Subgroups — #16854660

Order 3968.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 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	#14521076	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#14521005	C₂	1	2	yes	maximal
7	C₄×#92474	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
9	C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₃₁×#125	C₄	1	2	yes	—
12	C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₃₁×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#117	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×#92474	C₂²	1	1	yes	characteristic
19	C₂×#92474	C₂²	1	1	yes	characteristic
20	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
23	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
24	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
26	#11758391	C₂²	1	2	yes	—
27	#11758391	C₂²	1	4	yes	—
28	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
29	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
30	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₄²×C₃₁	C₂³	1	4	yes	—
46	C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	#92474	—	2	2	no	—
48	#92474	Q₈	1	4	yes	complemented
49	C₂×C₃₁⋊C₈	—	2	4	no	—
50	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
51	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
52	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
53	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₃₁	—	2	4	no	—
56	Q₈×C₃₁	—	4	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	4	1	no	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₃₁	#41	1	2	yes	—
68	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₃₁⋊C₈	—	2	2	no	—
70	C₃₁⋊C₈	—	2	2	no	—
71	C₃₁⋊C₈	—	2	2	no	—
72	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
73	#2962	—	31	1	no	maximal, Sylow
74	C₂²×C₃₁	—	2	1	no	—
75	C₂²×C₃₁	—	2	1	no	—
76	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
77	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₃₁	—	2	1	no	—
79	C₄×C₃₁	—	2	4	no	—
80	C₄×C₃₁	—	2	1	no	—
81	C₄×C₃₁	—	2	1	no	—
82	C₄×C₃₁	—	2	4	no	—
83	C₄×C₃₁	C₄×Q₈	1	2	yes	—
84	C₄×C₃₁	#131	1	2	yes	—
85	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
87	#533	D₃₁	1	1	yes	characteristic
88	#446	—	31	2	no	—
89	#446	—	31	4	no	—
90	C₈×Q₈	—	31	2	no	—
91	#424	—	31	2	no	—
92	C₂×C₄⋊C₈	—	31	1	no	—
93	#405	—	31	2	no	—
94	C₄×M₁₆	—	31	1	no	—
95	C₂×C₃₁	—	2	2	no	—
96	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
98	C₂×C₃₁	#568	1	1	yes	characteristic
99	C₂×M₁₆	—	31	1	no	—
100	C₂×M₁₆	—	31	1	no	—
101	C₂²×C₈	—	31	2	no	—
102	C₄⋊Q₈	Dic₃₁	1	1	yes	characteristic
103	C₄⋊Q₈	Dic₃₁	1	1	yes	characteristic
104	#125	Dic₃₁	1	2	yes	—
105	C₂²⋊Q₈	Dic₃₁	1	2	yes	—
106	C₂²⋊Q₈	Dic₃₁	1	2	yes	—
107	C₄×Q₈	C₂×D₃₁	1	2	yes	—
108	C₄×Q₈	C₂×D₃₁	1	2	yes	—
109	#117	C₂×D₃₁	1	2	yes	—
110	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
111	C₄⋊C₈	—	31	1	no	—
112	C₄⋊C₈	—	31	1	no	—
113	C₄⋊C₈	—	31	2	no	—
114	C₄⋊C₈	—	31	4	no	—
115	C₄⋊C₈	—	31	4	no	—
116	#97	—	31	2	no	—
117	#97	—	31	4	no	—
118	C₄×C₈	—	31	2	no	—
119	C₄×C₈	—	31	4	no	—
120	C₃₁	#2962	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
122	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
123	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	M₁₆	—	31	4	no	—
127	M₁₆	—	62	2	no	—
128	C₂×C₈	—	31	2	no	—
129	C₂×C₈	—	31	2	no	—
130	C₂×C₈	—	31	2	no	—
131	C₂×C₈	—	31	2	no	—
132	C₂×C₈	—	62	4	no	—
133	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
135	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
136	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
137	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
139	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
140	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
141	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
142	C₄²	C₂²×D₃₁	1	1	yes	characteristic
143	C₄²	C₂²×D₃₁	1	4	yes	—
144	C₄²	C₂²×D₃₁	1	1	yes	characteristic
145	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	Q₈×D₃₁	1	2	yes	—
151	C₂×C₄	#92499	1	2	yes	—
152	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
155	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
156	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
157	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
160	C₈	—	31	4	no	—
161	C₈	—	62	2	no	—
162	C₈	—	62	2	no	—
163	C₈	—	62	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×Dic₃₁	1	2	yes	—
167	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	Q₈×Dic₃₁	1	2	yes	—
174	C₄	#11758525	1	2	yes	—
175	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
176	C₄	C₂×#92499	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
179	C₂	C₂×#11758525	1	1	yes	characteristic
180	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	#2962
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
48. #92474	Q₈	semidirect
120. C₃₁	#2962	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
#14521076	2
C₂×C₄⋊(C₃₁⋊C₈)	2
#14521005	2
C₄×#92474	2
C₃₁×#533	2
#2962	31