Finite Group Databasebeta

Subgroups — #16845376

Order 3968.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14521425	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521425	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521421	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521396	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521395	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈×C₃₁	—	2	2	no	—
9	C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₂×#92480	—	2	2	no	—
12	C₂×#92479	—	2	2	no	—
13	C₂×#92479	—	2	2	no	—
14	#11758451	C₂²	1	1	yes	characteristic
15	#11758446	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758420	—	2	2	no	—
18	#11758409	—	2	2	no	—
19	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₃₁	—	2	1	no	—
22	C₂×D₄×C₃₁	—	2	1	no	—
23	C₂×D₄×C₃₁	—	2	1	no	—
24	C₂×D₄×C₃₁	—	2	1	no	—
25	C₂×D₄×C₃₁	—	2	1	no	—
26	C₂×D₄×C₃₁	—	2	1	no	—
27	C₂×D₁₂₄	—	2	1	no	—
28	C₂×D₁₂₄	—	2	1	no	—
29	C₂×D₁₂₄	—	2	1	no	—
30	C₂×Dic₆₂	—	2	1	no	—
31	D₈×C₃₁	—	4	4	no	—
32	C₂×C₈×C₃₁	D₄	1	2	yes	—
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	#92480	—	4	4	no	—
35	#92479	—	4	4	no	—
36	#92479	—	4	4	no	—
37	C₄⋊Dic₃₁	—	2	1	no	—
38	C₄⋊Dic₃₁	—	2	1	no	—
39	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
40	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
41	C₂³×C₃₁	—	4	1	no	—
42	C₂³×C₃₁	—	4	1	no	—
43	C₂²×D₃₁	—	4	1	no	—
44	D₄×C₃₁	—	4	2	no	—
45	D₄×C₃₁	—	4	2	no	—
46	D₄×C₃₁	—	4	2	no	—
47	D₄×C₃₁	—	4	2	no	—
48	D₄×C₃₁	—	4	2	no	—
49	D₄×C₃₁	—	4	2	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	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₃₁	—	4	1	no	—
54	D₁₂₄	—	4	2	no	—
55	D₁₂₄	—	4	2	no	—
56	D₁₂₄	—	4	2	no	—
57	Dic₆₂	—	4	2	no	—
58	C₈×C₃₁	—	2	2	no	—
59	C₃₁⋊C₈	—	2	2	no	—
60	C₃₁⋊C₈	D₈	1	4	yes	complemented
61	#1721	—	31	1	no	maximal, Sylow
62	C₂²×C₃₁	—	8	1	no	—
63	C₂²×C₃₁	—	8	1	no	—
64	C₂²×C₃₁	—	8	1	no	—
65	C₂²×C₃₁	—	8	1	no	—
66	C₂²×C₃₁	—	8	1	no	—
67	C₂²×C₃₁	—	8	1	no	—
68	C₂²×C₃₁	#127	1	1	yes	characteristic
69	C₂×D₃₁	—	8	1	no	—
70	C₂×D₃₁	—	8	1	no	—
71	C₂×D₃₁	—	8	1	no	—
72	C₄×C₃₁	C₂×D₈	1	2	yes	—
73	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
75	Dic₃₁	—	8	1	no	—
76	#493	—	31	1	no	—
77	#493	—	31	1	no	—
78	#493	D₃₁	1	1	yes	characteristic, complemented
79	#492	—	31	1	no	—
80	#486	—	31	1	no	—
81	#486	—	31	1	no	—
82	#322	—	31	1	no	—
83	C₂×C₃₁	—	16	1	no	—
84	C₂×C₃₁	—	16	1	no	—
85	C₂×C₃₁	#493	1	1	yes	characteristic
86	C₂×C₃₁	#496	1	1	yes	characteristic
87	C₂×C₃₁	#496	1	1	yes	characteristic
88	D₃₁	—	16	1	no	—
89	C₂×SD₁₆	—	62	2	no	—
90	C₂×D₈	—	2	2	no	—
91	C₂×D₈	—	62	2	no	—
92	C₂×D₈	—	62	2	no	—
93	C₄⋊Q₈	—	31	1	no	—
94	#127	—	31	1	no	—
95	#127	C₂×D₃₁	1	1	yes	characteristic
96	#127	C₂×D₃₁	1	1	yes	characteristic
97	#102	—	62	2	no	—
98	#102	—	62	2	no	—
99	C₄×C₈	—	31	1	no	—
100	C₄×C₈	—	31	1	no	—
101	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
102	C₃₁	#1721	1	1	yes	characteristic, complemented, Sylow
103	C₂×Q₈	—	62	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	62	1	no	—
111	C₂×D₄	—	62	1	no	—
112	C₂×D₄	—	62	1	no	—
113	SD₁₆	—	124	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	124	4	no	—
116	D₈	—	124	4	no	—
117	C₂×C₈	—	31	2	no	—
118	C₂×C₈	—	31	2	no	—
119	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
120	C₄⋊C₄	—	62	1	no	—
121	C₄⋊C₄	—	62	1	no	—
122	C₄²	C₂²×D₃₁	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	124	1	no	—
126	Q₈	—	124	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	124	2	no	—
134	D₄	—	124	2	no	—
135	D₄	—	124	2	no	—
136	C₂×C₄	—	124	1	no	—
137	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
138	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	31	4	no	—
142	C₈	—	62	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	248	1	no	—
150	C₂²	—	248	1	no	—
151	C₂²	—	248	1	no	—
152	C₂²	#11758517	1	1	yes	characteristic
153	C₄	—	248	1	no	—
154	C₄	D₈×D₃₁	1	2	yes	—
155	C₄	#11758488	1	2	yes	—
156	C₄	#11758509	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	496	1	no	—
160	C₂	#14521424	1	1	yes	characteristic
161	C₂	#14521436	1	1	yes	characteristic
162	C₂	#14521437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#493	C₂
socle	C₂²×C₃₁	#127
hypercenter	#493	D₃₁
hypocenter	C₃₁	#1721
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #14521425	C₂	semidirect
2. #14521425	C₂	semidirect
3. #14521421	C₂	semidirect
4. #14521396	C₂	semidirect
5. #14521395	C₂	semidirect
6. #14520771	C₂	semidirect
7. C₃₁×#493	C₂	semidirect
60. C₃₁⋊C₈	D₈	semidirect
78. #493	D₃₁	semidirect
102. C₃₁	#1721	semidirect

Maximal subgroups

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

subgroup	index
#14521425	2
#14521421	2
#14521396	2
#14521395	2
#14520771	2
C₃₁×#493	2
#1721	31