Subgroups — C₃₁×#26298 #21483057

Order 7936.

826 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2547	C₂	1	1	yes	maximal, characteristic
2	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2077	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2077	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1853	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1824	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#397	—	2	2	no	—
25	C₃₁×#391	—	2	1	no	—
26	C₃₁×#389	—	2	1	no	—
27	C₃₁×#385	—	2	2	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₃₁×#380	C₂²	1	2	yes	—
31	C₃₁×#380	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×#124	—	4	4	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	4	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×#106	—	4	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	4	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	4	2	no	—
89	C₂×Q₈×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	SD₁₆	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	4	no	—
107	C₂²×C₄×C₃₁	—	8	2	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	#41	1	2	yes	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	8	1	no	—
137	C₄⋊C₄×C₃₁	#41	1	2	yes	—
138	C₂²⋊C₄×C₃₁	—	2	4	no	—
139	C₂²⋊C₄×C₃₁	—	4	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₄²×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	#41	1	2	yes	—
145	C₄²×C₃₁	#41	1	2	yes	—
146	#26298	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
154	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₃₁	—	8	4	no	—
156	D₄×C₃₁	—	2	8	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	8	8	no	—
185	C₂×C₄×C₃₁	#123	1	2	yes	—
186	C₂×C₄×C₃₁	#123	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	—	4	2	no	—
198	#2547	C₂×C₃₁	1	1	yes	characteristic
199	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
200	#2077	C₂×C₃₁	1	2	yes	complemented
201	#2077	C₂×C₃₁	1	2	yes	complemented
202	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1954	C₂×C₃₁	1	1	yes	characteristic
205	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1853	C₂×C₃₁	1	2	yes	complemented
207	#1824	C₂×C₃₁	1	2	yes	complemented
208	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	4	2	no	—
213	C₂²×C₃₁	—	4	2	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
217	C₂²×C₃₁	#546	1	1	yes	characteristic
218	C₂²×C₃₁	#547	1	1	yes	characteristic
219	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
220	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₃₁	#577	1	1	yes	characteristic
222	C₂²×C₃₁	#578	1	1	yes	characteristic
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	—	8	1	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	#481	1	4	yes	—
232	C₄×C₃₁	#529	1	1	yes	characteristic
233	C₄×C₃₁	#542	1	1	yes	characteristic
234	C₄×C₃₁	#556	1	1	yes	characteristic
235	C₄×C₃₁	#559	1	1	yes	characteristic
236	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#124	—	2	1	no	—
238	C₂×C₄×D₄	—	2	1	no	—
239	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
241	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#106	C₂²×C₃₁	1	2	yes	—
244	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	#397	—	2	2	no	—
249	#391	—	2	1	no	—
250	#389	—	2	1	no	—
251	#385	—	2	2	no	—
252	#384	C₂²×C₃₁	1	1	yes	characteristic
253	#380	C₂²×C₃₁	1	2	yes	—
254	#380	C₂²×C₃₁	1	2	yes	—
255	#380	C₂²×C₃₁	1	2	yes	—
256	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
257	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	#340	C₂²×C₃₁	1	2	yes	—
259	#340	C₂²×C₃₁	1	1	yes	characteristic
260	#340	C₂²×C₃₁	1	2	yes	—
261	#340	C₂²×C₃₁	1	2	yes	—
262	#340	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	1	yes	characteristic
264	#340	C₂²×C₃₁	1	2	yes	—
265	C₂×C₃₁	—	4	4	no	—
266	C₂×C₃₁	#2447	1	1	yes	characteristic
267	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
268	C₂×C₃₁	#3097	1	1	yes	characteristic
269	C₂×C₃₁	#3298	1	1	yes	characteristic
270	C₂×C₃₁	#3324	1	1	yes	characteristic
271	C₂×C₃₁	#3335	1	1	yes	characteristic
272	C₂×C₃₁	#3353	1	1	yes	characteristic
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
275	C₂³×C₄	—	2	2	no	—
276	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	—	2	2	no	—
282	#124	—	4	4	no	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	—	2	2	no	—
285	C₄×D₄	—	4	4	no	—
286	C₂×C₄⋊C₄	—	2	4	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	4	2	no	—
292	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
303	C₂×C₄²	—	2	1	no	—
304	C₂×C₄²	—	2	1	no	—
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	#106	—	2	2	no	—
307	#106	—	2	2	no	—
308	#106	—	4	2	no	—
309	C₄⋊C₈	—	2	2	no	—
310	C₂²⋊C₈	—	2	4	no	—
311	C₂²⋊C₈	—	4	2	no	—
312	C₄×C₈	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	1	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	C₃₁	#26298	1	1	yes	characteristic, direct product factor, Sylow
319	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	SD₁₆×C₃₁	1	4	yes	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	4	4	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	8	2	no	—
341	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	#41×C₃₁	1	2	yes	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	8	1	no	—
370	C₄⋊C₄	#41×C₃₁	1	2	yes	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	4	2	no	—
376	C₄²	—	4	2	no	—
377	C₄²	#41×C₃₁	1	2	yes	—
378	C₄²	#41×C₃₁	1	2	yes	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
386	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	Q₈	—	8	4	no	—
388	D₄	—	2	8	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	8	8	no	—
417	C₂×C₄	C₃₁×#123	1	2	yes	—
418	C₂×C₄	C₃₁×#123	1	2	yes	—
419	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
420	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
421	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
426	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	4	1	no	—
429	C₈	—	4	2	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
438	C₂²	C₃₁×#546	1	1	yes	characteristic
439	C₂²	C₃₁×#547	1	1	yes	characteristic
440	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
441	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
442	C₂²	C₃₁×#577	1	1	yes	characteristic
443	C₂²	C₃₁×#578	1	1	yes	characteristic
444	C₄	—	2	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	2	no	—
449	C₄	—	8	1	no	—
450	C₄	—	8	1	no	—
451	C₄	—	8	2	no	—
452	C₄	C₃₁×#481	1	4	yes	—
453	C₄	C₃₁×#529	1	1	yes	characteristic
454	C₄	C₃₁×#542	1	1	yes	characteristic
455	C₄	C₃₁×#556	1	1	yes	characteristic
456	C₄	C₃₁×#559	1	1	yes	characteristic
457	C₂	—	4	4	no	—
458	C₂	C₃₁×#2447	1	1	yes	characteristic
459	C₂	C₂×C₃₁×#481	1	1	yes	characteristic
460	C₂	C₃₁×#3097	1	1	yes	characteristic
461	C₂	C₃₁×#3298	1	1	yes	characteristic
462	C₂	C₃₁×#3324	1	1	yes	characteristic
463	C₂	C₃₁×#3335	1	1	yes	characteristic
464	C₂	C₃₁×#3353	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2077	C₂	semidirect
4. C₃₁×#2077	C₂	semidirect
5. C₃₁×#1990	C₂	semidirect
6. C₃₁×#1988	C₂	semidirect
8. C₃₁×#1948	C₂	semidirect
9. C₃₁×#1853	C₂	semidirect
10. C₃₁×#1824	C₂	semidirect
11. C₄×C₃₁×#106	C₂	semidirect
146. #26298	C₃₁	direct (N × H)
200. #2077	C₂×C₃₁	semidirect
201. #2077	C₂×C₃₁	semidirect
202. #1990	C₂×C₃₁	semidirect
203. #1988	C₂×C₃₁	semidirect
205. #1948	C₂×C₃₁	semidirect
206. #1853	C₂×C₃₁	semidirect
207. #1824	C₂×C₃₁	semidirect
208. C₄×#106	C₂×C₃₁	semidirect
318. C₃₁	#26298	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×#2547	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2077	2
C₃₁×#1990	2
C₃₁×#1988	2
C₃₁×#1954	2
C₃₁×#1948	2
C₃₁×#1853	2
C₃₁×#1824	2
C₄×C₃₁×#106	2
#26298	31

Subgroups — C31×#26298 #21483057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26298 #21483057