Subgroups — C₂×C₃₁×#3239 #21517609

Order 7936.

2018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3239	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#501	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#484	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#481	C₂	1	2	yes	maximal
8	C₂×C₃₁×#475	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#474	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
14	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	8	yes	complemented
18	C₃₁×#529	—	2	4	no	—
19	C₃₁×#529	C₂²	1	8	yes	—
20	C₂×C₃₁×#126	—	2	1	no	—
21	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#123	—	2	1	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₃₁	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#117	—	2	1	no	—
30	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#501	C₂²	1	16	yes	complemented
34	C₃₁×#489	C₂²	1	16	yes	complemented
35	C₃₁×#484	C₂²	1	16	yes	complemented
36	C₃₁×#481	C₂²	1	16	yes	—
37	C₃₁×#475	C₂²	1	16	yes	complemented
38	C₃₁×#474	C₂²	1	16	yes	complemented
39	C₃₁×#432	C₂²	1	8	yes	complemented
40	C₂×C₃₁×#107	C₂²	1	4	yes	—
41	C₂×C₃₁×#106	C₂²	1	4	yes	—
42	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
43	C₂×C₃₁×#103	C₂²	1	4	yes	—
44	C₂×C₃₁×#102	C₂²	1	4	yes	—
45	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
46	C₂×C₃₁×#97	C₂²	1	2	yes	—
47	C₂²×Q₈×C₃₁	—	2	2	no	—
48	C₂²×Q₈×C₃₁	—	2	2	no	—
49	C₂²×Q₈×C₃₁	—	2	8	no	—
50	C₂²×Q₈×C₃₁	—	4	8	no	—
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	—	2	2	no	—
55	C₂³×C₄×C₃₁	—	2	1	no	—
56	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
60	C₃₁×C₄⋊Q₈	—	2	4	no	—
61	C₃₁×C₄⋊Q₈	—	2	4	no	—
62	C₃₁×C₄⋊Q₈	—	2	8	no	—
63	C₃₁×C₄⋊Q₈	—	2	8	no	—
64	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
66	C₃₁×#126	—	2	8	no	—
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	C₂³	1	4	yes	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	C₂³	1	4	yes	—
71	C₃₁×#123	—	2	8	no	—
72	C₃₁×C₂²⋊Q₈	—	2	2	no	—
73	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	C₂³	1	4	yes	—
76	C₄×Q₈×C₃₁	—	2	2	no	—
77	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
78	C₄×D₄×C₃₁	—	2	8	no	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₄×D₄×C₃₁	C₂³	1	4	yes	—
81	C₃₁×#117	—	2	8	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
100	C₂×C₄²×C₃₁	—	2	1	no	—
101	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	D₄	1	4	yes	—
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₃₁×#107	C₂³	1	16	yes	—
108	C₃₁×#106	C₂³	1	16	yes	—
109	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
110	C₃₁×#103	C₂³	1	16	yes	—
111	C₃₁×#102	C₂³	1	16	yes	—
112	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
113	C₃₁×#97	C₂³	1	8	yes	—
114	C₂⁴×C₃₁	—	2	1	no	—
115	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	2	8	no	—
118	C₂×Q₈×C₃₁	—	2	8	no	—
119	C₂×Q₈×C₃₁	—	2	4	no	—
120	C₂×Q₈×C₃₁	—	4	8	no	—
121	C₂×Q₈×C₃₁	—	4	32	no	—
122	C₂×Q₈×C₃₁	—	4	8	no	—
123	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	4	8	no	—
139	C₂²×C₄×C₃₁	—	4	4	no	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂×C₈×C₃₁	—	2	8	no	—
151	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	2	8	no	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
163	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₂²⋊C₄×C₃₁	—	2	4	no	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₄²×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	C₂×D₄	1	4	yes	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	C₂×D₄	1	2	yes	—
174	C₄²×C₃₁	C₂×D₄	1	4	yes	—
175	C₄²×C₃₁	C₂⁴	1	2	yes	—
176	C₄²×C₃₁	C₂⁴	1	2	yes	—
177	C₂×#3239	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	#137	1	4	yes	—
185	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
186	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	4	no	—
188	Q₈×C₃₁	—	4	16	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	Q₈×C₃₁	—	4	16	no	—
191	D₄×C₃₁	—	2	4	no	—
192	D₄×C₃₁	—	4	8	no	—
193	D₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	8	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	8	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	4	16	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
220	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
221	C₈×C₃₁	—	2	8	no	—
222	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
224	#3239	C₂×C₃₁	1	16	yes	direct product factor
225	C₂×#501	C₂×C₃₁	1	2	yes	complemented
226	C₂×#489	C₂×C₃₁	1	2	yes	complemented
227	C₂×#484	C₂×C₃₁	1	2	yes	complemented
228	C₂×#481	C₂×C₃₁	1	2	yes	—
229	C₂×#475	C₂×C₃₁	1	2	yes	complemented
230	C₂×#474	C₂×C₃₁	1	2	yes	complemented
231	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	8	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	8	4	no	—
239	C₂²×C₃₁	#536	1	2	yes	—
240	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₂²×C₃₁	C₂×#137	1	4	yes	—
242	C₂²×C₃₁	C₂×#137	1	2	yes	—
243	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	8	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	#536	1	2	yes	—
253	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
254	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
255	C₂³×Q₈	—	2	1	no	—
256	C₂×C₄⋊Q₈	—	2	1	no	—
257	C₂×C₄⋊Q₈	—	2	1	no	—
258	C₂×C₄⋊Q₈	—	2	4	no	—
259	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
260	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
262	#529	—	2	4	no	—
263	#529	C₂²×C₃₁	1	8	yes	—
264	C₂×#126	—	2	1	no	—
265	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#123	—	2	1	no	—
268	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#117	—	2	1	no	—
274	C₂²×C₄⋊C₄	—	2	1	no	—
275	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
276	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
277	#501	C₂²×C₃₁	1	16	yes	complemented
278	#489	C₂²×C₃₁	1	16	yes	complemented
279	#484	C₂²×C₃₁	1	16	yes	complemented
280	#481	C₂²×C₃₁	1	16	yes	—
281	#475	C₂²×C₃₁	1	16	yes	complemented
282	#474	C₂²×C₃₁	1	16	yes	complemented
283	#432	C₂²×C₃₁	1	8	yes	complemented
284	C₂×#107	C₂²×C₃₁	1	4	yes	—
285	C₂×#106	C₂²×C₃₁	1	4	yes	—
286	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
287	C₂×#103	C₂²×C₃₁	1	4	yes	—
288	C₂×#102	C₂²×C₃₁	1	4	yes	—
289	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
290	C₂×#97	C₂²×C₃₁	1	2	yes	—
291	C₂×C₃₁	—	2	4	no	—
292	C₂×C₃₁	—	8	2	no	—
293	C₂×C₃₁	#3239	1	4	yes	direct product factor
294	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
295	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
297	C₂²×Q₈	—	2	2	no	—
298	C₂²×Q₈	—	2	2	no	—
299	C₂²×Q₈	—	2	8	no	—
300	C₂²×Q₈	—	4	8	no	—
301	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂³×C₄	—	2	2	no	—
305	C₂³×C₄	—	2	1	no	—
306	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
310	C₄⋊Q₈	—	2	4	no	—
311	C₄⋊Q₈	—	2	4	no	—
312	C₄⋊Q₈	—	2	8	no	—
313	C₄⋊Q₈	—	2	8	no	—
314	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
315	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
316	#126	—	2	8	no	—
317	#125	—	2	2	no	—
318	#125	C₂³×C₃₁	1	4	yes	—
319	#124	—	2	2	no	—
320	#124	C₂³×C₃₁	1	4	yes	—
321	#123	—	2	8	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
324	#121	—	2	2	no	—
325	#121	C₂³×C₃₁	1	4	yes	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	C₂³×C₃₁	1	4	yes	—
331	#117	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	8	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₃₁	1	4	yes	—
354	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
357	#107	C₂³×C₃₁	1	16	yes	—
358	#106	C₂³×C₃₁	1	16	yes	—
359	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
360	#103	C₂³×C₃₁	1	16	yes	—
361	#102	C₂³×C₃₁	1	16	yes	—
362	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
363	#97	C₂³×C₃₁	1	8	yes	—
364	C₃₁	C₂×#3239	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	8	no	—
369	C₂×Q₈	—	2	8	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	4	8	no	—
372	C₂×Q₈	—	4	32	no	—
373	C₂×Q₈	—	4	8	no	—
374	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	4	8	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂×C₈	—	2	8	no	—
402	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
414	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	4	no	—
421	C₄²	—	2	4	no	—
422	C₄²	C₂×D₄×C₃₁	1	4	yes	—
423	C₄²	C₂×D₄×C₃₁	1	2	yes	—
424	C₄²	C₂×D₄×C₃₁	1	2	yes	—
425	C₄²	C₂×D₄×C₃₁	1	4	yes	—
426	C₄²	C₂⁴×C₃₁	1	2	yes	—
427	C₄²	C₂⁴×C₃₁	1	2	yes	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	C₃₁×#137	1	4	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
436	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	Q₈	—	2	4	no	—
438	Q₈	—	4	16	no	—
439	Q₈	—	4	2	no	—
440	Q₈	—	4	16	no	—
441	D₄	—	2	4	no	—
442	D₄	—	4	8	no	—
443	D₄	—	4	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	4	16	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	8	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
470	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
471	C₈	—	2	8	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	8	4	no	—
479	C₂²	C₃₁×#536	1	2	yes	—
480	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
481	C₂²	C₂×C₃₁×#137	1	4	yes	—
482	C₂²	C₂×C₃₁×#137	1	2	yes	—
483	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	8	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	C₃₁×#536	1	2	yes	—
493	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
494	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
495	C₂	—	2	4	no	—
496	C₂	—	8	2	no	—
497	C₂	C₃₁×#3239	1	4	yes	direct product factor
498	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
499	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
500	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#3239	C₂	direct (N × H)
4. C₂×C₃₁×#501	C₂	semidirect
5. C₂×C₃₁×#489	C₂	semidirect
6. C₂×C₃₁×#484	C₂	semidirect
8. C₂×C₃₁×#475	C₂	semidirect
9. C₂×C₃₁×#474	C₂	semidirect
10. C₂×C₃₁×#432	C₂	semidirect
17. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
33. C₃₁×#501	C₂²	semidirect
34. C₃₁×#489	C₂²	semidirect
35. C₃₁×#484	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
38. C₃₁×#474	C₂²	semidirect
39. C₃₁×#432	C₂²	semidirect
177. C₂×#3239	C₃₁	direct (N × H)
222. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
224. #3239	C₂×C₃₁	direct (N × H)
225. C₂×#501	C₂×C₃₁	semidirect
226. C₂×#489	C₂×C₃₁	semidirect
227. C₂×#484	C₂×C₃₁	semidirect
229. C₂×#475	C₂×C₃₁	semidirect
230. C₂×#474	C₂×C₃₁	semidirect
231. C₂×#432	C₂×C₃₁	semidirect
261. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
277. #501	C₂²×C₃₁	semidirect
278. #489	C₂²×C₃₁	semidirect
279. #484	C₂²×C₃₁	semidirect
281. #475	C₂²×C₃₁	semidirect
282. #474	C₂²×C₃₁	semidirect
283. #432	C₂²×C₃₁	semidirect
293. C₂×C₃₁	#3239	direct (N × H)
364. C₃₁	C₂×#3239	direct (N × H)
497. C₂	C₃₁×#3239	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₂×C₃₁×#529	2
C₃₁×#3239	2
C₂×C₃₁×#501	2
C₂×C₃₁×#489	2
C₂×C₃₁×#484	2
C₂×C₃₁×#481	2
C₂×C₃₁×#475	2
C₂×C₃₁×#474	2
C₂×C₃₁×#432	2
C₂×#3239	31

Subgroups — C2×C31×#3239 #21517609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3239 #21517609