Subgroups — C₃₁×#22722 #21479481

Order 7936.

832 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1940	C₂	1	4	yes	maximal
4	C₃₁×#1939	C₂	1	2	yes	maximal
5	C₃₁×#1842	C₂	1	2	yes	maximal
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1767	C₂	1	2	yes	maximal
8	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#446	—	2	8	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#97	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄³×C₃₁	C₂²	1	1	yes	characteristic
29	C₄³×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	4	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₄×Q₈×C₃₁	—	2	4	no	—
37	C₄×Q₈×C₃₁	—	2	4	no	—
38	C₄×Q₈×C₃₁	D₄	1	4	yes	—
39	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	8	no	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×C₄⋊C₈	—	2	4	no	—
63	C₃₁×C₄⋊C₈	—	4	4	no	—
64	C₃₁×C₄⋊C₈	—	4	2	no	—
65	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
66	C₃₁×#97	—	2	8	no	—
67	C₄×C₈×C₃₁	—	2	4	no	—
68	C₃₁×#95	C₂×C₄	1	2	yes	—
69	C₃₁×#95	C₂×C₄	1	2	yes	—
70	C₃₁×#95	C₂×C₄	1	2	yes	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
98	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	4	no	—
103	C₄²×C₃₁	—	2	4	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	4	1	no	—
110	C₄²×C₃₁	—	4	2	no	—
111	C₄²×C₃₁	M₁₆	1	4	yes	—
112	C₄²×C₃₁	C₂×D₄	1	2	yes	—
113	C₄²×C₃₁	C₂×D₄	1	4	yes	—
114	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
115	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
116	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
117	C₄²×C₃₁	#41	1	2	yes	—
118	#22722	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₃₁	—	2	8	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
142	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
150	C₂×C₄×C₃₁	#131	1	2	yes	—
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
157	C₈×C₃₁	—	2	4	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₈×C₃₁	—	4	4	no	—
160	C₂×#446	C₂×C₃₁	1	2	yes	—
161	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
162	#1940	C₂×C₃₁	1	4	yes	—
163	#1939	C₂×C₃₁	1	2	yes	—
164	#1842	C₂×C₃₁	1	1	yes	characteristic
165	#1842	C₂×C₃₁	1	2	yes	—
166	#1767	C₂×C₃₁	1	2	yes	—
167	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
168	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₂²×C₃₁	#520	1	1	yes	characteristic
171	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
172	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
173	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
174	C₂²×C₃₁	#568	1	1	yes	characteristic
175	C₄×C₃₁	—	2	4	no	—
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	2	4	no	—
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	2	2	no	—
180	C₄×C₃₁	—	2	4	no	—
181	C₄×C₃₁	—	4	2	no	—
182	C₄×C₃₁	—	4	1	no	—
183	C₄×C₃₁	#423	1	4	yes	—
184	C₄×C₃₁	#446	1	4	yes	complemented
185	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
186	C₄×C₃₁	#550	1	1	yes	characteristic
187	C₄×C₃₁	#557	1	1	yes	characteristic
188	C₄×C₃₁	Q₈²	1	1	yes	characteristic
189	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
192	#446	—	2	8	no	—
193	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
197	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
198	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×#97	C₂²×C₃₁	1	4	yes	—
201	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
202	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
203	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
205	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
208	C₄³	C₂²×C₃₁	1	1	yes	characteristic
209	C₄³	C₂²×C₃₁	1	2	yes	—
210	#336	C₂²×C₃₁	1	4	yes	—
211	#336	C₂²×C₃₁	1	2	yes	—
212	C₂×C₃₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
214	C₂×C₃₁	#2906	1	1	yes	characteristic
215	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
216	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
217	C₂×C₃₁	#2989	1	1	yes	characteristic
218	C₂×C₃₁	#2990	1	1	yes	characteristic
219	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
221	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₄×Q₈	—	2	4	no	—
224	C₄×Q₈	—	2	4	no	—
225	C₄×Q₈	D₄×C₃₁	1	4	yes	—
226	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
234	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
238	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
242	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
245	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
246	C₄⋊C₈	—	2	4	no	—
247	C₄⋊C₈	—	2	8	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	2	4	no	—
250	C₄⋊C₈	—	4	4	no	—
251	C₄⋊C₈	—	4	2	no	—
252	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
253	#97	—	2	8	no	—
254	C₄×C₈	—	2	4	no	—
255	#95	C₂×C₄×C₃₁	1	2	yes	—
256	#95	C₂×C₄×C₃₁	1	2	yes	—
257	#95	C₂×C₄×C₃₁	1	2	yes	—
258	C₃₁	#22722	1	1	yes	characteristic, direct product factor, Sylow
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
262	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
264	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
265	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
266	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
286	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
287	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	2	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	4	1	no	—
298	C₄²	—	4	2	no	—
299	C₄²	M₁₆×C₃₁	1	4	yes	—
300	C₄²	C₂×D₄×C₃₁	1	2	yes	—
301	C₄²	C₂×D₄×C₃₁	1	4	yes	—
302	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
303	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
304	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
305	C₄²	#41×C₃₁	1	2	yes	—
306	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
328	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
329	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
330	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
331	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
332	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
333	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
334	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
335	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
336	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
337	C₂×C₄	C₃₁×#131	1	2	yes	—
338	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
340	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
342	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
343	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
344	C₈	—	2	4	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	4	no	—
347	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
348	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
349	C₂²	C₃₁×#520	1	1	yes	characteristic
350	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
351	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
352	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
353	C₂²	C₃₁×#568	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	4	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	C₃₁×#423	1	4	yes	—
363	C₄	C₃₁×#446	1	4	yes	complemented
364	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
365	C₄	C₃₁×#550	1	1	yes	characteristic
366	C₄	C₃₁×#557	1	1	yes	characteristic
367	C₄	Q₈²×C₃₁	1	1	yes	characteristic
368	C₂	Q₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
369	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
370	C₂	C₃₁×#2906	1	1	yes	characteristic
371	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
372	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
373	C₂	C₃₁×#2989	1	1	yes	characteristic
374	C₂	C₃₁×#2990	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
65. C₃₁×C₄⋊C₈	Q₈	semidirect
118. #22722	C₃₁	direct (N × H)
184. C₄×C₃₁	#446	semidirect
252. C₄⋊C₈	Q₈×C₃₁	semidirect
258. C₃₁	#22722	direct (N × H)
363. C₄	C₃₁×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#446	2
C₄²×Q₈×C₃₁	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1842	2
C₃₁×#1767	2
C₄×C₃₁×C₄⋊C₈	2
#22722	31

Subgroups — C31×#22722 #21479481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22722 #21479481