Finite Group Databasebeta

Subgroups — #26022268

Order 11392.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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	#19152340	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151771	C₂	1	2	yes	maximal, complemented
3	#19151395	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151394	C₂	1	2	yes	maximal, complemented
5	#454×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668037	—	2	2	no	—
7	#15668001	C₂²	1	2	yes	—
8	#15668000	—	2	1	no	—
9	#15667998	—	2	2	no	—
10	#15667982	—	2	2	no	—
11	#15667980	—	2	1	no	—
12	#15667939	—	2	2	no	—
13	#15667931	C₂²	1	2	yes	complemented
14	2₋¹⁺⁴×C₈₉	C₂²	1	1	yes	characteristic, complemented
15	AGL(1,ℤ/8)×C₈₉	—	2	1	no	—
16	#124×C₈₉	C₂²	1	1	yes	characteristic
17	#104×C₈₉	—	2	1	no	—
18	#101×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₈₉	—	2	1	no	—
20	C₂×Dic₁₇₈	—	2	1	no	—
21	C₂×D₃₅₆	—	2	1	no	—
22	C₂×D₃₅₆	D₄	1	2	yes	—
23	C₂×C₂²⋊D₈₉	—	2	2	no	—
24	#13083561	—	4	2	no	—
25	#13083559	—	4	2	no	—
26	#13083558	—	2	2	no	—
27	C₂²⋊Dic₈₉	—	2	1	no	—
28	#13083551	—	2	2	no	—
29	#13083551	—	2	2	no	—
30	#13083551	—	2	2	no	—
31	C₄×Dic₈₉	D₄	1	2	yes	—
32	#41×C₈₉	—	2	2	no	—
33	#41×C₈₉	—	2	1	no	—
34	#41×C₈₉	—	4	1	no	—
35	C₂×Q₈×C₈₉	—	4	1	no	—
36	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
38	SD₁₆×C₈₉	—	4	1	no	—
39	D₈×C₈₉	—	4	1	no	—
40	M₁₆×C₈₉	—	2	1	no	—
41	C₂²⋊C₄×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂³×C₈₉	—	2	1	no	—
44	C₂²×D₈₉	—	2	2	no	—
45	C₂²×D₈₉	—	8	2	no	—
46	C₂²×D₈₉	—	8	1	no	—
47	Q₈×C₈₉	—	2	1	no	—
48	Q₈×C₈₉	—	2	2	no	—
49	Q₈×C₈₉	—	4	1	no	—
50	D₄×C₈₉	—	2	1	no	—
51	D₄×C₈₉	—	2	2	no	—
52	D₄×C₈₉	—	4	1	no	—
53	D₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	4	2	no	—
56	C₂×C₄×C₈₉	—	4	1	no	—
57	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₈₉	—	8	2	no	—
60	C₂×Dic₈₉	—	2	2	no	—
61	D₃₅₆	—	4	2	no	—
62	D₃₅₆	—	8	1	no	—
63	Dic₁₇₈	—	8	1	no	—
64	C₈×C₈₉	—	4	1	no	—
65	#10623494	—	4	2	no	—
66	C₂²×C₈₉	—	4	2	no	—
67	C₂²×C₈₉	—	4	1	no	—
68	C₂²×C₈₉	—	4	1	no	—
69	C₂²×C₈₉	#120	1	1	yes	characteristic
70	C₂×D₈₉	—	4	2	no	—
71	C₂×D₈₉	—	4	2	no	—
72	C₂×D₈₉	—	4	2	no	—
73	C₂×D₈₉	—	8	4	no	—
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	4	1	no	—
77	C₄×C₈₉	—	4	1	no	—
78	Dic₈₉	—	4	2	no	—
79	C₂×C₈₉	—	2	1	no	—
80	C₂×C₈₉	—	8	1	no	—
81	C₂×C₈₉	—	8	1	no	—
82	C₂×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
83	D₈₉	—	8	2	no	—
84	#2202	—	89	1	no	maximal, Sylow
85	C₈₉	#2202	1	1	yes	characteristic, complemented, Sylow
86	H(3,𝔽₄)	—	89	1	no	—
87	#454	—	89	2	no	—
88	#454	D₈₉	1	1	yes	characteristic, complemented
89	#355	—	89	1	no	—
90	#355	—	89	2	no	—
91	2₋¹⁺⁴	C₂×D₈₉	1	1	yes	characteristic, complemented
92	AGL(1,ℤ/8)	—	2	1	no	—
93	AGL(1,ℤ/8)	—	178	2	no	—
94	#124	—	89	2	no	—
95	#124	—	178	2	no	—
96	#124	—	178	1	no	—
97	#124	C₂×D₈₉	1	1	yes	characteristic
98	#120	—	178	2	no	—
99	#120	—	178	1	no	—
100	#104	—	2	1	no	—
101	#104	—	178	2	no	—
102	#101	—	89	2	no	—
103	#101	C₂×D₈₉	1	1	yes	characteristic, complemented
104	C₂⁴	—	178	1	no	—
105	#41	—	2	2	no	—
106	#41	—	2	1	no	—
107	#41	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	178	1	no	—
110	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×D₄	—	89	2	no	—
112	C₂×D₄	—	178	1	no	—
113	C₂×D₄	—	178	2	no	—
114	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
115	SD₁₆	—	4	1	no	—
116	SD₁₆	—	356	2	no	—
117	D₈	—	4	1	no	—
118	D₈	—	356	2	no	—
119	M₁₆	—	2	1	no	—
120	M₁₆	—	178	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	178	2	no	—
123	C₂²⋊C₄	—	178	1	no	—
124	C₂²⋊C₄	—	178	2	no	—
125	C₂²⋊C₄	—	178	2	no	—
126	C₄²	—	89	2	no	—
127	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
128	C₂³	—	2	1	no	—
129	C₂³	—	178	2	no	—
130	C₂³	—	712	2	no	—
131	C₂³	—	712	1	no	—
132	Q₈	—	2	1	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	712	1	no	—
136	D₄	—	2	1	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	356	2	no	—
141	D₄	—	712	1	no	—
142	D₄	—	712	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	178	2	no	—
147	C₂×C₄	D₄×D₈₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	356	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	356	2	no	—
155	C₂²	—	356	2	no	—
156	C₂²	—	356	2	no	—
157	C₂²	—	712	4	no	—
158	C₂²	#15667980	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	356	2	no	—
164	C₂	—	2	1	no	—
165	C₂	—	8	1	no	—
166	C₂	—	8	1	no	—
167	C₂	—	712	2	no	—
168	C₂	#19151784	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151784
commutator, cosocle	C₂×Q₈×C₈₉	C₂³
Frattini	C₂×Q₈	C₂²×D₈₉
Fitting	#454×C₈₉	C₂
socle	C₂×C₈₉	B(4,𝔽₂)
hypercenter	#454	D₈₉
hypocenter	C₈₉	#2202
LCS	G ▷ C₂×Q₈×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #454

Complements

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

N	H ≅ G/N	type
1. #19152340	C₂	semidirect
2. #19151771	C₂	semidirect
3. #19151395	C₂	semidirect
4. #19151394	C₂	semidirect
5. #454×C₈₉	C₂	semidirect
13. #15667931	C₂²	semidirect
14. 2₋¹⁺⁴×C₈₉	C₂²	semidirect
18. #101×C₈₉	C₂²	semidirect
85. C₈₉	#2202	semidirect
88. #454	D₈₉	semidirect
91. 2₋¹⁺⁴	C₂×D₈₉	semidirect
103. #101	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152340	2
#19151771	2
#19151395	2
#19151394	2
#454×C₈₉	2
#2202	89