Finite Group Databasebeta

Subgroups — #16857504

Order 3968.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14521855	C₂	1	4	yes	maximal, complemented
2	#14521309	C₂	1	4	yes	maximal, complemented
3	#14521231	C₂	1	2	yes	maximal, complemented
4	#14521216	C₂	1	2	yes	maximal, complemented
5	#14521167	C₂	1	2	yes	maximal, complemented
6	C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758564	C₂²	1	2	yes	—
8	#11758561	C₂²	1	2	yes	complemented
9	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
10	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₃₁×#127	C₂²	1	1	yes	characteristic
12	#11758519	C₂²	1	4	yes	complemented
13	#11758509	C₂²	1	4	yes	—
14	#11758493	—	2	4	no	—
15	#11758492	—	2	4	no	—
16	#11758491	—	2	4	no	—
17	SD₁₆×D₃₁	—	2	4	no	—
18	#11758481	C₂²	1	2	yes	—
19	#11758478	C₂²	1	4	yes	complemented
20	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	#11758424	C₂²	1	2	yes	complemented
22	#11758421	C₂²	1	2	yes	complemented
23	#11758407	C₂²	1	2	yes	complemented
24	#11758393	C₂²	1	4	yes	complemented
25	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
26	C₂×D₄×C₃₁	—	4	1	no	—
27	C₂×D₄×C₃₁	C₂³	1	2	yes	—
28	C₂×C₂²⋊D₃₁	—	4	2	no	—
29	#92499	—	2	4	no	—
30	Q₈×D₃₁	—	2	4	no	—
31	#92497	—	2	4	no	—
32	D₄×D₃₁	—	2	4	no	—
33	#92495	—	4	2	no	—
34	#92495	C₂³	1	4	yes	—
35	SD₁₆×C₃₁	—	2	4	no	—
36	SD₁₆×C₃₁	—	2	4	no	—
37	C₂×C₈×C₃₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₃₁	—	4	1	no	—
39	C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	#92482	—	2	4	no	—
41	#92481	—	2	4	no	—
42	#92480	—	2	4	no	—
43	#92479	—	2	4	no	—
44	#92474	C₂³	1	4	yes	—
45	Dic₁₂₄	D₄	1	2	yes	complemented
46	D₂₄₈	D₄	1	2	yes	complemented
47	#92470	D₄	1	4	yes	complemented
48	#92469	—	2	4	no	—
49	C₈×D₃₁	—	2	4	no	—
50	C₂³×C₃₁	—	2	2	no	—
51	C₂²×D₃₁	—	4	2	no	—
52	Q₈×C₃₁	—	2	4	no	—
53	D₄×C₃₁	—	2	4	no	—
54	D₄×C₃₁	—	8	2	no	—
55	C₂×C₄×C₃₁	—	2	1	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₃₁	—	2	4	no	—
59	C₂²⋊D₃₁	—	4	4	no	—
60	C₂×Dic₃₁	—	4	2	no	—
61	D₁₂₄	—	4	2	no	—
62	D₁₂₄	C₂×D₄	1	4	yes	—
63	C₄×D₃₁	—	2	4	no	—
64	C₄×D₃₁	—	4	4	no	—
65	Dic₆₂	—	4	2	no	—
66	Dic₆₂	C₂×D₄	1	4	yes	—
67	C₈×C₃₁	C₂×D₄	1	2	yes	—
68	C₈×C₃₁	C₂×D₄	1	2	yes	—
69	C₃₁⋊C₈	—	2	4	no	—
70	#3291	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	4	2	no	—
72	C₂²×C₃₁	—	8	2	no	—
73	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×D₃₁	—	2	4	no	—
75	C₂×D₃₁	—	16	2	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	—	4	2	no	—
78	C₄×C₃₁	C₂²×D₄	1	2	yes	—
79	Dic₃₁	—	2	4	no	—
80	#577	—	31	4	no	—
81	#492	D₃₁	1	1	yes	characteristic, complemented
82	#471	—	31	4	no	—
83	#456	—	31	2	no	—
84	#453	—	31	2	no	—
85	#443	—	31	2	no	—
86	C₂×C₃₁	—	2	1	no	—
87	C₂×C₃₁	—	8	2	no	—
88	C₂×C₃₁	D₄²	1	1	yes	characteristic
89	D₃₁	—	4	4	no	—
90	2₋¹⁺⁴	—	31	2	no	—
91	H(5,𝔽₂)	—	31	2	no	—
92	#137	—	31	4	no	—
93	#137	—	62	4	no	—
94	AGL(1,ℤ/8)	—	31	4	no	—
95	AGL(1,ℤ/8)	—	62	4	no	—
96	#135	—	31	2	no	—
97	#135	—	62	4	no	—
98	C₂×SD₁₆	—	62	4	no	—
99	C₂×SD₁₆	C₂×D₃₁	1	4	yes	complemented
100	#131	—	31	4	no	—
101	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
102	#127	C₂×D₃₁	1	1	yes	characteristic
103	#108	—	31	2	no	—
104	#104	—	31	4	no	—
105	#101	—	31	2	no	—
106	#100	—	31	2	no	—
107	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
108	C₃₁	#3291	1	1	yes	characteristic, complemented, Sylow
109	#41	—	31	4	no	—
110	#41	—	62	4	no	—
111	#41	—	62	4	no	—
112	#41	—	124	2	no	—
113	C₂×Q₈	—	62	4	no	—
114	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	62	4	no	—
117	C₂×D₄	—	124	2	no	—
118	C₂×D₄	C₂²×D₃₁	1	2	yes	—
119	Q₁₆	—	31	2	no	—
120	Q₁₆	—	62	4	no	—
121	SD₁₆	—	2	4	no	—
122	SD₁₆	—	2	4	no	—
123	SD₁₆	—	31	4	no	—
124	SD₁₆	—	62	4	no	—
125	SD₁₆	—	62	4	no	—
126	D₈	—	31	2	no	—
127	D₈	—	62	4	no	—
128	M₁₆	—	31	4	no	—
129	M₁₆	—	62	4	no	—
130	C₂×C₈	—	62	4	no	—
131	C₂×C₈	C₂²×D₃₁	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄²	C₂²×D₃₁	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	124	2	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	31	4	no	—
138	Q₈	—	124	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	31	4	no	—
142	D₄	—	62	4	no	—
143	D₄	—	124	2	no	—
144	D₄	—	124	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	62	4	no	—
148	C₂×C₄	—	124	4	no	—
149	C₂×C₄	—	124	2	no	—
150	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
151	C₈	—	62	4	no	—
152	C₈	D₄×D₃₁	1	2	yes	—
153	C₈	D₄×D₃₁	1	2	yes	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	62	4	no	—
157	C₂²	—	496	2	no	—
158	C₂²	#11758561	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	62	4	no	—
162	C₄	C₂×D₄×D₃₁	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	124	4	no	—
166	C₂	#14521684	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14521684
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#492	C₂
socle	C₂×C₃₁	D₄²
hypercenter	#492	D₃₁
hypocenter	C₃₁	#3291
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #14521855	C₂	semidirect
2. #14521309	C₂	semidirect
3. #14521231	C₂	semidirect
4. #14521216	C₂	semidirect
5. #14521167	C₂	semidirect
6. C₃₁×#492	C₂	semidirect
8. #11758561	C₂²	semidirect
9. C₂×SD₁₆×C₃₁	C₂²	semidirect
10. C₃₁×C₄⋊Q₈	C₂²	semidirect
12. #11758519	C₂²	semidirect
19. #11758478	C₂²	semidirect
20. C₄×C₈×C₃₁	C₂²	semidirect
21. #11758424	C₂²	semidirect
22. #11758421	C₂²	semidirect
23. #11758407	C₂²	semidirect
24. #11758393	C₂²	semidirect
45. Dic₁₂₄	D₄	semidirect
46. D₂₄₈	D₄	semidirect
47. #92470	D₄	semidirect
81. #492	D₃₁	semidirect
99. C₂×SD₁₆	C₂×D₃₁	semidirect
101. C₄⋊Q₈	C₂×D₃₁	semidirect
107. C₄×C₈	C₂×D₃₁	semidirect
108. C₃₁	#3291	semidirect

Maximal subgroups

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

subgroup	index
#14521855	2
#14521309	2
#14521231	2
#14521216	2
#14521167	2
C₃₁×#492	2
#3291	31