Subgroups — C₂×C₃₁×#1676 #21475876

Order 7936.

638 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#477	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1676	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#333	C₂	1	2	yes	maximal
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#500	—	2	2	no	—
10	C₃₁×#500	C₂²	1	4	yes	—
11	C₃₁×#477	—	2	4	no	—
12	C₃₁×#477	C₂²	1	8	yes	—
13	C₈×Q₈×C₃₁	—	2	2	no	—
14	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
15	C₂×C₃₁×#107	—	2	1	no	—
16	C₂×C₃₁×#107	—	2	1	no	—
17	C₂×C₃₁×#107	—	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×#333	C₂²	1	8	yes	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₃₁×C₄⋊Q₈	—	2	2	no	—
33	C₃₁×C₄⋊Q₈	—	2	2	no	—
34	C₃₁×C₄⋊Q₈	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
36	C₄×Q₈×C₃₁	—	2	1	no	—
37	C₄×Q₈×C₃₁	—	2	1	no	—
38	C₄×Q₈×C₃₁	D₄	1	2	yes	—
39	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#107	—	2	4	no	—
47	C₃₁×#107	—	2	4	no	—
48	C₃₁×#107	—	2	8	no	—
49	C₃₁×C₄⋊C₈	—	2	4	no	—
50	C₃₁×C₄⋊C₈	—	2	1	no	—
51	C₃₁×C₄⋊C₈	—	2	2	no	—
52	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
54	C₃₁×#103	—	2	8	no	—
55	C₄×C₈×C₃₁	—	2	4	no	—
56	C₄×C₈×C₃₁	D₄	1	2	yes	—
57	C₄×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂×Q₈×C₃₁	—	2	1	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	2	4	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	8	4	no	—
63	C₂×Q₈×C₃₁	Q₁₆	1	4	yes	—
64	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	4	2	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	4	4	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₄²×C₃₁	—	2	2	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₃₁	#41	1	1	yes	characteristic
92	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₂×#1676	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	#121	1	1	yes	characteristic
95	Q₈×C₃₁	—	2	2	no	—
96	Q₈×C₃₁	—	2	4	no	—
97	Q₈×C₃₁	—	8	4	no	—
98	Q₈×C₃₁	—	8	4	no	—
99	Q₈×C₃₁	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₃₁	#135	1	2	yes	—
113	C₂×C₄×C₃₁	#137	1	2	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	4	4	no	—
119	C₈×C₃₁	—	4	2	no	—
120	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
121	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#477	C₂×C₃₁	1	2	yes	—
123	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
124	#1676	C₂×C₃₁	1	8	yes	direct product factor
125	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#333	C₂×C₃₁	1	2	yes	—
127	C₂²×C₃₁	#462	1	2	yes	—
128	C₂²×C₃₁	#467	1	2	yes	—
129	C₂²×C₃₁	#472	1	2	yes	—
130	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	—	4	2	no	—
135	C₄×C₃₁	—	8	4	no	—
136	C₄×C₃₁	#462	1	2	yes	—
137	C₄×C₃₁	#467	1	2	yes	—
138	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
141	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
143	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
144	#500	—	2	2	no	—
145	#500	C₂²×C₃₁	1	4	yes	—
146	#477	—	2	4	no	—
147	#477	C₂²×C₃₁	1	8	yes	—
148	C₈×Q₈	—	2	2	no	—
149	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
150	C₂×#107	—	2	1	no	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	2	2	no	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#103	—	2	2	no	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	#335	C₂²×C₃₁	1	4	yes	—
160	#333	C₂²×C₃₁	1	8	yes	—
161	C₂×C₃₁	#1676	1	4	yes	direct product factor
162	C₂×C₃₁	C₂×#462	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
175	C₄×Q₈	—	2	1	no	—
176	C₄×Q₈	—	2	1	no	—
177	C₄×Q₈	D₄×C₃₁	1	2	yes	—
178	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	—	2	1	no	—
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	#107	—	2	4	no	—
186	#107	—	2	4	no	—
187	#107	—	2	8	no	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
192	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
193	#103	—	2	8	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	D₄×C₃₁	1	2	yes	—
196	C₄×C₈	C₂³×C₃₁	1	2	yes	—
197	C₃₁	C₂×#1676	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	8	4	no	—
203	C₂×Q₈	Q₁₆×C₃₁	1	4	yes	—
204	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₄²	#41×C₃₁	1	1	yes	characteristic
232	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₂³	C₃₁×#121	1	1	yes	characteristic
234	Q₈	—	2	4	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	8	4	no	—
237	Q₈	—	8	4	no	—
238	Q₈	C₂×Q₁₆×C₃₁	1	4	yes	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
247	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
248	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
250	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
251	C₂×C₄	C₃₁×#135	1	2	yes	—
252	C₂×C₄	C₃₁×#137	1	2	yes	—
253	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	2	no	—
259	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
260	C₂²	C₃₁×#462	1	2	yes	—
261	C₂²	C₃₁×#467	1	2	yes	—
262	C₂²	C₃₁×#472	1	2	yes	—
263	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	C₃₁×#462	1	2	yes	—
270	C₄	C₃₁×#467	1	2	yes	—
271	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
272	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
273	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
274	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
275	C₂	C₃₁×#1676	1	4	yes	direct product factor
276	C₂	C₂×C₃₁×#462	1	1	yes	characteristic
277	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
278	C₂	C₂×C₃₁×#472	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×#1676	C₂	direct (N × H)
93. C₂×#1676	C₃₁	direct (N × H)
124. #1676	C₂×C₃₁	direct (N × H)
161. C₂×C₃₁	#1676	direct (N × H)
197. C₃₁	C₂×#1676	direct (N × H)
275. C₂	C₃₁×#1676	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×#500	2
C₂×C₃₁×#477	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1676	2
C₂×C₃₁×#335	2
C₂×C₃₁×#333	2
C₂×#1676	31

Subgroups — C2×C31×#1676 #21475876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1676 #21475876