Subgroups — #16854806

Order 3968.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521180	C₂	1	2	yes	maximal, complemented
2	#14521180	C₂	1	4	yes	maximal, complemented
3	Q₈×C₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	#14521110	C₂	1	4	yes	maximal
5	#14521106	C₂	1	1	yes	maximal, characteristic
6	#14521102	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃₁×#125	C₄	1	2	yes	—
11	C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
18	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
19	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
23	#11758391	C₂²	1	2	yes	—
24	#11758391	C₂²	1	4	yes	—
25	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
28	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
36	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
40	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₄²×C₃₁	C₂³	1	4	yes	—
44	C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
46	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
47	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₃₁	—	2	4	no	—
50	Q₈×C₃₁	—	4	2	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	—	4	1	no	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₃₁⋊C₈	—	2	2	no	—
63	C₃₁⋊C₈	—	2	2	no	—
64	C₃₁⋊C₈	—	2	4	no	—
65	#2985	—	31	1	no	maximal, Sylow
66	C₂²×C₃₁	—	2	1	no	—
67	C₂²×C₃₁	—	2	1	no	—
68	C₂²×C₃₁	—	2	1	no	—
69	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₃₁	—	2	1	no	—
71	C₄×C₃₁	—	2	1	no	—
72	C₄×C₃₁	—	2	4	no	—
73	C₄×C₃₁	—	2	1	no	—
74	C₄×C₃₁	—	2	1	no	—
75	C₄×C₃₁	—	2	4	no	—
76	C₄×C₃₁	#131	1	2	yes	—
77	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
78	#533	D₃₁	1	1	yes	characteristic
79	#446	—	31	2	no	—
80	#446	—	31	4	no	—
81	C₈×Q₈	—	31	2	no	—
82	#433	—	31	4	no	—
83	#432	—	31	1	no	—
84	#431	—	31	1	no	—
85	C₂×C₃₁	—	4	1	no	—
86	C₂×C₃₁	#519	1	1	yes	characteristic
87	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
88	C₂×C₃₁	#568	1	1	yes	characteristic
89	C₄⋊Q₈	Dic₃₁	1	1	yes	characteristic
90	C₄⋊Q₈	Dic₃₁	1	1	yes	characteristic
91	#125	Dic₃₁	1	2	yes	—
92	C₂²⋊Q₈	Dic₃₁	1	2	yes	—
93	C₂²⋊Q₈	Dic₃₁	1	2	yes	—
94	C₄×Q₈	C₂×D₃₁	1	2	yes	—
95	C₄×Q₈	C₂×D₃₁	1	2	yes	—
96	#117	C₂×D₃₁	1	2	yes	—
97	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
98	C₄⋊C₈	—	31	2	no	—
99	C₄⋊C₈	—	31	2	no	—
100	C₄⋊C₈	—	31	4	no	—
101	C₄⋊C₈	—	31	4	no	—
102	C₂²⋊C₈	—	31	2	no	—
103	C₂²⋊C₈	—	31	2	no	—
104	#97	—	31	2	no	—
105	#97	—	31	4	no	—
106	C₄×C₈	—	31	1	no	—
107	C₄×C₈	—	31	1	no	—
108	C₄×C₈	—	31	4	no	—
109	C₃₁	#2985	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
111	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
112	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
115	C₂×C₈	—	31	2	no	—
116	C₂×C₈	—	31	2	no	—
117	C₂×C₈	—	31	4	no	—
118	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
119	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
120	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
121	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
122	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
123	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
124	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
125	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
126	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
127	C₄²	C₂²×D₃₁	1	1	yes	characteristic
128	C₄²	C₂²×D₃₁	1	4	yes	—
129	C₄²	C₂²×D₃₁	1	1	yes	characteristic
130	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
138	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
141	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
142	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
144	C₈	—	62	2	no	—
145	C₈	—	62	2	no	—
146	C₈	—	62	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	#11758525	1	2	yes	—
158	C₄	#11758564	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#14521527	1	1	yes	characteristic
161	C₂	C₂×#11758525	1	1	yes	characteristic
162	C₂	#14521816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#533	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#533	D₃₁
hypocenter	C₃₁	#2985
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14521180	C₂	semidirect
2. #14521180	C₂	semidirect
3. Q₈×C₃₁⋊C₈	C₂	semidirect
109. C₃₁	#2985	semidirect

Maximal subgroups

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

subgroup	index
#14521180	2
Q₈×C₃₁⋊C₈	2
#14521110	2
#14521106	2
#14521102	2
C₃₁×#533	2
#2985	31