Finite Group Databasebeta

Subgroups — #42121946

Order 27712.

198 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂×D₄×D₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₄₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20294082	C₂	1	1	yes	maximal, characteristic, complemented
4	#28357195	C₂	1	1	yes	maximal, characteristic, complemented
5	#28357140	C₂	1	1	yes	maximal, characteristic, complemented
6	#28357132	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄₃₃	—	2	1	no	—
9	C₂×D₁₇₃₂	—	2	1	no	—
10	C₂×D₁₇₃₂	C₂²	1	1	yes	characteristic
11	C₂×D₁₇₃₂	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₄₃₃	—	2	1	no	—
13	D₄×D₄₃₃	—	2	2	no	—
14	D₄×D₄₃₃	—	2	2	no	—
15	C₂×C₄×D₄₃₃	C₂²	1	1	yes	characteristic
16	D₃₄₆₄	—	2	2	no	—
17	#20294082	—	2	2	no	—
18	C₂×#16812358	C₂²	1	1	yes	characteristic, complemented
19	#20294074	—	2	1	no	—
20	C₂×D₄×C₄₃₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₄₃₃	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₄₃₃	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₄₃₃	—	2	4	no	—
24	C₂²×D₄₃₃	—	2	1	no	—
25	C₂²×D₄₃₃	—	2	1	no	—
26	C₂²×D₄₃₃	—	4	2	no	—
27	C₂²×D₄₃₃	—	4	2	no	—
28	C₂²×D₄₃₃	D₄	1	1	yes	characteristic
29	D₁₇₃₂	—	2	1	no	—
30	D₁₇₃₂	—	2	1	no	—
31	D₁₇₃₂	—	2	1	no	—
32	D₁₇₃₂	—	4	2	no	—
33	D₁₇₃₂	D₄	1	2	yes	complemented
34	C₂²⋊D₄₃₃	—	2	4	no	—
35	C₂×Dic₄₃₃	D₄	1	1	yes	characteristic
36	C₄×D₄₃₃	—	2	1	no	—
37	C₄×D₄₃₃	—	2	1	no	—
38	#16812358	—	2	1	no	—
39	C₂³×C₄₃₃	—	2	1	no	—
40	D₄×C₄₃₃	—	2	1	no	—
41	D₄×C₄₃₃	D₄	1	2	yes	complemented
42	C₂×C₄×C₄₃₃	—	2	1	no	—
43	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
44	C₈×C₄₃₃	—	2	1	no	—
45	C₂²×C₄₃₃	—	2	2	no	—
46	C₂²×C₄₃₃	—	4	1	no	—
47	C₂²×C₄₃₃	—	4	1	no	—
48	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₄₃₃	—	2	1	no	—
50	C₂×D₄₃₃	—	2	1	no	—
51	C₂×D₄₃₃	—	2	2	no	—
52	C₂×D₄₃₃	—	4	8	no	—
53	C₂×D₄₃₃	—	4	1	no	—
54	C₂×D₄₃₃	—	4	1	no	—
55	C₂×D₄₃₃	—	4	1	no	—
56	C₂×D₄₃₃	—	4	1	no	—
57	C₂×D₄₃₃	—	4	1	no	—
58	C₂×D₄₃₃	D₈	1	2	yes	complemented
59	C₄×C₄₃₃	—	4	1	no	—
60	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
61	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
62	Dic₄₃₃	—	2	1	no	—
63	C₂×C₄₃₃	—	4	2	no	—
64	C₂×C₄₃₃	#120	1	1	yes	characteristic
65	C₂×C₄₃₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	D₄₃₃	—	2	2	no	—
68	D₄₃₃	—	4	2	no	—
69	D₄₃₃	—	8	1	no	—
70	C₄₃₃	#447	1	1	yes	characteristic, complemented, Sylow
71	#447	—	433	1	no	maximal, Sylow
72	C₂²×D₄	—	433	1	no	—
73	C₂×D₈	—	433	1	no	—
74	C₂×D₈	—	433	1	no	—
75	#121	—	433	1	no	—
76	#102	—	433	1	no	—
77	#102	D₄₃₃	1	1	yes	characteristic, complemented
78	C₂²⋊C₈	—	433	1	no	—
79	C₂⁴	—	866	1	no	—
80	C₂×D₄	—	866	1	no	—
81	C₂×D₄	—	866	2	no	—
82	C₂×D₄	—	866	2	no	—
83	C₂×D₄	—	866	1	no	—
84	C₂×D₄	—	433	1	no	—
85	C₂×D₄	—	433	1	no	—
86	C₂×D₄	C₂×D₄₃₃	1	1	yes	characteristic
87	C₂²×C₄	—	433	1	no	—
88	D₈	—	866	2	no	—
89	D₈	—	866	2	no	—
90	C₂×C₈	—	433	1	no	—
91	C₂×C₈	C₂×D₄₃₃	1	1	yes	characteristic, complemented
92	C₄⋊C₄	C₂×D₄₃₃	1	1	yes	characteristic, complemented
93	C₂²⋊C₄	—	866	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	866	4	no	—
96	C₂³	—	866	1	no	—
97	C₂³	—	866	1	no	—
98	C₂³	—	433	1	no	—
99	C₂³	—	1732	2	no	—
100	C₂³	—	1732	2	no	—
101	D₄	—	2	1	no	—
102	D₄	—	866	4	no	—
103	D₄	—	866	1	no	—
104	D₄	—	866	1	no	—
105	D₄	—	866	1	no	—
106	D₄	—	433	2	no	—
107	D₄	—	1732	2	no	—
108	D₄	D₁₇₃₂	1	2	yes	complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	866	1	no	—
111	C₂×C₄	—	866	1	no	—
112	C₂×C₄	—	433	1	no	—
113	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	866	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	866	1	no	—
120	C₂²	—	866	1	no	—
121	C₂²	—	866	2	no	—
122	C₂²	—	433	2	no	—
123	C₂²	—	1732	8	no	—
124	C₂²	—	1732	1	no	—
125	C₂²	—	1732	1	no	—
126	C₂²	—	1732	1	no	—
127	C₂²	—	1732	1	no	—
128	C₂²	—	1732	1	no	—
129	C₂²	D₄×D₄₃₃	1	1	yes	characteristic
130	C₄	—	4	1	no	—
131	C₄	—	866	1	no	—
132	C₄	D₄×D₄₃₃	1	1	yes	characteristic
133	C₄	C₂×D₁₇₃₂	1	1	yes	characteristic
134	C₂	—	4	2	no	—
135	C₂	—	866	2	no	—
136	C₂	—	3464	1	no	—
137	C₂	—	1732	2	no	—
138	C₂	#28357190	1	1	yes	characteristic
139	C₂	D₈×D₄₃₃	1	1	yes	characteristic
140	C₂	#28357247	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₃₃
commutator, cosocle	C₂×C₄×C₄₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	#102×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂×D₄
hypercenter	#102	D₄₃₃
hypocenter	C₄₃₃	#447
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₄₃₃	C₂	semidirect
2. C₂×D₃₄₆₄	C₂	semidirect
3. C₂×#20294082	C₂	semidirect
4. #28357195	C₂	semidirect
5. #28357140	C₂	semidirect
6. #28357132	C₂	semidirect
7. #102×C₄₃₃	C₂	semidirect
11. C₂×D₁₇₃₂	C₂²	semidirect
18. C₂×#16812358	C₂²	semidirect
21. C₂×C₈×C₄₃₃	C₂²	semidirect
22. C₄⋊C₄×C₄₃₃	C₂²	semidirect
33. D₁₇₃₂	D₄	semidirect
41. D₄×C₄₃₃	D₄	semidirect
58. C₂×D₄₃₃	D₈	semidirect
70. C₄₃₃	#447	semidirect
77. #102	D₄₃₃	semidirect
91. C₂×C₈	C₂×D₄₃₃	semidirect
92. C₄⋊C₄	C₂×D₄₃₃	semidirect
108. D₄	D₁₇₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₄₃₃	2
C₂×D₃₄₆₄	2
C₂×#20294082	2
#28357195	2
#28357140	2
#28357132	2
#102×C₄₃₃	2
#447	433