Subgroups — C₃₁×#13265 #21470024

Order 7936.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1379	C₂	1	4	yes	maximal
5	C₃₁×#533	C₄	1	1	yes	characteristic
6	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#446	—	2	1	no	—
9	C₃₁×#446	—	2	1	no	—
10	C₈×Q₈×C₃₁	—	2	1	no	—
11	C₈×Q₈×C₃₁	—	2	1	no	—
12	C₃₁×#424	C₄	1	2	yes	—
13	C₃₁×#423	C₄	1	1	yes	characteristic
14	C₃₁×#405	C₂²	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×#346	—	2	1	no	—
19	C₄×C₁₆×C₃₁	—	2	1	no	—
20	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₃₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₃₁×#125	C₈	1	2	yes	—
26	C₃₁×C₂²⋊Q₈	—	2	1	no	—
27	C₃₁×C₂²⋊Q₈	—	2	1	no	—
28	C₄×Q₈×C₃₁	—	2	1	no	—
29	C₄×Q₈×C₃₁	—	2	1	no	—
30	C₃₁×#117	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₃₁×M₃₂	—	2	2	no	—
33	C₃₁×M₃₂	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₂×C₁₆×C₃₁	—	2	2	no	—
36	C₂×C₁₆×C₃₁	—	2	4	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	1	no	—
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×#97	—	2	2	no	—
42	C₄×C₈×C₃₁	—	2	2	no	—
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₃₁	—	2	1	no	—
48	C₂×Q₈×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₃₁	#41	1	2	yes	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
66	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
73	C₁₆×C₃₁	—	2	4	no	—
74	C₁₆×C₃₁	—	4	2	no	—
75	#13265	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
77	Q₈×C₃₁	—	4	2	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	4	1	no	—
83	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
89	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	#131	1	2	yes	—
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	2	1	no	—
93	C₈×C₃₁	—	4	2	no	—
94	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
95	C₈×C₃₁	#124	1	1	yes	characteristic
96	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
97	#2960	C₂×C₃₁	1	1	yes	characteristic
98	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
99	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
100	#1379	C₂×C₃₁	1	4	yes	—
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₃₁	#446	1	2	yes	—
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	#350	1	2	yes	—
111	C₄×C₃₁	#391	1	1	yes	characteristic
112	C₄×C₃₁	#408	1	1	yes	characteristic
113	C₄×C₃₁	C₈×Q₈	1	2	yes	—
114	#533	C₄×C₃₁	1	1	yes	characteristic
115	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
116	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
117	#446	—	2	1	no	—
118	#446	—	2	1	no	—
119	C₈×Q₈	—	2	1	no	—
120	C₈×Q₈	—	2	1	no	—
121	#424	C₄×C₃₁	1	2	yes	—
122	#423	C₄×C₃₁	1	1	yes	characteristic
123	#405	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₄⋊C₁₆	—	2	1	no	—
126	C₄⋊C₁₆	—	2	1	no	—
127	#346	—	2	1	no	—
128	C₄×C₁₆	—	2	1	no	—
129	C₂×C₃₁	—	2	2	no	—
130	C₂×C₃₁	#1984	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#350	1	1	yes	characteristic
132	C₂×C₃₁	#2117	1	1	yes	characteristic
133	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
134	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
136	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
137	C₄⋊Q₈	C₈×C₃₁	1	1	yes	characteristic
138	#125	C₈×C₃₁	1	2	yes	—
139	C₂²⋊Q₈	—	2	1	no	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	#117	C₂×C₄×C₃₁	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
145	M₃₂	—	2	2	no	—
146	M₃₂	Q₈×C₃₁	1	4	yes	complemented
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
159	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
160	C₃₁	#13265	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
166	M₁₆	—	2	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
174	C₂×C₈	#41×C₃₁	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
180	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
184	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
185	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
186	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
187	C₁₆	—	2	4	no	—
188	C₁₆	—	4	2	no	—
189	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
199	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
202	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₃₁×#131	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
208	C₈	C₃₁×#124	1	1	yes	characteristic
209	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
213	C₂²	C₃₁×#446	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	C₃₁×#350	1	2	yes	—
220	C₄	C₃₁×#391	1	1	yes	characteristic
221	C₄	C₃₁×#408	1	1	yes	characteristic
222	C₄	C₈×Q₈×C₃₁	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₃₁×#1984	1	1	yes	characteristic
225	C₂	C₂×C₃₁×#350	1	1	yes	characteristic
226	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	C₂³×C₃₁	C₂×M₁₆
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
33. C₃₁×M₃₂	Q₈	semidirect
75. #13265	C₃₁	direct (N × H)
146. M₃₂	Q₈×C₃₁	semidirect
160. C₃₁	#13265	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
C₂×C₃₁×C₄⋊C₁₆	2
C₄×C₃₁×M₃₂	2
C₃₁×#1379	2
#13265	31

Subgroups — C31×#13265 #21470024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13265 #21470024