Subgroups — C₃₁×#21337 #21478096

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1998	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1997	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1742	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#533	—	2	1	no	—
10	C₃₁×#533	—	2	1	no	—
11	C₃₁×#533	—	2	1	no	—
12	C₃₁×#533	—	2	1	no	—
13	C₃₁×#533	—	2	2	no	—
14	C₃₁×#533	—	2	2	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#125	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	C₄	1	4	yes	—
18	C₂×C₄×Q₈×C₃₁	C₄	1	4	yes	—
19	C₂×C₃₁×#117	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#423	—	2	2	no	—
22	C₃₁×#423	C₂²	1	2	yes	—
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
25	C₄×M₁₆×C₃₁	—	2	1	no	—
26	C₄×M₁₆×C₃₁	—	2	1	no	—
27	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	—	2	1	no	—
30	C₂×C₄×C₈×C₃₁	—	2	1	no	—
31	C₃₁×#343	C₂²	1	16	yes	—
32	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂×M₁₆×C₃₁	—	2	4	no	—
37	C₂×M₁₆×C₃₁	—	2	2	no	—
38	C₂×M₁₆×C₃₁	—	2	8	no	—
39	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
40	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
41	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×#125	—	2	2	no	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#125	—	2	4	no	—
51	C₃₁×C₂²⋊Q₈	—	2	4	no	—
52	C₃₁×C₂²⋊Q₈	—	2	4	no	—
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	2	4	no	—
55	C₄×Q₈×C₃₁	—	2	4	no	—
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	2	4	no	—
59	C₃₁×#117	—	2	2	no	—
60	C₃₁×#117	—	2	2	no	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	2	yes	—
74	C₂×C₄²×C₃₁	D₄	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	—	2	4	no	—
78	C₃₁×C₄⋊C₈	—	2	4	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₃₁×#97	—	2	2	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Q₈×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	—	2	4	no	—
86	C₂×Q₈×C₃₁	—	4	8	no	—
87	C₂²×C₄×C₃₁	—	4	4	no	—
88	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₃₁	#41	1	4	yes	—
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₃₁	—	2	8	no	—
104	M₁₆×C₃₁	—	2	4	no	—
105	M₁₆×C₃₁	—	2	8	no	—
106	M₁₆×C₃₁	—	4	4	no	—
107	M₁₆×C₃₁	C₂×Q₈	1	8	yes	—
108	C₂×C₈×C₃₁	—	2	8	no	—
109	C₂×C₈×C₃₁	—	2	4	no	—
110	C₂×C₈×C₃₁	—	2	4	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₄⋊C₄×C₃₁	—	2	8	no	—
113	C₄⋊C₄×C₃₁	—	2	8	no	—
114	C₄⋊C₄×C₃₁	—	2	4	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₂²⋊C₄×C₃₁	—	2	4	no	—
119	C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₄²×C₃₁	—	2	4	no	—
121	C₄²×C₃₁	—	2	4	no	—
122	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
123	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
124	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
125	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
126	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	C₂×D₄	1	2	yes	—
131	C₄²×C₃₁	C₂×D₄	1	2	yes	—
132	#21337	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
137	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
138	Q₈×C₃₁	—	4	8	no	—
139	Q₈×C₃₁	—	4	8	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	4	no	—
142	C₂×C₄×C₃₁	—	2	4	no	—
143	C₂×C₄×C₃₁	—	2	8	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	8	no	—
148	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
155	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
157	C₂×C₄×C₃₁	#124	1	2	yes	—
158	C₂×C₄×C₃₁	#124	1	2	yes	—
159	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
160	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
161	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
164	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
165	C₈×C₃₁	—	2	8	no	—
166	C₈×C₃₁	—	4	4	no	—
167	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
168	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
169	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
170	#1998	C₂×C₃₁	1	4	yes	complemented
171	#1997	C₂×C₃₁	1	4	yes	complemented
172	#1742	C₂×C₃₁	1	4	yes	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	2	no	—
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	—	4	4	no	—
177	C₂²×C₃₁	#391	1	2	yes	—
178	C₂²×C₃₁	#391	1	2	yes	—
179	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₃₁	C₂×C₄×Q₈	1	2	yes	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	—	2	2	no	—
184	C₄×C₃₁	—	4	8	no	—
185	C₄×C₃₁	#391	1	2	yes	—
186	C₄×C₃₁	#391	1	2	yes	—
187	C₄×C₃₁	#413	1	4	yes	—
188	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
189	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
190	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
192	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
193	#533	—	2	1	no	—
194	#533	—	2	1	no	—
195	#533	—	2	1	no	—
196	#533	—	2	1	no	—
197	#533	—	2	2	no	—
198	#533	—	2	2	no	—
199	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
200	C₂×#125	C₂²×C₃₁	1	2	yes	—
201	C₂×C₂²⋊Q₈	C₄×C₃₁	1	4	yes	—
202	C₂×C₄×Q₈	C₄×C₃₁	1	4	yes	—
203	C₂×#117	C₂²×C₃₁	1	2	yes	—
204	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
205	#423	—	2	2	no	—
206	#423	C₂²×C₃₁	1	2	yes	—
207	#423	C₂²×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₈	—	2	2	no	—
209	C₄×M₁₆	—	2	1	no	—
210	C₄×M₁₆	—	2	1	no	—
211	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
212	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
213	C₂×#97	—	2	1	no	—
214	C₂×C₄×C₈	—	2	1	no	—
215	#343	C₂²×C₃₁	1	16	yes	—
216	C₂×C₃₁	—	2	2	no	—
217	C₂×C₃₁	—	2	2	no	—
218	C₂×C₃₁	—	2	2	no	—
219	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
220	C₂×C₃₁	C₂×#413	1	2	yes	—
221	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
222	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
223	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	4	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	8	no	—
228	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
229	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
230	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	—	2	2	no	—
233	C₄⋊Q₈	—	2	2	no	—
234	C₄⋊Q₈	—	2	2	no	—
235	C₄⋊Q₈	—	2	2	no	—
236	C₄⋊Q₈	—	2	2	no	—
237	#125	—	2	2	no	—
238	#125	—	2	2	no	—
239	#125	—	2	4	no	—
240	C₂²⋊Q₈	—	2	4	no	—
241	C₂²⋊Q₈	—	2	4	no	—
242	C₂²⋊Q₈	—	2	4	no	—
243	C₂²⋊Q₈	—	2	4	no	—
244	C₄×Q₈	—	2	4	no	—
245	C₄×Q₈	—	2	4	no	—
246	C₄×Q₈	—	2	4	no	—
247	C₄×Q₈	—	2	4	no	—
248	#117	—	2	2	no	—
249	#117	—	2	2	no	—
250	#117	—	2	2	no	—
251	#117	—	2	2	no	—
252	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
253	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
254	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
255	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
256	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
257	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
258	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄²	D₄×C₃₁	1	2	yes	—
263	C₂×C₄²	D₄×C₃₁	1	2	yes	—
264	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	4	no	—
268	#97	—	2	2	no	—
269	#97	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₃₁	#21337	1	1	yes	characteristic, direct product factor, Sylow
273	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	4	8	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
279	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
280	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
281	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
284	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
291	C₂²×C₄	#41×C₃₁	1	4	yes	—
292	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
293	M₁₆	—	2	8	no	—
294	M₁₆	—	2	4	no	—
295	M₁₆	—	2	8	no	—
296	M₁₆	—	4	4	no	—
297	M₁₆	C₂×Q₈×C₃₁	1	8	yes	—
298	C₂×C₈	—	2	8	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	4	4	no	—
302	C₄⋊C₄	—	2	8	no	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	4	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₂²⋊C₄	—	2	4	no	—
309	C₂²⋊C₄	—	2	4	no	—
310	C₄²	—	2	4	no	—
311	C₄²	—	2	4	no	—
312	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
313	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
314	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
315	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
316	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₃₁	1	2	yes	—
321	C₄²	C₂×D₄×C₃₁	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
325	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
326	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
327	Q₈	—	4	8	no	—
328	Q₈	—	4	8	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	8	no	—
337	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
338	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
339	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
340	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
341	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
344	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
345	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
346	C₂×C₄	C₃₁×#124	1	2	yes	—
347	C₂×C₄	C₃₁×#124	1	2	yes	—
348	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
349	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
350	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
353	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
354	C₈	—	2	8	no	—
355	C₈	—	4	4	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	4	no	—
360	C₂²	C₃₁×#391	1	2	yes	—
361	C₂²	C₃₁×#391	1	2	yes	—
362	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
363	C₂²	C₂×C₄×Q₈×C₃₁	1	2	yes	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	8	no	—
368	C₄	C₃₁×#391	1	2	yes	—
369	C₄	C₃₁×#391	1	2	yes	—
370	C₄	C₃₁×#413	1	4	yes	—
371	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
372	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
373	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	2	2	no	—
376	C₂	—	2	2	no	—
377	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
378	C₂	C₂×C₃₁×#413	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1998	C₂	semidirect
5. C₃₁×#1997	C₂	semidirect
39. C₂×M₁₆×C₃₁	Q₈	semidirect
132. #21337	C₃₁	direct (N × H)
170. #1998	C₂×C₃₁	semidirect
171. #1997	C₂×C₃₁	semidirect
228. C₂×M₁₆	Q₈×C₃₁	semidirect
272. C₃₁	#21337	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#533	2
C₂×C₃₁×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1998	2
C₃₁×#1997	2
C₃₁×#1742	2
#21337	31

Subgroups — C31×#21337 #21478096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21337 #21478096