Subgroups — C₂×C₃₁×#1670 #21475872

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#475	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1670	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#332	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#494	—	2	2	no	—
12	C₃₁×#494	C₂²	1	4	yes	—
13	C₃₁×#475	—	2	4	no	—
14	C₃₁×#475	C₂²	1	8	yes	—
15	C₈×Q₈×C₃₁	—	2	2	no	—
16	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
17	C₂×C₃₁×#106	—	2	2	no	—
18	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	—	2	2	no	—
22	C₂×C₄×C₈×C₃₁	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#335	C₂²	1	4	yes	—
25	C₃₁×#332	C₂²	1	8	yes	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₃₁	—	2	4	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	4	4	no	—
31	C₂×Q₁₆×C₃₁	—	4	4	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄×Q₈×C₃₁	—	2	1	no	—
41	C₄×Q₈×C₃₁	—	2	1	no	—
42	C₄×Q₈×C₃₁	D₄	1	2	yes	—
43	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#106	—	2	8	no	—
51	C₃₁×C₄⋊C₈	—	2	4	no	—
52	C₃₁×C₄⋊C₈	—	2	1	no	—
53	C₃₁×C₄⋊C₈	—	2	2	no	—
54	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
56	C₃₁×#103	—	2	8	no	—
57	C₄×C₈×C₃₁	—	2	4	no	—
58	C₄×C₈×C₃₁	D₄	1	2	yes	—
59	C₄×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂×Q₈×C₃₁	—	2	2	no	—
62	C₂×Q₈×C₃₁	—	2	4	no	—
63	C₂×Q₈×C₃₁	—	4	2	no	—
64	C₂×Q₈×C₃₁	—	4	4	no	—
65	C₂×Q₈×C₃₁	—	4	4	no	—
66	C₂×Q₈×C₃₁	Q₁₆	1	4	yes	complemented
67	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	4	8	no	—
75	Q₁₆×C₃₁	—	4	8	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	4	4	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	#41	1	1	yes	characteristic
97	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₂×#1670	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	#121	1	1	yes	characteristic
100	Q₈×C₃₁	—	2	2	no	—
101	Q₈×C₃₁	—	2	4	no	—
102	Q₈×C₃₁	—	4	4	no	—
103	Q₈×C₃₁	—	4	4	no	—
104	Q₈×C₃₁	—	8	4	no	—
105	Q₈×C₃₁	C₂×Q₁₆	1	4	yes	complemented
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	8	4	no	—
113	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₃₁	#135	1	2	yes	—
119	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₈×C₃₁	—	2	2	no	—
124	C₈×C₃₁	—	4	4	no	—
125	C₈×C₃₁	—	4	2	no	—
126	C₈×C₃₁	C₂×SD₁₆	1	4	yes	—
127	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
128	C₂×#475	C₂×C₃₁	1	2	yes	—
129	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
130	#1670	C₂×C₃₁	1	8	yes	direct product factor
131	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
132	C₂×#332	C₂×C₃₁	1	2	yes	—
133	C₂²×C₃₁	#460	1	2	yes	—
134	C₂²×C₃₁	#467	1	2	yes	—
135	C₂²×C₃₁	#473	1	2	yes	—
136	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	8	4	no	—
142	C₄×C₃₁	#460	1	2	yes	—
143	C₄×C₃₁	#467	1	2	yes	—
144	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
145	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
146	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
147	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂²×Q₁₆	—	2	1	no	—
149	C₂²×Q₁₆	—	2	1	no	—
150	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
151	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
152	#494	—	2	2	no	—
153	#494	C₂²×C₃₁	1	4	yes	—
154	#475	—	2	4	no	—
155	#475	C₂²×C₃₁	1	8	yes	—
156	C₈×Q₈	—	2	2	no	—
157	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
158	C₂×#106	—	2	2	no	—
159	C₂×C₄⋊C₈	—	2	1	no	—
160	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#103	—	2	2	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#335	C₂²×C₃₁	1	4	yes	—
166	#332	C₂²×C₃₁	1	8	yes	—
167	C₂×C₃₁	#1670	1	4	yes	direct product factor
168	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
169	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
170	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
171	C₂²×Q₈	—	2	2	no	—
172	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	—	2	4	no	—
175	C₂×Q₁₆	—	4	4	no	—
176	C₂×Q₁₆	—	4	4	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	—	2	1	no	—
187	C₄×Q₈	D₄×C₃₁	1	2	yes	—
188	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	#106	—	2	8	no	—
196	C₄⋊C₈	—	2	4	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
200	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
201	#103	—	2	8	no	—
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	D₄×C₃₁	1	2	yes	—
204	C₄×C₈	C₂³×C₃₁	1	2	yes	—
205	C₃₁	C₂×#1670	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	Q₁₆×C₃₁	1	4	yes	complemented
213	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
220	Q₁₆	—	4	8	no	—
221	Q₁₆	—	4	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
229	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₄²	#41×C₃₁	1	1	yes	characteristic
243	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
244	C₂³	C₃₁×#121	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	2	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	8	4	no	—
250	Q₈	C₂×Q₁₆×C₃₁	1	4	yes	complemented
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
259	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
261	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
263	C₂×C₄	C₃₁×#135	1	2	yes	—
264	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₈	C₂×SD₁₆×C₃₁	1	4	yes	—
272	C₂²	C₃₁×#460	1	2	yes	—
273	C₂²	C₃₁×#467	1	2	yes	—
274	C₂²	C₃₁×#473	1	2	yes	—
275	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	8	4	no	—
281	C₄	C₃₁×#460	1	2	yes	—
282	C₄	C₃₁×#467	1	2	yes	—
283	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
284	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
285	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
286	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₂	C₃₁×#1670	1	4	yes	direct product factor
288	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
289	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
290	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1670	C₂	direct (N × H)
66. C₂×Q₈×C₃₁	Q₁₆	semidirect
98. C₂×#1670	C₃₁	direct (N × H)
105. Q₈×C₃₁	C₂×Q₁₆	semidirect
130. #1670	C₂×C₃₁	direct (N × H)
167. C₂×C₃₁	#1670	direct (N × H)
205. C₃₁	C₂×#1670	direct (N × H)
212. C₂×Q₈	Q₁₆×C₃₁	semidirect
250. Q₈	C₂×Q₁₆×C₃₁	semidirect
287. C₂	C₃₁×#1670	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×#494	2
C₂×C₃₁×#475	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1670	2
C₂×C₃₁×#335	2
C₂×C₃₁×#332	2
C₂×#1670	31

Subgroups — C2×C31×#1670 #21475872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1670 #21475872