Finite Group Databasebeta

Subgroups — #41002975

Order 26944.

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×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20269756	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212341	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212286	C₂	1	1	yes	maximal, characteristic, complemented
6	#27212278	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄₂₁	—	2	1	no	—
9	C₂×D₁₆₈₄	—	2	1	no	—
10	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₄₂₁	—	2	1	no	—
13	D₄×D₄₂₁	—	2	2	no	—
14	D₄×D₄₂₁	—	2	2	no	—
15	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
16	D₃₃₆₈	—	2	2	no	—
17	#20269756	—	2	2	no	—
18	C₂×#16809890	C₂²	1	1	yes	characteristic, complemented
19	#20269748	—	2	1	no	—
20	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₄₂₁	—	2	4	no	—
24	C₂²×D₄₂₁	—	2	1	no	—
25	C₂²×D₄₂₁	—	2	1	no	—
26	C₂²×D₄₂₁	—	4	2	no	—
27	C₂²×D₄₂₁	—	4	2	no	—
28	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
29	D₁₆₈₄	—	2	1	no	—
30	D₁₆₈₄	—	2	1	no	—
31	D₁₆₈₄	—	2	1	no	—
32	D₁₆₈₄	—	4	2	no	—
33	D₁₆₈₄	D₄	1	2	yes	complemented
34	C₂²⋊D₄₂₁	—	2	4	no	—
35	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
36	C₄×D₄₂₁	—	2	1	no	—
37	C₄×D₄₂₁	—	2	1	no	—
38	#16809890	—	2	1	no	—
39	C₂³×C₄₂₁	—	2	1	no	—
40	D₄×C₄₂₁	—	2	1	no	—
41	D₄×C₄₂₁	D₄	1	2	yes	complemented
42	C₂×C₄×C₄₂₁	—	2	1	no	—
43	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
44	C₈×C₄₂₁	—	2	1	no	—
45	C₂²×C₄₂₁	—	2	2	no	—
46	C₂²×C₄₂₁	—	4	1	no	—
47	C₂²×C₄₂₁	—	4	1	no	—
48	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
49	C₂×D₄₂₁	—	2	1	no	—
50	C₂×D₄₂₁	—	2	1	no	—
51	C₂×D₄₂₁	—	2	2	no	—
52	C₂×D₄₂₁	—	4	8	no	—
53	C₂×D₄₂₁	—	4	1	no	—
54	C₂×D₄₂₁	—	4	1	no	—
55	C₂×D₄₂₁	—	4	1	no	—
56	C₂×D₄₂₁	—	4	1	no	—
57	C₂×D₄₂₁	—	4	1	no	—
58	C₂×D₄₂₁	D₈	1	2	yes	complemented
59	C₄×C₄₂₁	—	4	1	no	—
60	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
61	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
62	Dic₄₂₁	—	2	1	no	—
63	C₂×C₄₂₁	—	4	2	no	—
64	C₂×C₄₂₁	#120	1	1	yes	characteristic
65	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	D₄₂₁	—	2	2	no	—
68	D₄₂₁	—	4	2	no	—
69	D₄₂₁	—	8	1	no	—
70	C₄₂₁	#447	1	1	yes	characteristic, complemented, Sylow
71	#447	—	421	1	no	maximal, Sylow
72	C₂²×D₄	—	421	1	no	—
73	C₂×D₈	—	421	1	no	—
74	C₂×D₈	—	421	1	no	—
75	#121	—	421	1	no	—
76	#102	—	421	1	no	—
77	#102	D₄₂₁	1	1	yes	characteristic, complemented
78	C₂²⋊C₈	—	421	1	no	—
79	C₂⁴	—	842	1	no	—
80	C₂×D₄	—	842	1	no	—
81	C₂×D₄	—	842	2	no	—
82	C₂×D₄	—	842	2	no	—
83	C₂×D₄	—	842	1	no	—
84	C₂×D₄	—	421	1	no	—
85	C₂×D₄	—	421	1	no	—
86	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
87	C₂²×C₄	—	421	1	no	—
88	D₈	—	842	2	no	—
89	D₈	—	842	2	no	—
90	C₂×C₈	—	421	1	no	—
91	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic, complemented
92	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic, complemented
93	C₂²⋊C₄	—	842	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	842	4	no	—
96	C₂³	—	842	1	no	—
97	C₂³	—	842	1	no	—
98	C₂³	—	1684	2	no	—
99	C₂³	—	1684	2	no	—
100	C₂³	—	421	1	no	—
101	D₄	—	2	1	no	—
102	D₄	—	842	4	no	—
103	D₄	—	842	1	no	—
104	D₄	—	842	1	no	—
105	D₄	—	842	1	no	—
106	D₄	—	1684	2	no	—
107	D₄	—	421	2	no	—
108	D₄	D₁₆₈₄	1	2	yes	complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	842	1	no	—
111	C₂×C₄	—	842	1	no	—
112	C₂×C₄	—	421	1	no	—
113	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	842	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	842	1	no	—
120	C₂²	—	842	1	no	—
121	C₂²	—	842	2	no	—
122	C₂²	—	1684	8	no	—
123	C₂²	—	1684	1	no	—
124	C₂²	—	1684	1	no	—
125	C₂²	—	1684	1	no	—
126	C₂²	—	1684	1	no	—
127	C₂²	—	1684	1	no	—
128	C₂²	—	421	2	no	—
129	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
130	C₄	—	4	1	no	—
131	C₄	—	842	1	no	—
132	C₄	D₄×D₄₂₁	1	1	yes	characteristic
133	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
134	C₂	—	4	2	no	—
135	C₂	—	3368	1	no	—
136	C₂	—	842	2	no	—
137	C₂	—	1684	2	no	—
138	C₂	#27212336	1	1	yes	characteristic
139	C₂	D₈×D₄₂₁	1	1	yes	characteristic
140	C₂	#27212393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#102×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	#102	D₄₂₁
hypocenter	C₄₂₁	#447
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
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×D₄₂₁	C₂	semidirect
2. C₂×D₃₃₆₈	C₂	semidirect
3. C₂×#20269756	C₂	semidirect
4. #27212341	C₂	semidirect
5. #27212286	C₂	semidirect
6. #27212278	C₂	semidirect
7. #102×C₄₂₁	C₂	semidirect
11. C₂×D₁₆₈₄	C₂²	semidirect
18. C₂×#16809890	C₂²	semidirect
21. C₂×C₈×C₄₂₁	C₂²	semidirect
22. C₄⋊C₄×C₄₂₁	C₂²	semidirect
33. D₁₆₈₄	D₄	semidirect
41. D₄×C₄₂₁	D₄	semidirect
58. C₂×D₄₂₁	D₈	semidirect
70. C₄₂₁	#447	semidirect
77. #102	D₄₂₁	semidirect
91. C₂×C₈	C₂×D₄₂₁	semidirect
92. C₄⋊C₄	C₂×D₄₂₁	semidirect
108. D₄	D₁₆₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₄₂₁	2
C₂×D₃₃₆₈	2
C₂×#20269756	2
#27212341	2
#27212286	2
#27212278	2
#102×C₄₂₁	2
#447	421