Subgroups — C₃₁×#11962 #21468721

Order 7936.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 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	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1362	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₃₁	—	2	1	no	—
6	C₂²×Q₁₆×C₃₁	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
8	C₃₁×#494	—	2	1	no	—
9	C₃₁×#494	—	2	1	no	—
10	C₃₁×#494	C₄	1	2	yes	—
11	C₃₁×#494	C₂²	1	2	yes	—
12	C₃₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₃₁	—	2	1	no	—
17	C₃₁×#335	—	2	1	no	—
18	C₃₁×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₁	—	2	2	no	—
20	C₂×Q₁₆×C₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₃₁	—	2	2	no	—
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	4	4	no	—
24	C₂×Q₁₆×C₃₁	—	4	4	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₃₁	—	4	1	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×C₄⋊Q₈	—	2	2	no	—
31	C₃₁×C₄⋊Q₈	—	4	1	no	—
32	C₃₁×C₄⋊Q₈	—	4	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₃₁	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	4	1	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂×Q₈×C₃₁	—	4	4	no	—
45	C₂×Q₈×C₃₁	—	4	4	no	—
46	C₂×Q₈×C₃₁	—	4	2	no	—
47	C₂²×C₄×C₃₁	—	4	2	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₃₁	—	4	4	no	—
52	Q₁₆×C₃₁	—	4	4	no	—
53	Q₁₆×C₃₁	—	4	4	no	—
54	Q₁₆×C₃₁	—	4	4	no	—
55	M₁₆×C₃₁	—	8	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₃₁	D₈	1	1	yes	characteristic
69	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₁₆×C₃₁	—	4	2	no	—
72	#11962	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	#102	1	1	yes	characteristic
74	Q₈×C₃₁	—	4	4	no	—
75	Q₈×C₃₁	—	4	4	no	—
76	Q₈×C₃₁	—	8	4	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	8	4	no	—
82	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
86	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₃₁	—	2	1	no	—
91	C₈×C₃₁	—	2	1	no	—
92	C₈×C₃₁	—	8	2	no	—
93	C₈×C₃₁	#100	1	2	yes	—
94	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
96	#2162	C₂×C₃₁	1	1	yes	characteristic
97	#1567	C₂×C₃₁	1	1	yes	characteristic
98	#1362	C₂×C₃₁	1	4	yes	complemented
99	C₂²×C₃₁	—	2	1	no	—
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	#358	1	2	yes	—
102	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	8	4	no	—
106	C₄×C₃₁	#358	1	2	yes	—
107	C₄×C₃₁	#362	1	2	yes	—
108	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
109	C₄×C₃₁	#418	1	1	yes	characteristic
110	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
111	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
112	C₂²×Q₁₆	—	2	1	no	—
113	C₂²×Q₁₆	C₄×C₃₁	1	2	yes	—
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	#494	—	2	1	no	—
116	#494	—	2	1	no	—
117	#494	C₄×C₃₁	1	2	yes	—
118	#494	C₂²×C₃₁	1	2	yes	—
119	#423	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
122	C₂²⋊C₁₆	—	2	1	no	—
123	C₄×C₁₆	—	2	1	no	—
124	#335	—	2	1	no	—
125	#335	C₂²×C₃₁	1	2	yes	—
126	C₂×C₃₁	—	2	2	no	—
127	C₂×C₃₁	#1545	1	1	yes	characteristic
128	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
129	C₂×C₃₁	C₂×#362	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	4	4	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
137	C₂×M₁₆	—	4	1	no	—
138	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	4	1	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
151	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
152	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
154	C₃₁	#11962	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	8	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
180	C₄²	D₈×C₃₁	1	1	yes	characteristic
181	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
182	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₁₆	—	4	2	no	—
184	C₂³	C₃₁×#102	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
194	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
195	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
196	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
197	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	8	2	no	—
204	C₈	C₃₁×#100	1	2	yes	—
205	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₃₁×#358	1	2	yes	—
209	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	C₃₁×#358	1	2	yes	—
214	C₄	C₃₁×#362	1	2	yes	—
215	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
216	C₄	C₃₁×#418	1	1	yes	characteristic
217	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
218	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₃₁×#1545	1	1	yes	characteristic
221	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
222	C₂	C₂×C₃₁×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1362	C₂	semidirect
72. #11962	C₃₁	direct (N × H)
98. #1362	C₂×C₃₁	semidirect
154. C₃₁	#11962	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
C₃₁×#2162	2
C₃₁×#1567	2
C₃₁×#1362	2
#11962	31

Subgroups — C31×#11962 #21468721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11962 #21468721