Subgroups — C₂×C₃₁×#3076 #21517238

Order 7936.

2162 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3076	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#479	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#427	C₂	1	2	yes	maximal, complemented
11	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#416	C₂	1	2	yes	maximal
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#533	—	2	4	no	—
16	C₃₁×#533	C₂²	1	8	yes	complemented
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#125	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
22	C₄×#41×C₃₁	—	2	4	no	—
23	C₄×#41×C₃₁	C₂²	1	8	yes	—
24	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
25	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₃₁	—	2	1	no	—
29	C₂×C₄×D₄×C₃₁	—	2	1	no	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#117	—	2	1	no	—
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	—	2	1	no	—
36	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#479	C₂²	1	16	yes	complemented
38	C₃₁×#478	C₂²	1	16	yes	complemented
39	C₃₁×#477	C₂²	1	8	yes	complemented
40	C₃₁×#476	C₂²	1	8	yes	complemented
41	C₃₁×#475	C₂²	1	8	yes	complemented
42	C₃₁×#474	C₂²	1	8	yes	complemented
43	C₃₁×#427	C₂²	1	16	yes	complemented
44	C₂×C₃₁×#107	C₂²	1	2	yes	—
45	C₂×C₃₁×#107	C₂²	1	2	yes	—
46	C₂×C₃₁×#106	C₂²	1	2	yes	—
47	C₂×C₃₁×#106	C₂²	1	2	yes	—
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	complemented
52	C₃₁×#416	C₂²	1	16	yes	—
53	C₂×C₃₁×#103	C₂²	1	2	yes	—
54	C₂×C₃₁×#103	C₂²	1	2	yes	—
55	C₂×C₃₁×#102	C₂²	1	2	yes	—
56	C₂×C₃₁×#102	C₂²	1	2	yes	—
57	C₂×#41×C₃₁	—	2	2	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	Q₈	1	8	yes	complemented
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂²×Q₈×C₃₁	—	2	1	no	—
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	—	2	1	no	—
67	C₂³×C₄×C₃₁	—	2	1	no	—
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	—	2	8	no	—
73	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
76	C₃₁×C₄⋊Q₈	—	2	2	no	—
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
80	C₃₁×#125	—	2	4	no	—
81	C₃₁×#125	C₂³	1	8	yes	—
82	C₃₁×C₂²⋊Q₈	—	2	8	no	—
83	C₃₁×C₂²⋊Q₈	—	2	8	no	—
84	C₄×Q₈×C₃₁	—	2	8	no	—
85	C₄×Q₈×C₃₁	—	2	8	no	—
86	C₄×Q₈×C₃₁	—	2	2	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
89	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₄×D₄×C₃₁	—	2	8	no	—
91	C₄×D₄×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	C₂³	1	4	yes	—
95	C₄×D₄×C₃₁	C₂³	1	4	yes	—
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	—	2	4	no	—
98	C₃₁×#117	—	2	8	no	—
99	C₃₁×#117	C₂³	1	8	yes	—
100	C₃₁×#117	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄²×C₃₁	—	2	1	no	—
116	C₂×C₄²×C₃₁	—	2	1	no	—
117	C₂×C₄²×C₃₁	—	2	1	no	—
118	C₂×C₄²×C₃₁	—	2	2	no	—
119	C₂×C₄²×C₃₁	—	2	8	no	—
120	C₂×C₄²×C₃₁	D₄	1	4	yes	—
121	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₃₁×#107	C₂³	1	8	yes	—
127	C₃₁×#107	C₂³	1	8	yes	—
128	C₃₁×#106	C₂³	1	8	yes	—
129	C₃₁×#106	C₂³	1	8	yes	—
130	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
131	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
132	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×#103	C₂³	1	8	yes	—
135	C₃₁×#102	C₂³	1	8	yes	—
136	C₃₁×#102	C₂³	1	8	yes	—
137	C₂⁴×C₃₁	—	2	1	no	—
138	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
139	#41×C₃₁	—	2	16	no	—
140	#41×C₃₁	—	2	8	no	—
141	#41×C₃₁	C₂×Q₈	1	16	yes	complemented
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	4	no	—
144	C₂×Q₈×C₃₁	—	2	4	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	4	4	no	—
147	C₂×Q₈×C₃₁	—	4	4	no	—
148	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
149	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
150	C₂×D₄×C₃₁	—	2	8	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	1	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
155	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	2	4	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	1	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	4	2	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	4	4	no	—
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
178	C₂²×C₄×C₃₁	#41	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂×C₈×C₃₁	—	2	4	no	—
182	C₂×C₈×C₃₁	—	2	4	no	—
183	C₂×C₈×C₃₁	—	2	16	no	—
184	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
185	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
186	C₄⋊C₄×C₃₁	—	2	4	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	8	no	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	2	4	no	—
204	C₄²×C₃₁	—	2	4	no	—
205	C₄²×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	8	no	—
208	C₄²×C₃₁	C₂×D₄	1	4	yes	—
209	C₄²×C₃₁	C₂×D₄	1	4	yes	—
210	C₄²×C₃₁	C₂×D₄	1	2	yes	—
211	C₄²×C₃₁	C₂×D₄	1	2	yes	—
212	C₄²×C₃₁	C₂⁴	1	2	yes	—
213	C₄²×C₃₁	C₂⁴	1	2	yes	—
214	C₂×#3076	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	4	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	—	4	2	no	—
220	C₂³×C₃₁	—	4	2	no	—
221	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
222	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₃₁	—	2	4	no	—
225	Q₈×C₃₁	—	2	2	no	—
226	Q₈×C₃₁	—	4	8	no	—
227	Q₈×C₃₁	—	4	8	no	—
228	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
229	D₄×C₃₁	—	2	16	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	2	2	no	—
232	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	16	no	—
239	C₂×C₄×C₃₁	—	2	8	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	4	no	—
248	C₂×C₄×C₃₁	—	4	2	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	8	no	—
251	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
252	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
253	C₂×C₄×C₃₁	#135	1	4	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
261	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
262	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
263	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
264	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
265	C₈×C₃₁	—	2	4	no	—
266	C₈×C₃₁	—	2	4	no	—
267	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
268	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
269	#3076	C₂×C₃₁	1	16	yes	direct product factor
270	C₂×#479	C₂×C₃₁	1	2	yes	complemented
271	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
273	C₂×#478	C₂×C₃₁	1	2	yes	complemented
274	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂×#474	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂×#427	C₂×C₃₁	1	2	yes	complemented
277	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₂×#416	C₂×C₃₁	1	2	yes	—
279	C₂²×C₃₁	—	2	8	no	—
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	2	4	no	—
283	C₂²×C₃₁	—	4	8	no	—
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
287	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
288	C₂²×C₃₁	C₂×#135	1	2	yes	—
289	C₂²×C₃₁	#575	1	2	yes	—
290	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	—	4	4	no	—
297	C₄×C₃₁	—	4	2	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	C₂×C₂²⋊Q₈	1	4	yes	—
300	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
301	C₄×C₃₁	C₂×#135	1	4	yes	—
302	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
303	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
304	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
305	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
306	#533	—	2	4	no	—
307	#533	C₂²×C₃₁	1	8	yes	complemented
308	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×#125	C₂²×C₃₁	1	2	yes	—
311	C₂×C₂²⋊Q₈	—	2	1	no	—
312	C₂×C₂²⋊Q₈	—	2	1	no	—
313	C₄×#41	—	2	4	no	—
314	C₄×#41	C₂²×C₃₁	1	8	yes	—
315	C₂×C₄×Q₈	—	2	1	no	—
316	C₂×C₄×Q₈	—	2	1	no	—
317	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×D₄	—	2	1	no	—
320	C₂×C₄×D₄	—	2	1	no	—
321	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#117	—	2	1	no	—
324	C₂×#117	C₂²×C₃₁	1	2	yes	—
325	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
326	C₂²×C₄²	—	2	1	no	—
327	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
328	#479	C₂²×C₃₁	1	16	yes	complemented
329	#478	C₂²×C₃₁	1	16	yes	complemented
330	#477	C₂²×C₃₁	1	8	yes	complemented
331	#476	C₂²×C₃₁	1	8	yes	complemented
332	#475	C₂²×C₃₁	1	8	yes	complemented
333	#474	C₂²×C₃₁	1	8	yes	complemented
334	#427	C₂²×C₃₁	1	16	yes	complemented
335	C₂×#107	C₂²×C₃₁	1	2	yes	—
336	C₂×#107	C₂²×C₃₁	1	2	yes	—
337	C₂×#106	C₂²×C₃₁	1	2	yes	—
338	C₂×#106	C₂²×C₃₁	1	2	yes	—
339	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	complemented
343	#416	C₂²×C₃₁	1	16	yes	—
344	C₂×#103	C₂²×C₃₁	1	2	yes	—
345	C₂×#103	C₂²×C₃₁	1	2	yes	—
346	C₂×#102	C₂²×C₃₁	1	2	yes	—
347	C₂×#102	C₂²×C₃₁	1	2	yes	—
348	C₂×C₃₁	—	2	4	no	—
349	C₂×C₃₁	—	4	4	no	—
350	C₂×C₃₁	#3076	1	4	yes	direct product factor
351	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
352	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
353	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	2	4	no	—
356	C₂×#41	Q₈×C₃₁	1	8	yes	complemented
357	C₂×#41	C₂³×C₃₁	1	4	yes	—
358	C₂²×Q₈	—	2	1	no	—
359	C₂²×Q₈	—	2	1	no	—
360	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
367	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×C₈	—	2	8	no	—
370	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
371	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
372	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
373	C₄⋊Q₈	—	2	2	no	—
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
376	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
377	#125	—	2	4	no	—
378	#125	C₂³×C₃₁	1	8	yes	—
379	C₂²⋊Q₈	—	2	8	no	—
380	C₂²⋊Q₈	—	2	8	no	—
381	C₄×Q₈	—	2	8	no	—
382	C₄×Q₈	—	2	8	no	—
383	C₄×Q₈	—	2	2	no	—
384	C₄×Q₈	—	2	2	no	—
385	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
386	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	8	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	C₂³×C₃₁	1	4	yes	—
392	C₄×D₄	C₂³×C₃₁	1	4	yes	—
393	#117	—	2	2	no	—
394	#117	—	2	4	no	—
395	#117	—	2	8	no	—
396	#117	C₂³×C₃₁	1	4	yes	—
397	#117	C₂³×C₃₁	1	8	yes	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	2	no	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	—	2	8	no	—
417	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₃₁	1	4	yes	—
420	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
423	#107	C₂³×C₃₁	1	8	yes	—
424	#107	C₂³×C₃₁	1	8	yes	—
425	#106	C₂³×C₃₁	1	8	yes	—
426	#106	C₂³×C₃₁	1	8	yes	—
427	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
428	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
429	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
430	#103	C₂³×C₃₁	1	8	yes	—
431	#103	C₂³×C₃₁	1	8	yes	—
432	#102	C₂³×C₃₁	1	8	yes	—
433	#102	C₂³×C₃₁	1	8	yes	—
434	C₃₁	C₂×#3076	1	1	yes	characteristic, direct product factor, Sylow
435	C₂⁴	—	2	1	no	—
436	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
437	#41	—	2	16	no	—
438	#41	—	2	8	no	—
439	#41	C₂×Q₈×C₃₁	1	16	yes	complemented
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	4	no	—
442	C₂×Q₈	—	2	4	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	4	4	no	—
445	C₂×Q₈	—	4	4	no	—
446	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
447	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
453	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	8	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
476	C₂²×C₄	#41×C₃₁	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
479	C₂×C₈	—	2	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₂×C₈	—	2	16	no	—
482	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
483	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	8	no	—
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
495	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
497	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₄²	—	2	4	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	8	no	—
506	C₄²	C₂×D₄×C₃₁	1	4	yes	—
507	C₄²	C₂×D₄×C₃₁	1	4	yes	—
508	C₄²	C₂×D₄×C₃₁	1	2	yes	—
509	C₄²	C₂×D₄×C₃₁	1	2	yes	—
510	C₄²	C₂⁴×C₃₁	1	2	yes	—
511	C₄²	C₂⁴×C₃₁	1	2	yes	—
512	C₂³	—	2	2	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	4	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
519	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
520	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	Q₈	—	2	4	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	4	8	no	—
524	Q₈	—	4	8	no	—
525	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
526	D₄	—	2	16	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	2	no	—
529	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	16	no	—
536	C₂×C₄	—	2	8	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	8	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	8	no	—
548	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
549	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
550	C₂×C₄	C₃₁×#135	1	4	yes	—
551	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
558	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
559	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
560	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
561	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
562	C₈	—	2	4	no	—
563	C₈	—	2	4	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	4	8	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	2	no	—
571	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
572	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
573	C₂²	C₂×C₃₁×#135	1	2	yes	—
574	C₂²	C₃₁×#575	1	2	yes	—
575	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
576	C₄	—	2	4	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	4	no	—
584	C₄	C₂×C₃₁×C₂²⋊Q₈	1	4	yes	—
585	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
586	C₄	C₂×C₃₁×#135	1	4	yes	—
587	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
588	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
589	C₂	—	2	4	no	—
590	C₂	—	4	4	no	—
591	C₂	C₃₁×#3076	1	4	yes	direct product factor
592	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
593	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
594	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#533	C₂	semidirect
3. C₃₁×#3076	C₂	direct (N × H)
4. C₂×C₃₁×#479	C₂	semidirect
5. C₂×C₃₁×#477	C₂	semidirect
6. C₂×C₃₁×#475	C₂	semidirect
7. C₂×C₃₁×#478	C₂	semidirect
8. C₂×C₃₁×#476	C₂	semidirect
9. C₂×C₃₁×#474	C₂	semidirect
10. C₂×C₃₁×#427	C₂	semidirect
11. C₂²×C₃₁×C₄⋊C₈	C₂	semidirect
16. C₃₁×#533	C₂²	semidirect
37. C₃₁×#479	C₂²	semidirect
38. C₃₁×#478	C₂²	semidirect
39. C₃₁×#477	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
43. C₃₁×#427	C₂²	semidirect
51. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
59. C₂×#41×C₃₁	Q₈	semidirect
141. #41×C₃₁	C₂×Q₈	semidirect
214. C₂×#3076	C₃₁	direct (N × H)
267. C₂×#533	C₂×C₃₁	semidirect
269. #3076	C₂×C₃₁	direct (N × H)
270. C₂×#479	C₂×C₃₁	semidirect
271. C₂×#477	C₂×C₃₁	semidirect
272. C₂×#475	C₂×C₃₁	semidirect
273. C₂×#478	C₂×C₃₁	semidirect
274. C₂×#476	C₂×C₃₁	semidirect
275. C₂×#474	C₂×C₃₁	semidirect
276. C₂×#427	C₂×C₃₁	semidirect
277. C₂²×C₄⋊C₈	C₂×C₃₁	semidirect
307. #533	C₂²×C₃₁	semidirect
328. #479	C₂²×C₃₁	semidirect
329. #478	C₂²×C₃₁	semidirect
330. #477	C₂²×C₃₁	semidirect
331. #476	C₂²×C₃₁	semidirect
332. #475	C₂²×C₃₁	semidirect
333. #474	C₂²×C₃₁	semidirect
334. #427	C₂²×C₃₁	semidirect
342. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
350. C₂×C₃₁	#3076	direct (N × H)
356. C₂×#41	Q₈×C₃₁	semidirect
434. C₃₁	C₂×#3076	direct (N × H)
439. #41	C₂×Q₈×C₃₁	semidirect
591. C₂	C₃₁×#3076	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×#533	2
C₂×C₄×#41×C₃₁	2
C₃₁×#3076	2
C₂×C₃₁×#479	2
C₂×C₃₁×#477	2
C₂×C₃₁×#475	2
C₂×C₃₁×#478	2
C₂×C₃₁×#476	2
C₂×C₃₁×#474	2
C₂×C₃₁×#427	2
C₂²×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#416	2
C₂×#3076	31

Subgroups — C2×C31×#3076 #21517238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3076 #21517238