Subgroups — #16854740

Order 3968.

456 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521121	C₂	1	8	yes	maximal, complemented
2	#14521106	C₂	1	2	yes	maximal, complemented
3	#14521013	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₃₁	C₄	1	4	yes	—
10	C₄×D₄×C₃₁	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×#92474	C₂²	1	4	yes	complemented
16	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	complemented
17	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
18	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
19	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
21	#11758391	C₂²	1	4	yes	complemented
22	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₃₁	—	2	4	no	—
25	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	—	4	2	no	—
27	C₂²×C₄×C₃₁	—	4	2	no	—
28	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
37	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
40	C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₄²×C₃₁	C₂³	1	2	yes	—
42	#92474	—	4	4	no	—
43	C₂×C₃₁⋊C₈	—	4	4	no	—
44	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	—	4	1	no	—
47	C₂³×C₃₁	—	4	1	no	—
48	C₂³×C₃₁	—	4	1	no	—
49	C₂³×C₃₁	—	4	1	no	—
50	C₂³×C₃₁	M₁₆	1	4	yes	complemented
51	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	D₄×C₃₁	—	2	8	no	—
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	—	2	4	no	—
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	4	2	no	—
60	C₂×C₄×C₃₁	—	4	2	no	—
61	C₂×C₄×C₃₁	—	4	1	no	—
62	C₂×C₄×C₃₁	—	4	1	no	—
63	C₂×C₄×C₃₁	—	4	1	no	—
64	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
70	C₃₁⋊C₈	—	2	8	no	—
71	#2973	—	31	1	no	maximal, Sylow
72	C₂²×C₃₁	—	2	1	no	—
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	2	1	no	—
75	C₂²×C₃₁	—	2	2	no	—
76	C₂²×C₃₁	—	2	2	no	—
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	—	2	2	no	—
79	C₂²×C₃₁	—	4	4	no	—
80	C₂²×C₃₁	—	4	4	no	—
81	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
82	C₂²×C₃₁	C₂×M₁₆	1	4	yes	—
83	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₃₁	—	2	1	no	—
85	C₄×C₃₁	—	2	2	no	—
86	C₄×C₃₁	—	2	2	no	—
87	C₄×C₃₁	—	2	2	no	—
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
90	C₂×C₄×D₄	D₃₁	1	1	yes	characteristic
91	#435	—	31	8	no	—
92	#432	—	31	2	no	—
93	#407	—	31	2	no	—
94	C₂×C₂²⋊C₈	—	31	2	no	—
95	C₂×C₃₁	—	2	2	no	—
96	C₂×C₃₁	—	2	4	no	—
97	C₂×C₃₁	—	4	2	no	—
98	C₂×C₃₁	#518	1	1	yes	characteristic
99	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
100	C₂×C₃₁	#568	1	1	yes	characteristic
101	C₂²×D₄	Dic₃₁	1	1	yes	characteristic
102	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
104	C₂×M₁₆	—	31	4	no	—
105	C₂²×C₈	—	31	4	no	—
106	C₄×D₄	Dic₃₁	1	4	yes	—
107	C₄×D₄	C₂×D₃₁	1	4	yes	—
108	C₂×C₄⋊C₄	Dic₃₁	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	Dic₃₁	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	Dic₃₁	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
112	C₄⋊C₈	—	31	4	no	—
113	C₂²⋊C₈	—	31	4	no	—
114	C₂²⋊C₈	—	31	4	no	—
115	C₂²⋊C₈	—	31	4	no	—
116	#97	—	31	4	no	—
117	C₃₁	#2973	1	1	yes	characteristic, complemented, Sylow
118	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
119	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
125	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
129	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
130	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
132	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
133	M₁₆	—	124	4	no	—
134	C₂×C₈	—	31	8	no	—
135	C₂×C₈	—	124	4	no	—
136	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
137	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
138	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
139	C₄²	C₂×Dic₃₁	1	2	yes	—
140	C₄²	C₂²×D₃₁	1	2	yes	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	#92474	1	4	yes	complemented
147	C₂³	C₂²×Dic₃₁	1	2	yes	—
148	C₂³	C₂²×Dic₃₁	1	2	yes	—
149	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
150	D₄	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
162	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
163	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
164	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
165	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
166	C₈	—	62	8	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂×#92474	1	2	yes	—
177	C₂²	C₂×#92474	1	4	yes	—
178	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	#11758561	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	#14521524	1	1	yes	characteristic
189	C₂	C₂²×#92474	1	1	yes	characteristic
190	C₂	#14521816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₃₁	C₂
prime radical	C₂²×D₄×C₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₃₁
hypocenter	C₃₁	#2973
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14521121	C₂	semidirect
2. #14521106	C₂	semidirect
3. #14521013	C₂	semidirect
4. C₂×C₂²⋊(C₃₁⋊C₈)	C₂	semidirect
15. C₂×#92474	C₂²	semidirect
16. C₂²×C₃₁⋊C₈	C₂²	semidirect
18. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
19. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
20. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
21. #11758391	C₂²	semidirect
50. C₂³×C₃₁	M₁₆	semidirect
117. C₃₁	#2973	semidirect
146. C₂³	#92474	semidirect

Maximal subgroups

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

subgroup	index
#14521121	2
#14521106	2
#14521013	2
C₂×C₂²⋊(C₃₁⋊C₈)	2
C₂×C₄×D₄×C₃₁	2
#2973	31