Subgroups — C₂×C₃₁×#3397 #21518153

Order 7936.

2250 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3397	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#496	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#440	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	—	2	1	no	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#556	—	2	4	no	—
20	C₃₁×#556	C₂²	1	8	yes	complemented
21	C₃₁×#550	—	2	4	no	—
22	C₃₁×#550	C₂²	1	8	yes	complemented
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#127	C₂²	1	2	yes	—
26	C₂×C₃₁×#126	—	2	2	no	—
27	C₂×C₃₁×#124	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
29	C₂×C₃₁×#121	—	2	1	no	—
30	C₂×C₃₁×#121	—	2	2	no	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₃₁×#117	—	2	1	no	—
37	C₃₁×#496	C₂²	1	16	yes	complemented
38	C₃₁×#492	C₂²	1	8	yes	complemented
39	C₃₁×#463	C₂²	1	16	yes	complemented
40	C₃₁×#461	C₂²	1	8	yes	complemented
41	C₃₁×#460	C₂²	1	16	yes	complemented
42	C₃₁×#459	C₂²	1	8	yes	complemented
43	C₃₁×#446	C₂²	1	8	yes	complemented
44	C₃₁×#440	C₂²	1	16	yes	complemented
45	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	2	yes	—
50	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	4	8	no	—
57	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	—	2	1	no	—
61	C₂²×D₄×C₃₁	—	2	2	no	—
62	C₂²×D₄×C₃₁	—	2	1	no	—
63	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂³×C₄×C₃₁	—	2	2	no	—
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂×SD₁₆×C₃₁	—	2	8	no	—
70	C₂×SD₁₆×C₃₁	—	4	4	no	—
71	C₂×SD₁₆×C₃₁	—	4	4	no	—
72	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
73	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂×D₈×C₃₁	—	2	4	no	—
76	C₂×D₈×C₃₁	—	4	4	no	—
77	C₂×D₈×C₃₁	C₂³	1	4	yes	—
78	C₂×D₈×C₃₁	C₂³	1	8	yes	—
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	—	2	2	no	—
83	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
84	C₃₁×#127	—	2	2	no	—
85	C₃₁×#127	—	2	4	no	—
86	C₃₁×#127	C₂³	1	4	yes	—
87	C₃₁×#127	C₂³	1	8	yes	—
88	C₃₁×#126	—	2	16	no	—
89	C₃₁×#124	—	2	4	no	—
90	C₃₁×#124	C₂³	1	8	yes	—
91	C₃₁×C₂²⋊Q₈	—	2	8	no	—
92	C₃₁×#121	—	2	8	no	—
93	C₃₁×#121	—	2	16	no	—
94	C₄×Q₈×C₃₁	—	2	4	no	—
95	C₄×Q₈×C₃₁	D₄	1	4	yes	—
96	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
97	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	C₂³	1	4	yes	—
102	C₄×D₄×C₃₁	C₂³	1	4	yes	—
103	C₄×D₄×C₃₁	C₂³	1	8	yes	—
104	C₃₁×#117	—	2	8	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄²×C₃₁	—	2	2	no	—
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₃₁×#107	C₂³	1	4	yes	—
121	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
122	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#102	C₂³	1	4	yes	—
125	C₃₁×#102	C₂³	1	4	yes	—
126	C₃₁×#102	C₂³	1	8	yes	—
127	C₃₁×#97	C₂³	1	8	yes	—
128	C₄×C₈×C₃₁	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	—	2	1	no	—
130	C₂⁴×C₃₁	—	2	1	no	—
131	C₂⁴×C₃₁	—	2	2	no	—
132	#41×C₃₁	—	4	32	no	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	8	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	4	8	no	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	8	4	no	—
155	C₂×D₄×C₃₁	—	8	2	no	—
156	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
158	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	4	8	no	—
166	C₂²×C₄×C₃₁	—	4	4	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	SD₁₆×C₃₁	—	2	16	no	—
174	SD₁₆×C₃₁	—	4	8	no	—
175	SD₁₆×C₃₁	—	4	8	no	—
176	D₈×C₃₁	—	2	8	no	—
177	D₈×C₃₁	—	4	8	no	—
178	C₂×C₈×C₃₁	—	2	4	no	—
179	C₂×C₈×C₃₁	—	2	2	no	—
180	C₂×C₈×C₃₁	#41	1	4	yes	complemented
181	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
182	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
183	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₃₁	—	2	4	no	—
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
192	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
193	C₂²⋊C₄×C₃₁	—	2	8	no	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	8	no	—
198	C₄²×C₃₁	—	2	8	no	—
199	C₄²×C₃₁	C₂×D₄	1	2	yes	—
200	C₄²×C₃₁	C₂×D₄	1	4	yes	—
201	C₄²×C₃₁	C₂⁴	1	2	yes	—
202	C₄²×C₃₁	C₂⁴	1	4	yes	—
203	C₂×#3397	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	4	no	—
207	C₂³×C₃₁	—	4	2	no	—
208	C₂³×C₃₁	—	4	2	no	—
209	C₂³×C₃₁	—	4	4	no	—
210	C₂³×C₃₁	—	4	4	no	—
211	C₂³×C₃₁	—	4	2	no	—
212	C₂³×C₃₁	—	4	2	no	—
213	C₂³×C₃₁	—	4	2	no	—
214	C₂³×C₃₁	—	4	4	no	—
215	C₂³×C₃₁	—	4	2	no	—
216	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₃₁	—	2	8	no	—
218	Q₈×C₃₁	—	2	4	no	—
219	Q₈×C₃₁	—	2	4	no	—
220	Q₈×C₃₁	—	4	4	no	—
221	D₄×C₃₁	—	2	2	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	—	2	4	no	—
224	D₄×C₃₁	—	2	2	no	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	2	2	no	—
227	D₄×C₃₁	—	4	16	no	—
228	D₄×C₃₁	—	4	2	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	4	8	no	—
232	D₄×C₃₁	—	8	8	no	—
233	D₄×C₃₁	—	8	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	4	16	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	8	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
253	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
254	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
255	C₈×C₃₁	—	2	4	no	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	C₂×#41	1	4	yes	complemented
258	C₂×#556	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#3397	C₂×C₃₁	1	16	yes	direct product factor
261	C₂×#496	C₂×C₃₁	1	2	yes	complemented
262	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₂×#460	C₂×C₃₁	1	2	yes	complemented
264	C₂×#463	C₂×C₃₁	1	2	yes	complemented
265	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂×#459	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
268	C₂×#440	C₂×C₃₁	1	2	yes	complemented
269	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	2	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	—	4	2	no	—
279	C₂²×C₃₁	—	8	4	no	—
280	C₂²×C₃₁	—	8	8	no	—
281	C₂²×C₃₁	—	8	4	no	—
282	C₂²×C₃₁	#550	1	2	yes	—
283	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
284	C₂²×C₃₁	#577	1	2	yes	—
285	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
286	C₄×C₃₁	—	2	4	no	—
287	C₄×C₃₁	—	2	4	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	4	4	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	#550	1	2	yes	—
293	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
294	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
295	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂²×#41	—	2	1	no	—
297	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
298	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
299	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
300	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
301	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
302	#556	—	2	4	no	—
303	#556	C₂²×C₃₁	1	8	yes	complemented
304	#550	—	2	4	no	—
305	#550	C₂²×C₃₁	1	8	yes	complemented
306	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#127	C₂²×C₃₁	1	2	yes	—
309	C₂×#126	—	2	2	no	—
310	C₂×#124	C₂²×C₃₁	1	2	yes	—
311	C₂×C₂²⋊Q₈	—	2	1	no	—
312	C₂×#121	—	2	1	no	—
313	C₂×#121	—	2	2	no	—
314	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
315	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
319	C₂×#117	—	2	1	no	—
320	#496	C₂²×C₃₁	1	16	yes	complemented
321	#492	C₂²×C₃₁	1	8	yes	complemented
322	#463	C₂²×C₃₁	1	16	yes	complemented
323	#461	C₂²×C₃₁	1	8	yes	complemented
324	#460	C₂²×C₃₁	1	16	yes	complemented
325	#459	C₂²×C₃₁	1	8	yes	complemented
326	#446	C₂²×C₃₁	1	8	yes	complemented
327	#440	C₂²×C₃₁	1	16	yes	complemented
328	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
329	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
332	C₂×#103	C₂²×C₃₁	1	2	yes	—
333	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
335	C₂×#102	C₂²×C₃₁	1	2	yes	—
336	C₂×#97	C₂²×C₃₁	1	2	yes	—
337	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	—	8	4	no	—
340	C₂×C₃₁	—	8	2	no	—
341	C₂×C₃₁	#3397	1	4	yes	direct product factor
342	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
343	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
345	C₂×#41	—	2	8	no	—
346	C₂×#41	—	4	8	no	—
347	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
348	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	—	2	2	no	—
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
356	C₂³×C₄	—	2	1	no	—
357	C₂³×C₄	—	2	2	no	—
358	C₂³×C₄	—	2	1	no	—
359	C₂×SD₁₆	—	2	8	no	—
360	C₂×SD₁₆	—	4	4	no	—
361	C₂×SD₁₆	—	4	4	no	—
362	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
363	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
364	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
365	C₂×D₈	—	2	4	no	—
366	C₂×D₈	—	4	4	no	—
367	C₂×D₈	C₂³×C₃₁	1	4	yes	—
368	C₂×D₈	C₂³×C₃₁	1	8	yes	—
369	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
370	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
374	#127	—	2	2	no	—
375	#127	—	2	4	no	—
376	#127	C₂³×C₃₁	1	4	yes	—
377	#127	C₂³×C₃₁	1	8	yes	—
378	#126	—	2	16	no	—
379	#124	—	2	4	no	—
380	#124	C₂³×C₃₁	1	8	yes	—
381	C₂²⋊Q₈	—	2	8	no	—
382	#121	—	2	8	no	—
383	#121	—	2	16	no	—
384	C₄×Q₈	—	2	4	no	—
385	C₄×Q₈	D₄×C₃₁	1	4	yes	—
386	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
387	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	C₂³×C₃₁	1	4	yes	—
392	C₄×D₄	C₂³×C₃₁	1	8	yes	—
393	C₄×D₄	C₂³×C₃₁	1	4	yes	—
394	#117	—	2	8	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	2	no	—
407	C₂×C₄²	—	2	2	no	—
408	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
410	#107	C₂³×C₃₁	1	4	yes	—
411	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
412	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
413	#103	C₂³×C₃₁	1	8	yes	—
414	#102	C₂³×C₃₁	1	4	yes	—
415	#102	C₂³×C₃₁	1	4	yes	—
416	#102	C₂³×C₃₁	1	8	yes	—
417	#97	C₂³×C₃₁	1	8	yes	—
418	C₄×C₈	C₂³×C₃₁	1	4	yes	—
419	C₃₁	C₂×#3397	1	1	yes	characteristic, direct product factor, Sylow
420	C₂⁴	—	2	1	no	—
421	C₂⁴	—	2	1	no	—
422	C₂⁴	—	2	2	no	—
423	#41	—	4	32	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
429	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	8	4	no	—
446	C₂×D₄	—	8	2	no	—
447	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
448	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
449	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	8	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	4	8	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	4	no	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
464	SD₁₆	—	2	16	no	—
465	SD₁₆	—	4	8	no	—
466	SD₁₆	—	4	8	no	—
467	D₈	—	2	8	no	—
468	D₈	—	4	8	no	—
469	C₂×C₈	—	2	4	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	#41×C₃₁	1	4	yes	complemented
472	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
473	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
474	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	8	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
481	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
482	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
483	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₄²	—	2	8	no	—
490	C₄²	C₂×D₄×C₃₁	1	2	yes	—
491	C₄²	C₂×D₄×C₃₁	1	4	yes	—
492	C₄²	C₂⁴×C₃₁	1	2	yes	—
493	C₄²	C₂⁴×C₃₁	1	4	yes	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	4	no	—
500	C₂³	—	4	4	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	—	4	4	no	—
505	C₂³	—	4	2	no	—
506	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	Q₈	—	2	8	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	2	4	no	—
510	Q₈	—	4	4	no	—
511	D₄	—	2	2	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	2	no	—
515	D₄	—	2	4	no	—
516	D₄	—	2	2	no	—
517	D₄	—	4	16	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	8	no	—
522	D₄	—	8	8	no	—
523	D₄	—	8	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	4	16	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	8	no	—
535	C₂×C₄	—	4	8	no	—
536	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
537	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
538	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
540	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
543	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
544	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
545	C₈	—	2	4	no	—
546	C₈	—	2	2	no	—
547	C₈	C₂×#41×C₃₁	1	4	yes	complemented
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	8	4	no	—
558	C₂²	—	8	8	no	—
559	C₂²	—	8	4	no	—
560	C₂²	C₃₁×#550	1	2	yes	—
561	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
562	C₂²	C₃₁×#577	1	2	yes	—
563	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
564	C₄	—	2	4	no	—
565	C₄	—	2	4	no	—
566	C₄	—	2	2	no	—
567	C₄	—	4	4	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	2	no	—
570	C₄	C₃₁×#550	1	2	yes	—
571	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
572	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
573	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
574	C₂	—	8	2	no	—
575	C₂	—	8	4	no	—
576	C₂	—	8	2	no	—
577	C₂	C₃₁×#3397	1	4	yes	direct product factor
578	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
579	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
580	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#556	C₂	semidirect
2. C₂×C₃₁×#550	C₂	semidirect
3. C₃₁×#3397	C₂	direct (N × H)
4. C₂×C₃₁×#496	C₂	semidirect
5. C₂×C₃₁×#492	C₂	semidirect
6. C₂×C₃₁×#460	C₂	semidirect
7. C₂×C₃₁×#463	C₂	semidirect
8. C₂×C₃₁×#461	C₂	semidirect
9. C₂×C₃₁×#459	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
11. C₂×C₃₁×#440	C₂	semidirect
12. C₂×C₄×D₈×C₃₁	C₂	semidirect
20. C₃₁×#556	C₂²	semidirect
22. C₃₁×#550	C₂²	semidirect
37. C₃₁×#496	C₂²	semidirect
38. C₃₁×#492	C₂²	semidirect
39. C₃₁×#463	C₂²	semidirect
40. C₃₁×#461	C₂²	semidirect
41. C₃₁×#460	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#446	C₂²	semidirect
44. C₃₁×#440	C₂²	semidirect
45. C₄×D₈×C₃₁	C₂²	semidirect
180. C₂×C₈×C₃₁	#41	semidirect
203. C₂×#3397	C₃₁	direct (N × H)
257. C₈×C₃₁	C₂×#41	semidirect
258. C₂×#556	C₂×C₃₁	semidirect
259. C₂×#550	C₂×C₃₁	semidirect
260. #3397	C₂×C₃₁	direct (N × H)
261. C₂×#496	C₂×C₃₁	semidirect
262. C₂×#492	C₂×C₃₁	semidirect
263. C₂×#460	C₂×C₃₁	semidirect
264. C₂×#463	C₂×C₃₁	semidirect
265. C₂×#461	C₂×C₃₁	semidirect
266. C₂×#459	C₂×C₃₁	semidirect
267. C₂×#446	C₂×C₃₁	semidirect
268. C₂×#440	C₂×C₃₁	semidirect
269. C₂×C₄×D₈	C₂×C₃₁	semidirect
303. #556	C₂²×C₃₁	semidirect
305. #550	C₂²×C₃₁	semidirect
320. #496	C₂²×C₃₁	semidirect
321. #492	C₂²×C₃₁	semidirect
322. #463	C₂²×C₃₁	semidirect
323. #461	C₂²×C₃₁	semidirect
324. #460	C₂²×C₃₁	semidirect
325. #459	C₂²×C₃₁	semidirect
326. #446	C₂²×C₃₁	semidirect
327. #440	C₂²×C₃₁	semidirect
328. C₄×D₈	C₂²×C₃₁	semidirect
341. C₂×C₃₁	#3397	direct (N × H)
419. C₃₁	C₂×#3397	direct (N × H)
471. C₂×C₈	#41×C₃₁	semidirect
547. C₈	C₂×#41×C₃₁	semidirect
577. C₂	C₃₁×#3397	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×#556	2
C₂×C₃₁×#550	2
C₃₁×#3397	2
C₂×C₃₁×#496	2
C₂×C₃₁×#492	2
C₂×C₃₁×#460	2
C₂×C₃₁×#463	2
C₂×C₃₁×#461	2
C₂×C₃₁×#459	2
C₂×C₃₁×#446	2
C₂×C₃₁×#440	2
C₂×C₄×D₈×C₃₁	2
C₂×#3397	31

Subgroups — C2×C31×#3397 #21518153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3397 #21518153