Subgroups — C₃₁×#34026 #21490785

Order 7936.

2782 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2298	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2297	C₂	1	2	yes	maximal
4	C₃₁×#2296	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2295	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2293	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2290	C₂	1	8	yes	maximal, complemented
8	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#117	C₂	1	4	yes	maximal, complemented
10	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×#41×C₃₁	C₄	1	16	yes	complemented
13	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	8	yes	—
17	C₂×C₃₁×#117	C₂²	1	2	yes	—
18	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
21	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#391	C₂²	1	4	yes	—
25	C₃₁×#390	C₂²	1	4	yes	—
26	C₃₁×#389	C₂²	1	8	yes	—
27	C₃₁×#388	C₂²	1	16	yes	—
28	C₃₁×#387	C₂²	1	16	yes	—
29	C₃₁×#386	C₂²	1	8	yes	—
30	C₃₁×#385	C₂²	1	8	yes	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
34	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
37	C₃₁×#376	C₂²	1	8	yes	—
38	C₃₁×#376	C₂²	1	2	yes	—
39	C₄³×C₃₁	C₂²	1	4	yes	—
40	C₄³×C₃₁	C₂²	1	2	yes	—
41	C₄³×C₃₁	C₂²	1	2	yes	—
42	C₄³×C₃₁	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	—	2	8	no	—
44	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
51	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
54	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	—
55	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
56	C₃₁×#117	—	2	8	no	—
57	C₃₁×#117	—	2	32	no	—
58	C₃₁×#117	C₂×C₄	1	8	yes	—
59	C₃₁×#117	C₂×C₄	1	16	yes	complemented
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
65	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
66	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
68	C₂×C₄²×C₃₁	—	2	8	no	—
69	C₂×C₄²×C₃₁	—	2	16	no	—
70	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
71	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
72	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#95	C₂³	1	16	yes	—
85	C₃₁×#95	C₂³	1	16	yes	—
86	C₃₁×#95	C₂³	1	8	yes	—
87	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
88	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
89	#41×C₃₁	—	2	32	no	—
90	C₂×Q₈×C₃₁	—	2	4	no	—
91	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₃₁	—	2	8	no	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
95	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₃₁	—	2	8	no	—
97	C₂²×C₄×C₃₁	—	2	16	no	—
98	C₂²×C₄×C₃₁	—	2	8	no	—
99	C₂²×C₄×C₃₁	—	2	16	no	—
100	C₂²×C₄×C₃₁	—	2	8	no	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
103	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
104	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₃₁	#41	1	8	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₃₁	—	2	32	no	—
115	C₄⋊C₄×C₃₁	—	2	8	no	—
116	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
117	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
118	C₂²⋊C₄×C₃₁	—	2	8	no	—
119	C₂²⋊C₄×C₃₁	—	2	32	no	—
120	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
121	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
122	C₄²×C₃₁	—	2	16	no	—
123	C₄²×C₃₁	—	2	8	no	—
124	C₄²×C₃₁	—	2	8	no	—
125	C₄²×C₃₁	—	2	8	no	—
126	C₄²×C₃₁	—	2	8	no	—
127	C₄²×C₃₁	—	2	16	no	—
128	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
129	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
130	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
131	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
132	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
133	C₄²×C₃₁	#41	1	16	yes	complemented
134	#34026	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	2	4	no	—
136	C₂³×C₃₁	—	2	8	no	—
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
139	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
140	C₂³×C₃₁	C₂×#41	1	4	yes	—
141	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
142	Q₈×C₃₁	—	2	8	no	—
143	D₄×C₃₁	—	2	16	no	—
144	D₄×C₃₁	—	2	8	no	—
145	C₂×C₄×C₃₁	—	2	32	no	—
146	C₂×C₄×C₃₁	—	2	4	no	—
147	C₂×C₄×C₃₁	—	2	8	no	—
148	C₂×C₄×C₃₁	—	2	8	no	—
149	C₂×C₄×C₃₁	—	2	8	no	—
150	C₂×C₄×C₃₁	—	2	4	no	—
151	C₂×C₄×C₃₁	—	2	16	no	—
152	C₂×C₄×C₃₁	—	2	16	no	—
153	C₂×C₄×C₃₁	—	2	32	no	—
154	C₂×C₄×C₃₁	—	2	8	no	—
155	C₂×C₄×C₃₁	—	2	16	no	—
156	C₂×C₄×C₃₁	#117	1	16	yes	complemented
157	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
158	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
159	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
160	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
161	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
162	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
163	C₂×C₄×C₃₁	C₂×#41	1	16	yes	—
164	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
165	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
166	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
167	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
168	#2298	C₂×C₃₁	1	4	yes	complemented
169	#2297	C₂×C₃₁	1	2	yes	—
170	#2296	C₂×C₃₁	1	2	yes	complemented
171	#2295	C₂×C₃₁	1	4	yes	complemented
172	#2293	C₂×C₃₁	1	4	yes	complemented
173	#2290	C₂×C₃₁	1	8	yes	complemented
174	C₄×#117	C₂×C₃₁	1	4	yes	complemented
175	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
176	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂²×C₃₁	—	2	4	no	—
178	C₂²×C₃₁	—	2	8	no	—
179	C₂²×C₃₁	—	2	4	no	—
180	C₂²×C₃₁	—	2	8	no	—
181	C₂²×C₃₁	—	4	8	no	—
182	C₂²×C₃₁	C₂×#117	1	4	yes	—
183	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
184	C₂²×C₃₁	C₂²×#41	1	2	yes	—
185	C₂²×C₃₁	C₂²×#41	1	4	yes	—
186	C₄×C₃₁	—	2	4	no	—
187	C₄×C₃₁	—	2	4	no	—
188	C₄×C₃₁	—	2	8	no	—
189	C₄×C₃₁	—	2	16	no	—
190	C₄×C₃₁	—	4	8	no	—
191	C₄×C₃₁	C₂×#117	1	4	yes	—
192	C₄×C₃₁	C₂×#117	1	4	yes	—
193	C₄×C₃₁	C₄×#41	1	8	yes	—
194	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
195	C₄×#41	C₄×C₃₁	1	16	yes	complemented
196	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
198	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
199	C₂×#117	C₂²×C₃₁	1	8	yes	—
200	C₂×#117	C₂²×C₃₁	1	2	yes	—
201	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
202	C₂×#117	C₂²×C₃₁	1	2	yes	—
203	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
204	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
205	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
206	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
207	#391	C₂²×C₃₁	1	4	yes	—
208	#390	C₂²×C₃₁	1	4	yes	—
209	#389	C₂²×C₃₁	1	8	yes	—
210	#388	C₂²×C₃₁	1	16	yes	—
211	#387	C₂²×C₃₁	1	16	yes	—
212	#386	C₂²×C₃₁	1	8	yes	—
213	#385	C₂²×C₃₁	1	8	yes	—
214	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
215	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
216	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
217	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
218	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
219	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
220	#376	C₂²×C₃₁	1	8	yes	—
221	#376	C₂²×C₃₁	1	2	yes	—
222	C₄³	C₂²×C₃₁	1	2	yes	—
223	C₄³	C₂²×C₃₁	1	4	yes	—
224	C₄³	C₂²×C₃₁	1	2	yes	—
225	C₄³	C₂²×C₃₁	1	2	yes	—
226	C₂×C₃₁	—	2	4	no	—
227	C₂×C₃₁	—	4	4	no	—
228	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
229	C₂×C₃₁	C₂×C₄×#41	1	2	yes	—
230	C₂×C₃₁	#3444	1	4	yes	—
231	C₂×#41	—	2	8	no	—
232	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
233	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
234	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
235	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
236	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
237	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
238	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
239	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
240	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
241	C₄×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
242	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	—
243	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
244	#117	—	2	8	no	—
245	#117	—	2	32	no	—
246	#117	C₂×C₄×C₃₁	1	8	yes	—
247	#117	C₂×C₄×C₃₁	1	16	yes	complemented
248	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
249	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
250	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
251	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
254	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
255	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
256	C₂×C₄²	—	2	8	no	—
257	C₂×C₄²	—	2	16	no	—
258	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
259	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
260	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
261	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
262	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
263	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
264	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
265	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
270	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
272	#95	C₂³×C₃₁	1	16	yes	—
273	#95	C₂³×C₃₁	1	16	yes	—
274	#95	C₂³×C₃₁	1	8	yes	—
275	C₃₁	#34026	1	1	yes	characteristic, direct product factor, Sylow
276	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
277	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
278	#41	—	2	32	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
284	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
285	C₂²×C₄	—	2	8	no	—
286	C₂²×C₄	—	2	16	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	16	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
292	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
293	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
294	C₂²×C₄	#41×C₃₁	1	8	yes	—
295	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
296	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
297	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
298	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
299	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
300	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
302	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
303	C₄⋊C₄	—	2	32	no	—
304	C₄⋊C₄	—	2	8	no	—
305	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
306	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
307	C₂²⋊C₄	—	2	8	no	—
308	C₂²⋊C₄	—	2	32	no	—
309	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
310	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
311	C₄²	—	2	16	no	—
312	C₄²	—	2	8	no	—
313	C₄²	—	2	8	no	—
314	C₄²	—	2	8	no	—
315	C₄²	—	2	8	no	—
316	C₄²	—	2	16	no	—
317	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
318	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
319	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
320	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
321	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
322	C₄²	#41×C₃₁	1	16	yes	complemented
323	C₂³	—	2	4	no	—
324	C₂³	—	2	8	no	—
325	C₂³	—	2	4	no	—
326	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
327	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
328	C₂³	C₂×#41×C₃₁	1	4	yes	—
329	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
330	Q₈	—	2	8	no	—
331	D₄	—	2	16	no	—
332	D₄	—	2	8	no	—
333	C₂×C₄	—	2	32	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	8	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	16	no	—
340	C₂×C₄	—	2	16	no	—
341	C₂×C₄	—	2	32	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	16	no	—
344	C₂×C₄	C₃₁×#117	1	16	yes	complemented
345	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
346	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
347	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
348	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
349	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
350	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
351	C₂×C₄	C₂×#41×C₃₁	1	16	yes	—
352	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
353	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
354	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
355	C₂²	—	2	4	no	—
356	C₂²	—	2	8	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	8	no	—
359	C₂²	—	4	8	no	—
360	C₂²	C₂×C₃₁×#117	1	4	yes	—
361	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
362	C₂²	C₂²×#41×C₃₁	1	2	yes	—
363	C₂²	C₂²×#41×C₃₁	1	4	yes	—
364	C₄	—	2	4	no	—
365	C₄	—	2	4	no	—
366	C₄	—	2	8	no	—
367	C₄	—	2	16	no	—
368	C₄	—	4	8	no	—
369	C₄	C₂×C₃₁×#117	1	4	yes	—
370	C₄	C₂×C₃₁×#117	1	4	yes	—
371	C₄	C₄×#41×C₃₁	1	8	yes	—
372	C₂	—	2	4	no	—
373	C₂	—	4	4	no	—
374	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
375	C₂	C₂×C₄×#41×C₃₁	1	2	yes	—
376	C₂	C₃₁×#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2298	C₂	semidirect
4. C₃₁×#2296	C₂	semidirect
5. C₃₁×#2295	C₂	semidirect
6. C₃₁×#2293	C₂	semidirect
7. C₃₁×#2290	C₂	semidirect
8. C₄×C₃₁×#117	C₂	semidirect
9. C₄×C₃₁×#117	C₂	semidirect
10. C₂×C₄³×C₃₁	C₂	semidirect
12. C₄×#41×C₃₁	C₄	semidirect
53. C₄×Q₈×C₃₁	C₂×C₄	semidirect
55. C₄×D₄×C₃₁	C₂×C₄	semidirect
59. C₃₁×#117	C₂×C₄	semidirect
71. C₂×C₄²×C₃₁	C₂×C₄	semidirect
133. C₄²×C₃₁	#41	semidirect
134. #34026	C₃₁	direct (N × H)
156. C₂×C₄×C₃₁	#117	semidirect
168. #2298	C₂×C₃₁	semidirect
170. #2296	C₂×C₃₁	semidirect
171. #2295	C₂×C₃₁	semidirect
172. #2293	C₂×C₃₁	semidirect
173. #2290	C₂×C₃₁	semidirect
174. C₄×#117	C₂×C₃₁	semidirect
175. C₄×#117	C₂×C₃₁	semidirect
176. C₂×C₄³	C₂×C₃₁	semidirect
195. C₄×#41	C₄×C₃₁	semidirect
241. C₄×Q₈	C₂×C₄×C₃₁	semidirect
243. C₄×D₄	C₂×C₄×C₃₁	semidirect
247. #117	C₂×C₄×C₃₁	semidirect
259. C₂×C₄²	C₂×C₄×C₃₁	semidirect
275. C₃₁	#34026	direct (N × H)
322. C₄²	#41×C₃₁	semidirect
344. C₂×C₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₃₁×#2298	2
C₃₁×#2297	2
C₃₁×#2296	2
C₃₁×#2295	2
C₃₁×#2293	2
C₃₁×#2290	2
C₄×C₃₁×#117	2
C₂×C₄³×C₃₁	2
#34026	31

Subgroups — C31×#34026 #21490785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34026 #21490785