Finite Group Databasebeta

Subgroups — #26022255

Order 11392.

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

normal subgroups only

This group has 270 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	#19151786	C₂	1	2	yes	maximal, complemented
3	#19151775	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151384	C₂	1	2	yes	maximal, complemented
5	#354×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668057	C₂²	1	1	yes	characteristic
7	#15668043	—	2	1	no	—
8	#15668001	—	2	1	no	—
9	#15668000	—	2	1	no	—
10	#15667998	—	2	2	no	—
11	#15667982	—	2	2	no	—
12	#15667982	—	2	2	no	—
13	#15667980	C₂²	1	2	yes	—
14	#15667940	—	2	1	no	—
15	#15667929	C₂²	1	1	yes	characteristic
16	#15667926	—	2	2	no	—
17	#124×C₈₉	C₂²	1	1	yes	characteristic
18	#99×C₈₉	C₂²	1	2	yes	—
19	C₂³×D₈₉	D₄	1	2	yes	complemented
20	#13083574	—	2	1	no	—
21	#13083573	—	4	1	no	—
22	C₂×Dic₁₇₈	—	2	1	no	—
23	C₂×D₃₅₆	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	C₂×D₃₅₆	—	4	2	no	—
26	C₂×C₂²⋊D₈₉	—	2	2	no	—
27	C₂×C₂²⋊D₈₉	—	2	2	no	—
28	D₄×D₈₉	—	4	1	no	—
29	#13083564	—	4	1	no	—
30	#13083561	—	4	1	no	—
31	#13083558	—	2	1	no	—
32	C₂²⋊Dic₈₉	D₄	1	2	yes	complemented
33	#13083551	—	2	2	no	—
34	#13083551	—	2	2	no	—
35	#13083551	—	2	2	no	—
36	#13083551	—	4	2	no	—
37	C₄×Dic₈₉	—	2	1	no	—
38	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic, complemented
39	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²⋊C₄×C₈₉	—	2	2	no	—
41	C₂²⋊C₄×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	D₄	1	1	yes	characteristic, complemented
43	C₂³×C₈₉	C₂×D₄	1	2	yes	—
44	C₂²×D₈₉	—	2	2	no	—
45	C₂²×D₈₉	—	2	2	no	—
46	C₂²×D₈₉	—	4	2	no	—
47	C₂²×D₈₉	—	8	2	no	—
48	Q₈×C₈₉	—	4	1	no	—
49	D₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	4	2	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²⋊D₈₉	—	4	2	no	—
55	C₂²⋊D₈₉	—	4	1	no	—
56	C₂²⋊D₈₉	—	8	2	no	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	—	2	2	no	—
59	D₃₅₆	—	2	1	no	—
60	D₃₅₆	—	8	1	no	—
61	D₃₅₆	—	8	1	no	—
62	D₃₅₆	—	8	4	no	—
63	C₄×D₈₉	—	4	1	no	—
64	Dic₁₇₈	—	2	1	no	—
65	Dic₁₇₈	—	8	1	no	—
66	#10623494	—	4	1	no	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	—	4	2	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	4	no	—
73	C₂×D₈₉	—	4	1	no	—
74	C₂×D₈₉	—	4	2	no	—
75	C₂×D₈₉	—	8	2	no	—
76	C₂×D₈₉	—	8	2	no	—
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	—	4	1	no	—
79	C₄×C₈₉	—	4	1	no	—
80	C₄×C₈₉	—	8	2	no	—
81	Dic₈₉	—	4	2	no	—
82	Dic₈₉	—	4	1	no	—
83	C₂×C₈₉	—	2	1	no	—
84	C₂×C₈₉	—	4	2	no	—
85	C₂×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
86	D₈₉	—	8	2	no	—
87	D₈₉	—	8	1	no	—
88	#2199	—	89	1	no	maximal, Sylow
89	C₈₉	#2199	1	1	yes	characteristic, complemented, Sylow
90	H(3,𝔽₄)	—	89	1	no	—
91	B(4,𝔽₂)	—	89	2	no	—
92	#455	—	89	1	no	—
93	#354	D₈₉	1	1	yes	characteristic, complemented
94	AΣL(1,𝔽₁₆)	—	89	2	no	—
95	H(5,𝔽₂)	—	89	1	no	—
96	#137	—	178	1	no	—
97	#124	—	178	2	no	—
98	#124	—	178	1	no	—
99	#124	—	178	1	no	—
100	#124	C₂×D₈₉	1	1	yes	characteristic
101	#120	—	89	2	no	—
102	#120	—	178	2	no	—
103	#120	—	178	2	no	—
104	#104	—	178	1	no	—
105	#100	—	89	1	no	—
106	#99	—	178	2	no	—
107	#99	C₂×D₈₉	1	2	yes	—
108	C₂⁴	—	89	2	no	—
109	#41	—	356	1	no	—
110	#41	—	178	1	no	—
111	C₂×Q₈	—	178	1	no	—
112	C₂×Q₈	D₃₅₆	1	1	yes	characteristic, complemented
113	C₂×D₄	—	356	2	no	—
114	C₂×D₄	—	356	1	no	—
115	C₂×D₄	—	178	1	no	—
116	C₂×D₄	—	178	1	no	—
117	C₂×D₄	—	178	2	no	—
118	C₂×D₄	—	178	2	no	—
119	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
120	Q₁₆	—	356	1	no	—
121	SD₁₆	—	356	1	no	—
122	M₁₆	—	178	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	89	2	no	—
126	C₂²⋊C₄	—	356	2	no	—
127	C₂²⋊C₄	—	178	2	no	—
128	C₂²⋊C₄	—	178	2	no	—
129	C₂²⋊C₄	—	178	2	no	—
130	C₄²	—	178	1	no	—
131	C₄²	D₃₅₆	1	1	yes	characteristic, complemented
132	C₂³	—	356	2	no	—
133	C₂³	—	178	2	no	—
134	C₂³	—	178	2	no	—
135	C₂³	—	712	2	no	—
136	C₂³	D₄×D₈₉	1	2	yes	—
137	Q₈	—	4	1	no	—
138	Q₈	—	178	1	no	—
139	Q₈	—	712	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	356	2	no	—
142	D₄	—	356	1	no	—
143	D₄	—	178	1	no	—
144	D₄	—	712	1	no	—
145	D₄	—	712	1	no	—
146	D₄	—	712	4	no	—
147	D₄	—	712	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	356	1	no	—
152	C₂×C₄	—	178	2	no	—
153	C₂×C₄	—	178	2	no	—
154	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
155	C₈	—	356	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	356	2	no	—
159	C₂²	—	356	2	no	—
160	C₂²	—	356	4	no	—
161	C₂²	—	356	1	no	—
162	C₂²	—	356	2	no	—
163	C₂²	—	712	2	no	—
164	C₂²	—	712	2	no	—
165	C₂²	#15667982	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	8	2	no	—
170	C₄	—	356	2	no	—
171	C₄	—	356	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	4	2	no	—
174	C₂	—	712	2	no	—
175	C₂	—	712	1	no	—
176	C₂	#19151786	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151786
commutator, cosocle	C₂×D₄×C₈₉	C₂³
Frattini	C₂×D₄	C₂²×D₈₉
Fitting	#354×C₈₉	C₂
socle	C₂×C₈₉	B(4,𝔽₂)
hypercenter	#354	D₈₉
hypocenter	C₈₉	#2199
LCS	G ▷ C₂×D₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #354

Complements

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

N	H ≅ G/N	type
1. #19152340	C₂	semidirect
2. #19151786	C₂	semidirect
3. #19151775	C₂	semidirect
4. #19151384	C₂	semidirect
5. #354×C₈₉	C₂	semidirect
19. C₂³×D₈₉	D₄	semidirect
32. C₂²⋊Dic₈₉	D₄	semidirect
38. C₂×Q₈×C₈₉	D₄	semidirect
42. C₄²×C₈₉	D₄	semidirect
89. C₈₉	#2199	semidirect
93. #354	D₈₉	semidirect
112. C₂×Q₈	D₃₅₆	semidirect
131. C₄²	D₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#19152340	2
#19151786	2
#19151775	2
#19151384	2
#354×C₈₉	2
#2199	89