Finite Group Databasebeta

Subgroups — #16845246

Order 3968.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521397	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521244	C₂	1	1	yes	maximal, characteristic
3	#14521127	C₂	1	1	yes	maximal, characteristic, complemented
4	#14520801	C₂	1	1	yes	maximal, characteristic, complemented
5	#14520798	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×#92479	—	2	1	no	—
11	C₂×#92474	—	2	1	no	—
12	#11758451	C₂²	1	1	yes	characteristic
13	C₃₁×#106	—	2	1	no	—
14	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃₁×#102	—	2	1	no	—
16	C₂²⋊(C₃₁⋊C₈)	—	2	1	no	—
17	#11758397	—	2	1	no	—
18	#11758397	—	2	1	no	—
19	#11758397	—	2	1	no	—
20	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	—	2	1	no	—
26	C₂×D₁₂₄	—	2	1	no	—
27	C₂×D₁₂₄	—	2	1	no	—
28	C₂×D₁₂₄	—	2	1	no	—
29	C₂×C₈×C₃₁	—	2	1	no	—
30	C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
33	C₂²⋊C₄×C₃₁	—	2	1	no	—
34	C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	#92479	—	4	2	no	—
36	#92474	—	4	2	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₃₁	—	2	1	no	—
42	C₂²×D₃₁	—	4	1	no	—
43	Q₈×C₃₁	—	8	1	no	—
44	D₄×C₃₁	—	2	1	no	—
45	D₄×C₃₁	—	2	1	no	—
46	C₂×C₄×C₃₁	—	2	1	no	—
47	C₂×C₄×C₃₁	—	4	1	no	—
48	C₂×C₄×C₃₁	—	4	1	no	—
49	C₂×C₄×C₃₁	—	4	1	no	—
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
53	D₁₂₄	—	4	1	no	—
54	D₁₂₄	—	4	1	no	—
55	D₁₂₄	—	4	1	no	—
56	D₁₂₄	—	4	1	no	—
57	D₁₂₄	—	8	1	no	—
58	C₈×C₃₁	—	4	1	no	—
59	C₃₁⋊C₈	—	2	1	no	—
60	C₃₁⋊C₈	—	2	1	no	—
61	C₃₁⋊C₈	—	4	1	no	—
62	C₃₁⋊C₈	—	4	1	no	—
63	#1697	—	31	1	no	maximal, Sylow
64	C₂²×C₃₁	—	4	1	no	—
65	C₂²×C₃₁	—	4	1	no	—
66	C₂²×C₃₁	—	4	1	no	—
67	C₂²×C₃₁	#121	1	1	yes	characteristic
68	C₂×D₃₁	—	8	1	no	—
69	C₂×D₃₁	—	8	1	no	—
70	C₂×D₃₁	—	8	1	no	—
71	C₄×C₃₁	—	2	1	no	—
72	C₄×C₃₁	—	4	1	no	—
73	C₄×C₃₁	—	8	1	no	—
74	C₄×C₃₁	#135	1	1	yes	characteristic
75	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₃₁	#137	1	1	yes	characteristic
78	#486	—	31	1	no	—
79	#476	D₃₁	1	1	yes	characteristic, complemented
80	#459	—	31	1	no	—
81	#436	—	31	1	no	—
82	#335	—	31	1	no	—
83	#333	—	31	1	no	—
84	#331	—	31	1	no	—
85	C₂×C₃₁	—	8	1	no	—
86	C₂×C₃₁	#463	1	1	yes	characteristic
87	C₂×C₃₁	#468	1	1	yes	characteristic
88	C₂×C₃₁	#472	1	1	yes	characteristic
89	D₃₁	—	16	1	no	—
90	C₂×D₈	—	62	1	no	—
91	C₂×M₁₆	—	62	1	no	—
92	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
93	#127	—	31	1	no	—
94	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
95	#106	—	2	1	no	—
96	C₄⋊C₈	—	31	1	no	—
97	C₄⋊C₈	—	31	1	no	—
98	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
99	#102	—	2	1	no	—
100	#102	—	62	1	no	—
101	#102	—	62	1	no	—
102	#102	—	62	1	no	—
103	C₂²⋊C₈	—	62	1	no	—
104	C₄×C₈	—	31	1	no	—
105	C₃₁	#1697	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	62	1	no	—
108	C₂×D₄	—	62	1	no	—
109	C₂×D₄	—	62	1	no	—
110	C₂×D₄	C₂²⋊D₃₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	124	2	no	—
113	M₁₆	—	124	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	31	1	no	—
116	C₂×C₈	—	31	1	no	—
117	C₂×C₈	—	62	1	no	—
118	C₂×C₈	—	62	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₃₁	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	124	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	124	1	no	—
130	D₄	—	124	1	no	—
131	D₄	—	124	1	no	—
132	D₄	—	124	1	no	—
133	D₄	—	248	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
139	C₂×C₄	#92497	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	62	1	no	—
143	C₈	—	62	1	no	—
144	C₈	—	124	1	no	—
145	C₈	—	124	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	248	1	no	—
150	C₂²	—	248	1	no	—
151	C₂²	—	248	1	no	—
152	C₂²	#11758516	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	#11758491	1	1	yes	characteristic
157	C₄	#11758493	1	1	yes	characteristic
158	C₄	#11758519	1	1	yes	characteristic
159	C₄	#11758526	1	1	yes	characteristic
160	C₂	—	8	1	no	—
161	C₂	—	496	1	no	—
162	C₂	#14521266	1	1	yes	characteristic
163	C₂	#14521297	1	1	yes	characteristic
164	C₂	#14521317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11758516
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#476	C₂
socle	C₂²×C₃₁	#121
prime radical	#14521244	C₂
hypercenter	#476	D₃₁
hypocenter	C₃₁	#1697
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #14521397	C₂	semidirect
3. #14521127	C₂	semidirect
4. #14520801	C₂	semidirect
5. #14520798	C₂	semidirect
6. #14520792	C₂	semidirect
7. C₃₁×#476	C₂	semidirect
79. #476	D₃₁	semidirect
105. C₃₁	#1697	semidirect

Maximal subgroups

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

subgroup	index
#14521397	2
#14521244	2
#14521127	2
#14520801	2
#14520798	2
#14520792	2
C₃₁×#476	2
#1697	31