Subgroups — C₃₁×#13193 #21469952

Order 7936.

446 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#350	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#131	C₂²	1	2	yes	—
6	C₃₁×#532	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#436	—	2	1	no	—
10	C₃₁×#436	—	2	1	no	—
11	C₈×D₄×C₃₁	—	2	1	no	—
12	C₈×D₄×C₃₁	—	2	1	no	—
13	C₃₁×#423	C₄	1	1	yes	characteristic
14	C₃₁×#408	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×#350	—	2	8	no	—
19	C₃₁×#346	—	2	1	no	—
20	C₄×C₁₆×C₃₁	—	2	1	no	—
21	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
22	C₃₁×#131	—	2	4	no	—
23	C₃₁×#131	—	2	4	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₃₁×#127	C₈	1	1	yes	characteristic
31	C₃₁×#124	C₈	1	2	yes	—
32	C₃₁×#121	—	2	1	no	—
33	C₃₁×#121	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	—	2	1	no	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₃₁×M₃₂	—	2	2	no	—
38	C₃₁×M₃₂	D₄	1	4	yes	complemented
39	C₂×C₁₆×C₃₁	—	2	2	no	—
40	C₂×C₁₆×C₃₁	—	2	4	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	#41×C₃₁	—	2	4	no	—
51	#41×C₃₁	—	2	4	no	—
52	C₂×Q₈×C₃₁	C₂×C₈	1	2	yes	—
53	C₂×D₄×C₃₁	—	2	2	no	—
54	C₂×D₄×C₃₁	—	2	1	no	—
55	C₂×D₄×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₃₁	—	2	4	no	—
62	M₁₆×C₃₁	—	4	2	no	—
63	M₁₆×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	4	4	no	—
67	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	#41	1	2	yes	—
73	C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
81	C₁₆×C₃₁	—	2	4	no	—
82	C₁₆×C₃₁	—	4	2	no	—
83	#13193	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
86	Q₈×C₃₁	—	2	4	no	—
87	D₄×C₃₁	—	2	4	no	—
88	D₄×C₃₁	—	4	2	no	—
89	D₄×C₃₁	—	4	4	no	—
90	D₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	1	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
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	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#131	1	2	yes	—
104	C₈×C₃₁	—	2	1	no	—
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	4	2	no	—
107	C₈×C₃₁	#121	1	2	yes	—
108	C₈×C₃₁	#124	1	1	yes	characteristic
109	C₈×C₃₁	#127	1	1	yes	characteristic
110	#2931	C₂×C₃₁	1	1	yes	characteristic
111	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
112	C₂×#350	C₂×C₃₁	1	4	yes	complemented
113	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	—	4	2	no	—
119	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₃₁	#436	1	2	yes	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	#350	1	2	yes	complemented
126	C₄×C₃₁	#390	1	1	yes	characteristic
127	C₄×C₃₁	#408	1	1	yes	characteristic
128	C₄×C₃₁	C₈×D₄	1	2	yes	—
129	C₂×#131	C₂²×C₃₁	1	2	yes	—
130	#532	C₄×C₃₁	1	1	yes	characteristic
131	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
132	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
133	#436	—	2	1	no	—
134	#436	—	2	1	no	—
135	C₈×D₄	—	2	1	no	—
136	C₈×D₄	—	2	1	no	—
137	#423	C₄×C₃₁	1	1	yes	characteristic
138	#408	C₄×C₃₁	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	1	no	—
141	C₄⋊C₁₆	—	2	1	no	—
142	#350	—	2	8	no	—
143	#346	—	2	1	no	—
144	C₄×C₁₆	—	2	1	no	—
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	—	8	2	no	—
147	C₂×C₃₁	#1954	1	1	yes	characteristic
148	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
149	C₂×C₃₁	#2117	1	1	yes	characteristic
150	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
151	#131	—	2	4	no	—
152	#131	—	2	4	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
159	#127	C₈×C₃₁	1	1	yes	characteristic
160	#124	C₈×C₃₁	1	2	yes	—
161	#121	—	2	1	no	—
162	#121	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
166	M₃₂	—	2	2	no	—
167	M₃₂	D₄×C₃₁	1	4	yes	complemented
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	4	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
176	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
177	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
179	C₃₁	#13193	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	2	4	no	—
181	#41	—	2	4	no	—
182	C₂×Q₈	C₂×C₈×C₃₁	1	2	yes	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₈	#41×C₃₁	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
209	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
210	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₁₆	—	4	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
231	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₃₁×#131	1	2	yes	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	C₃₁×#121	1	2	yes	—
237	C₈	C₃₁×#124	1	1	yes	characteristic
238	C₈	C₃₁×#127	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
245	C₂²	C₃₁×#436	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	C₃₁×#350	1	2	yes	complemented
251	C₄	C₃₁×#390	1	1	yes	characteristic
252	C₄	C₃₁×#408	1	1	yes	characteristic
253	C₄	C₈×D₄×C₃₁	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	2	no	—
256	C₂	C₃₁×#1954	1	1	yes	characteristic
257	C₂	C₂×C₃₁×#350	1	1	yes	characteristic
258	C₂	C₃₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₃₁×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₃₁×#350	C₂	semidirect
4. C₄×C₃₁×M₃₂	C₂	semidirect
38. C₃₁×M₃₂	D₄	semidirect
83. #13193	C₃₁	direct (N × H)
111. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
112. C₂×#350	C₂×C₃₁	semidirect
113. C₄×M₃₂	C₂×C₃₁	semidirect
125. C₄×C₃₁	#350	semidirect
167. M₃₂	D₄×C₃₁	semidirect
179. C₃₁	#13193	direct (N × H)
250. C₄	C₃₁×#350	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2931	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#350	2
C₄×C₃₁×M₃₂	2
#13193	31

Subgroups — C31×#13193 #21469952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13193 #21469952