₂	—	2	2	no	—
478	C₂	—	4	4	no	—
479	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
480	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3503	C₂	semidirect
2. C₃₁×#3387	C₂	semidirect
3. C₃₁×#3384	C₂	semidirect
4. C₃₁×#3381	C₂	semidirect
5. Q₈×SD₁₆×C₃₁	C₂	semidirect
6. C₃₁×#3163	C₂	semidirect
7. C₂×C₃₁×#498	C₂	semidirect
8. C₃₁×#3084	C₂	semidirect
9. C₃₁×#3079	C₂	semidirect
10. C₃₁×#2960	C₂	semidirect
63. C₃₁×#135	Q₈	semidirect
133. #41×C₃₁	C₂×Q₈	semidirect
150. Q₁₆×C₃₁	C₂×Q₈	semidirect
151. SD₁₆×C₃₁	C₂×Q₈	semidirect
152. D₈×C₃₁	C₂×Q₈	semidirect
155. C₂×C₈×C₃₁	C₂×Q₈	semidirect
188. #61374	C₃₁	direct (N × H)
221. #3503	C₂×C₃₁	semidirect
222. #3387	C₂×C₃₁	semidirect
223. #3384	C₂×C₃₁	semidirect
224. #3381	C₂×C₃₁	semidirect
225. Q₈×SD₁₆	C₂×C₃₁	semidirect
226. #3163	C₂×C₃₁	semidirect
227. C₂×#498	C₂×C₃₁	semidirect
228. #3084	C₂×C₃₁	semidirect
229. #3079	C₂×C₃₁	semidirect
230. #2960	C₂×C₃₁	semidirect
303. #135	Q₈×C₃₁	semidirect
372. C₃₁	#61374	direct (N × H)
374. #41	C₂×Q₈×C₃₁	semidirect
391. Q₁₆	C₂×Q₈×C₃₁	semidirect
392. SD₁₆	C₂×Q₈×C₃₁	semidirect
393. D₈	C₂×Q₈×C₃₁	semidirect
396. C₂×C₈	C₂×Q₈×C₃₁	semidirect

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×#3387	2
C₃₁×#3384	2
C₃₁×#3381	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3163	2
C₂×C₃₁×#498	2
C₃₁×#3084	2
C₃₁×#3079	2
C₃₁×#2960	2
C₃₁×#2952	2
C₄×C₃₁×#135	2
#61374	31

Subgroups — C31×#61374 #21518133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61374 #21518133