Subgroups — C₃₁×#33476 #21490235

Order 7936.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2183	C₂	1	4	yes	maximal, complemented
4	Q₈×C₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2151	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2108	C₂	1	2	yes	maximal
8	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#533	C₄	1	1	yes	characteristic
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#446	C₄	1	2	yes	—
14	C₃₁×#446	C₄	1	2	yes	—
15	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#424	C₄	1	2	yes	—
18	C₃₁×#423	C₄	1	1	yes	characteristic
19	C₃₁×#405	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₃₁×#346	C₂²	1	2	yes	—
27	C₃₁×#346	C₂²	1	4	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
35	C₃₁×#125	C₈	1	2	yes	—
36	C₃₁×C₂²⋊Q₈	C₈	1	2	yes	—
37	C₃₁×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₃₁×#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₃₁×M₃₂	—	2	2	no	—
43	C₃₁×M₃₂	Q₈	1	4	yes	complemented
44	C₂×C₁₆×C₃₁	—	2	4	no	—
45	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
48	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₃₁×#97	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₃₁	C₂³	1	4	yes	—
59	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
60	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₃₁	#41	1	2	yes	—
72	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
74	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
75	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
76	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
77	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
80	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
85	C₁₆×C₃₁	—	2	2	no	—
86	C₁₆×C₃₁	—	2	2	no	—
87	C₁₆×C₃₁	—	2	2	no	—
88	C₁₆×C₃₁	C₂×Q₈	1	4	yes	—
89	#33476	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
91	Q₈×C₃₁	—	2	4	no	—
92	Q₈×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	4	1	no	—
95	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#131	1	2	yes	—
104	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	4	no	—
108	C₈×C₃₁	C₄×Q₈	1	2	yes	—
109	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
110	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
111	#2960	C₂×C₃₁	1	1	yes	characteristic
112	#2183	C₂×C₃₁	1	2	yes	complemented
113	#2183	C₂×C₃₁	1	4	yes	complemented
114	Q₈×C₁₆	C₂×C₃₁	1	2	yes	complemented
115	#2151	C₂×C₃₁	1	2	yes	—
116	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
117	#2108	C₂×C₃₁	1	2	yes	—
118	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
122	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	4	no	—
128	C₄×C₃₁	C₈×Q₈	1	2	yes	—
129	C₄×C₃₁	#504	1	2	yes	—
130	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
132	#533	C₄×C₃₁	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
136	#446	C₄×C₃₁	1	2	yes	—
137	#446	C₄×C₃₁	1	2	yes	—
138	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
139	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
140	#424	C₄×C₃₁	1	2	yes	—
141	#423	C₄×C₃₁	1	1	yes	characteristic
142	#405	C₂²×C₃₁	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
145	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
146	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
147	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
148	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
149	#346	C₂²×C₃₁	1	2	yes	—
150	#346	C₂²×C₃₁	1	4	yes	—
151	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
152	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
153	C₂×C₃₁	—	2	2	no	—
154	C₂×C₃₁	C₂×C₈×Q₈	1	1	yes	characteristic
155	C₂×C₃₁	C₂×#504	1	1	yes	characteristic
156	C₂×C₃₁	#3407	1	1	yes	characteristic
157	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
161	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
162	#125	C₈×C₃₁	1	2	yes	—
163	C₂²⋊Q₈	C₈×C₃₁	1	2	yes	—
164	C₂²⋊Q₈	C₈×C₃₁	1	2	yes	—
165	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
166	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
167	#117	C₂×C₄×C₃₁	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	Q₈×C₃₁	1	4	yes	complemented
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
173	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
174	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
175	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
176	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
177	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
178	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
179	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
180	#97	C₂×C₄×C₃₁	1	4	yes	—
181	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₃₁	1	4	yes	—
185	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₃₁	#33476	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
188	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
197	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
199	C₂×C₈	#41×C₃₁	1	2	yes	—
200	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
202	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
203	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
204	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
205	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
206	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
207	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
208	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
209	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	2	no	—
216	C₁₆	C₂×Q₈×C₃₁	1	4	yes	—
217	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
223	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
227	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
228	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₃₁×#131	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	C₄×Q₈×C₃₁	1	2	yes	—
236	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
237	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
241	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	C₈×Q₈×C₃₁	1	2	yes	—
248	C₄	C₃₁×#504	1	2	yes	—
249	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
250	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×C₈×Q₈×C₃₁	1	1	yes	characteristic
253	C₂	C₂×C₃₁×#504	1	1	yes	characteristic
254	C₂	C₃₁×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2183	C₂	semidirect
3. C₃₁×#2183	C₂	semidirect
4. Q₈×C₁₆×C₃₁	C₂	semidirect
43. C₃₁×M₃₂	Q₈	semidirect
89. #33476	C₃₁	direct (N × H)
112. #2183	C₂×C₃₁	semidirect
113. #2183	C₂×C₃₁	semidirect
114. Q₈×C₁₆	C₂×C₃₁	semidirect
170. M₃₂	Q₈×C₃₁	semidirect
186. C₃₁	#33476	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2960	2
C₃₁×#2183	2
Q₈×C₁₆×C₃₁	2
C₃₁×#2151	2
C₂×C₃₁×C₄⋊C₁₆	2
C₃₁×#2108	2
C₄×C₃₁×M₃₂	2
#33476	31

Subgroups — C31×#33476 #21490235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33476 #21490235