Subgroups — C₃₁×#24728 #21481487

Order 7936.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2541	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2041	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	—	2	1	no	—
17	C₂×C₃₁×#121	—	2	1	no	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#460	—	2	8	no	—
22	C₃₁×#460	—	2	4	no	—
23	C₄×SD₁₆×C₃₁	—	2	4	no	—
24	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#392	—	2	2	no	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#384	—	2	1	no	—
39	C₃₁×#384	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	4	no	—
57	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#127	—	2	4	no	—
62	C₃₁×#127	—	2	2	no	—
63	C₃₁×#127	—	2	2	no	—
64	C₃₁×#127	—	4	2	no	—
65	C₃₁×#121	—	4	4	no	—
66	C₃₁×#121	—	4	4	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	4	no	—
69	C₄×Q₈×C₃₁	—	4	2	no	—
70	C₄×Q₈×C₃₁	D₄	1	4	yes	complemented
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	—	2	2	no	—
88	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#106	—	2	2	no	—
91	C₃₁×C₄⋊C₈	—	2	4	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×#103	—	2	2	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×#102	—	2	4	no	—
98	C₃₁×#102	—	4	2	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₃₁×#95	—	2	2	no	—
101	C₃₁×#95	D₄	1	2	yes	—
102	C₂⁴×C₃₁	—	2	1	no	—
103	C₂⁴×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₃₁	#41	1	2	yes	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	4	4	no	—
122	C₂×D₄×C₃₁	—	4	4	no	—
123	C₂×D₄×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	4	2	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	2	no	—
128	C₂×D₄×C₃₁	—	4	2	no	—
129	C₂×D₄×C₃₁	—	8	1	no	—
130	C₂×D₄×C₃₁	—	8	2	no	—
131	C₂×D₄×C₃₁	—	8	1	no	—
132	C₂×D₄×C₃₁	—	8	2	no	—
133	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	2	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	SD₁₆×C₃₁	—	2	8	no	—
147	SD₁₆×C₃₁	—	4	8	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	4	4	no	—
165	C₄⋊C₄×C₃₁	—	4	2	no	—
166	C₄⋊C₄×C₃₁	—	8	1	no	—
167	C₄⋊C₄×C₃₁	—	8	2	no	—
168	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
169	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₃₁	—	4	4	no	—
171	C₂²⋊C₄×C₃₁	—	4	2	no	—
172	C₂²⋊C₄×C₃₁	—	4	4	no	—
173	C₄²×C₃₁	—	2	4	no	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	—	4	4	no	—
177	C₄²×C₃₁	C₂×D₄	1	2	yes	—
178	C₄²×C₃₁	#41	1	2	yes	—
179	#24728	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	4	1	no	—
182	C₂³×C₃₁	—	4	1	no	—
183	C₂³×C₃₁	—	4	1	no	—
184	C₂³×C₃₁	—	4	1	no	—
185	C₂³×C₃₁	—	4	1	no	—
186	C₂³×C₃₁	—	4	1	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	—	4	2	no	—
191	C₂³×C₃₁	—	4	2	no	—
192	C₂³×C₃₁	—	4	2	no	—
193	C₂³×C₃₁	—	4	2	no	—
194	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	#121	1	4	yes	complemented
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	—	4	1	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	8	8	no	—
208	D₄×C₃₁	—	8	4	no	—
209	D₄×C₃₁	—	8	8	no	—
210	D₄×C₃₁	—	8	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	8	4	no	—
231	C₂×C₄×C₃₁	#121	1	2	yes	—
232	C₂×C₄×C₃₁	#121	1	2	yes	—
233	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
234	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
235	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	4	2	no	—
244	C₈×C₃₁	—	4	1	no	—
245	C₂×#460	C₂×C₃₁	1	2	yes	complemented
246	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#2541	C₂×C₃₁	1	1	yes	characteristic
249	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#2065	C₂×C₃₁	1	2	yes	complemented
251	#2041	C₂×C₃₁	1	2	yes	complemented
252	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1880	C₂×C₃₁	1	2	yes	complemented
255	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	8	1	no	—
260	C₂²×C₃₁	—	8	1	no	—
261	C₂²×C₃₁	—	8	1	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	—	8	2	no	—
264	C₂²×C₃₁	—	8	2	no	—
265	C₂²×C₃₁	—	8	2	no	—
266	C₂²×C₃₁	—	8	2	no	—
267	C₂²×C₃₁	—	8	2	no	—
268	C₂²×C₃₁	—	8	2	no	—
269	C₂²×C₃₁	—	8	2	no	—
270	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
271	C₂²×C₃₁	D₄²	1	1	yes	characteristic
272	C₂²×C₃₁	#548	1	1	yes	characteristic
273	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
274	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂²×C₃₁	#576	1	1	yes	characteristic
276	C₂²×C₃₁	#577	1	1	yes	characteristic
277	C₄×C₃₁	—	2	4	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	4	2	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	8	2	no	—
283	C₄×C₃₁	—	8	1	no	—
284	C₄×C₃₁	#460	1	4	yes	complemented
285	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
286	C₄×C₃₁	#537	1	1	yes	characteristic
287	C₄×C₃₁	#547	1	1	yes	characteristic
288	C₄×C₃₁	#553	1	1	yes	characteristic
289	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
290	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#127	C₂²×C₃₁	1	2	yes	—
293	C₂×#121	—	2	1	no	—
294	C₂×#121	—	2	1	no	—
295	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
296	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
298	#460	—	2	8	no	—
299	#460	—	2	4	no	—
300	C₄×SD₁₆	—	2	4	no	—
301	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
304	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#102	C₂²×C₃₁	1	2	yes	—
306	C₂×#102	C₂²×C₃₁	1	2	yes	—
307	C₂×#102	C₂²×C₃₁	1	2	yes	—
308	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×#102	C₂²×C₃₁	1	2	yes	—
310	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
311	#392	—	2	2	no	—
312	#390	C₂²×C₃₁	1	2	yes	—
313	#389	—	2	2	no	—
314	#389	C₂²×C₃₁	1	2	yes	—
315	#384	—	2	1	no	—
316	#384	C₂²×C₃₁	1	1	yes	characteristic
317	C₄×C₄⋊C₄	—	2	1	no	—
318	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
319	#340	C₂²×C₃₁	1	2	yes	—
320	#336	C₂²×C₃₁	1	2	yes	—
321	C₂×C₃₁	—	8	2	no	—
322	C₂×C₃₁	—	16	2	no	—
323	C₂×C₃₁	#2416	1	1	yes	characteristic
324	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
325	C₂×C₃₁	#3045	1	1	yes	characteristic
326	C₂×C₃₁	#3274	1	1	yes	characteristic
327	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
328	C₂×C₃₁	#3362	1	1	yes	characteristic
329	C₂×C₃₁	#3368	1	1	yes	characteristic
330	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
336	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	—	2	1	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	4	no	—
341	C₂×SD₁₆	—	2	4	no	—
342	C₂×SD₁₆	—	2	4	no	—
343	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
344	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
345	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	#127	—	2	4	no	—
348	#127	—	2	2	no	—
349	#127	—	2	2	no	—
350	#127	—	4	2	no	—
351	#121	—	4	4	no	—
352	#121	—	4	4	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	—	2	4	no	—
355	C₄×Q₈	—	4	2	no	—
356	C₄×Q₈	D₄×C₃₁	1	4	yes	complemented
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₂²⋊C₄	—	2	1	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	—	2	2	no	—
374	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
375	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
376	#106	—	2	2	no	—
377	C₄⋊C₈	—	2	4	no	—
378	C₄⋊C₈	—	2	2	no	—
379	#103	—	2	2	no	—
380	#102	—	2	4	no	—
381	#102	—	2	4	no	—
382	#102	—	2	2	no	—
383	#102	—	2	4	no	—
384	#102	—	4	2	no	—
385	C₄×C₈	—	2	2	no	—
386	#95	—	2	2	no	—
387	#95	D₄×C₃₁	1	2	yes	—
388	C₃₁	#24728	1	1	yes	characteristic, direct product factor, Sylow
389	C₂⁴	—	2	1	no	—
390	C₂⁴	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
396	C₂×Q₈	#41×C₃₁	1	2	yes	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	8	1	no	—
417	C₂×D₄	—	8	2	no	—
418	C₂×D₄	—	8	1	no	—
419	C₂×D₄	—	8	2	no	—
420	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
433	SD₁₆	—	2	8	no	—
434	SD₁₆	—	4	8	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	4	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	—	8	1	no	—
454	C₄⋊C₄	—	8	2	no	—
455	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
456	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
457	C₂²⋊C₄	—	4	4	no	—
458	C₂²⋊C₄	—	4	2	no	—
459	C₂²⋊C₄	—	4	4	no	—
460	C₄²	—	2	4	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	4	4	no	—
464	C₄²	C₂×D₄×C₃₁	1	2	yes	—
465	C₄²	#41×C₃₁	1	2	yes	—
466	C₂³	—	2	2	no	—
467	C₂³	—	4	1	no	—
468	C₂³	—	4	1	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	2	no	—
476	C₂³	—	4	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	Q₈	—	2	2	no	—
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	C₃₁×#121	1	4	yes	complemented
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	2	no	—
493	D₄	—	8	8	no	—
494	D₄	—	8	4	no	—
495	D₄	—	8	8	no	—
496	D₄	—	8	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	8	4	no	—
517	C₂×C₄	C₃₁×#121	1	2	yes	—
518	C₂×C₄	C₃₁×#121	1	2	yes	—
519	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
520	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
521	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
527	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
528	C₈	—	2	2	no	—
529	C₈	—	4	2	no	—
530	C₈	—	4	1	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	8	1	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	1	no	—
537	C₂²	—	8	1	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	8	2	no	—
542	C₂²	—	8	2	no	—
543	C₂²	—	8	2	no	—
544	C₂²	—	8	2	no	—
545	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
546	C₂²	D₄²×C₃₁	1	1	yes	characteristic
547	C₂²	C₃₁×#548	1	1	yes	characteristic
548	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
549	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
550	C₂²	C₃₁×#576	1	1	yes	characteristic
551	C₂²	C₃₁×#577	1	1	yes	characteristic
552	C₄	—	2	4	no	—
553	C₄	—	2	4	no	—
554	C₄	—	2	2	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	2	no	—
557	C₄	—	8	2	no	—
558	C₄	—	8	1	no	—
559	C₄	C₃₁×#460	1	4	yes	complemented
560	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
561	C₄	C₃₁×#537	1	1	yes	characteristic
562	C₄	C₃₁×#547	1	1	yes	characteristic
563	C₄	C₃₁×#553	1	1	yes	characteristic
564	C₂	—	8	2	no	—
565	C₂	—	16	2	no	—
566	C₂	C₃₁×#2416	1	1	yes	characteristic
567	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
568	C₂	C₃₁×#3045	1	1	yes	characteristic
569	C₂	C₃₁×#3274	1	1	yes	characteristic
570	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
571	C₂	C₃₁×#3362	1	1	yes	characteristic
572	C₂	C₃₁×#3368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#460	C₂	semidirect
2. C₂×C₃₁×#460	C₂	semidirect
3. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
5. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
7. C₃₁×#2041	C₂	semidirect
8. C₃₁×#1957	C₂	semidirect
9. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1880	C₂	semidirect
11. C₃₁×#1802	C₂	semidirect
57. C₂×SD₁₆×C₃₁	D₄	semidirect
70. C₄×Q₈×C₃₁	D₄	semidirect
168. C₄⋊C₄×C₃₁	SD₁₆	semidirect
179. #24728	C₃₁	direct (N × H)
198. Q₈×C₃₁	#121	semidirect
245. C₂×#460	C₂×C₃₁	semidirect
246. C₂×#460	C₂×C₃₁	semidirect
247. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
249. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
250. #2065	C₂×C₃₁	semidirect
251. #2041	C₂×C₃₁	semidirect
252. #1957	C₂×C₃₁	semidirect
253. #1939	C₂×C₃₁	semidirect
254. #1880	C₂×C₃₁	semidirect
255. #1802	C₂×C₃₁	semidirect
284. C₄×C₃₁	#460	semidirect
343. C₂×SD₁₆	D₄×C₃₁	semidirect
356. C₄×Q₈	D₄×C₃₁	semidirect
388. C₃₁	#24728	direct (N × H)
455. C₄⋊C₄	SD₁₆×C₃₁	semidirect
484. Q₈	C₃₁×#121	semidirect
559. C₄	C₃₁×#460	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#460	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2541	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2065	2
C₃₁×#2041	2
C₃₁×#1957	2
C₃₁×#1939	2
C₃₁×#1880	2
C₃₁×#1802	2
#24728	31

Subgroups — C31×#24728 #21481487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24728 #21481487