Finite Group Databasebeta

Subgroups — #26019786

Order 11392.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19151945	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151889	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151794	C₂	1	1	yes	maximal, characteristic
4	#19151677	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151346	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151341	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇₁₂	—	2	1	no	—
9	C₂×#13083557	—	2	1	no	—
10	#15668012	C₂²	1	1	yes	characteristic
11	#15668009	C₂²	1	1	yes	characteristic
12	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
13	#15667944	—	2	1	no	—
14	C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	#15667934	—	2	1	no	—
16	#15667934	—	2	1	no	—
17	#15667934	—	2	1	no	—
18	#15667934	—	2	1	no	—
19	#15667924	—	2	1	no	—
20	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	C₂×D₃₅₆	—	2	1	no	—
26	C₂×D₃₅₆	—	2	1	no	—
27	C₂×D₃₅₆	D₄	1	1	yes	characteristic
28	C₂×C₄×D₈₉	—	2	1	no	—
29	D₇₁₂	—	4	2	no	—
30	#13083557	—	4	2	no	—
31	C₂×#10623494	—	2	1	no	—
32	C₄⋊Dic₈₉	—	2	1	no	—
33	C₄⋊Dic₈₉	—	2	1	no	—
34	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	#13083551	—	2	1	no	—
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×D₈₉	—	2	1	no	—
42	C₂²×D₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×Dic₈₉	—	2	1	no	—
47	C₂×Dic₈₉	—	4	1	no	—
48	D₃₅₆	—	2	1	no	—
49	D₃₅₆	—	2	1	no	—
50	D₃₅₆	—	4	1	no	—
51	D₃₅₆	—	4	1	no	—
52	D₃₅₆	—	4	1	no	—
53	D₃₅₆	—	4	1	no	—
54	D₃₅₆	—	8	1	no	—
55	C₄×D₈₉	—	4	1	no	—
56	C₄×D₈₉	—	4	1	no	—
57	Dic₁₇₈	—	8	1	no	—
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	—	2	1	no	—
60	C₈×C₈₉	—	4	1	no	—
61	C₈×C₈₉	—	4	1	no	—
62	#10623494	—	4	1	no	—
63	C₂²×C₈₉	#121	1	1	yes	characteristic
64	C₂×D₈₉	—	4	1	no	—
65	C₂×D₈₉	—	4	1	no	—
66	C₂×D₈₉	—	4	1	no	—
67	C₂×D₈₉	—	8	1	no	—
68	C₂×D₈₉	—	8	1	no	—
69	C₂×D₈₉	—	8	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	#135	1	1	yes	characteristic
72	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₈₉	#137	1	1	yes	characteristic
75	Dic₈₉	—	4	1	no	—
76	Dic₈₉	—	8	1	no	—
77	C₂×C₈₉	#463	1	1	yes	characteristic
78	C₂×C₈₉	#468	1	1	yes	characteristic
79	C₂×C₈₉	#472	1	1	yes	characteristic
80	D₈₉	—	8	1	no	—
81	D₈₉	—	16	1	no	—
82	#1697	—	89	1	no	maximal, Sylow
83	C₈₉	#1697	1	1	yes	characteristic, complemented, Sylow
84	#486	—	89	1	no	—
85	#476	—	89	1	no	—
86	#459	—	89	1	no	—
87	#436	—	89	1	no	—
88	#335	D₈₉	1	1	yes	characteristic, complemented
89	#333	—	89	1	no	—
90	#331	—	89	1	no	—
91	C₂×D₈	—	178	1	no	—
92	C₂×M₁₆	—	178	1	no	—
93	C₄⋊Q₈	—	89	1	no	—
94	#127	—	89	1	no	—
95	C₄×D₄	—	89	1	no	—
96	#106	—	178	1	no	—
97	C₄⋊C₈	—	89	1	no	—
98	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
100	#102	—	178	1	no	—
101	#102	—	178	1	no	—
102	#102	—	178	1	no	—
103	#102	—	178	1	no	—
104	C₂²⋊C₈	—	178	1	no	—
105	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
106	C₂×Q₈	—	178	1	no	—
107	C₂×D₄	—	89	1	no	—
108	C₂×D₄	—	178	1	no	—
109	C₂×D₄	—	178	1	no	—
110	C₂×D₄	—	178	1	no	—
111	C₂²×C₄	—	178	1	no	—
112	D₈	—	356	2	no	—
113	M₁₆	—	356	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	178	1	no	—
117	C₂×C₈	D₃₅₆	1	1	yes	characteristic
118	C₂×C₈	D₃₅₆	1	1	yes	characteristic
119	C₄⋊C₄	—	89	1	no	—
120	C₄⋊C₄	—	178	1	no	—
121	C₄⋊C₄	—	178	1	no	—
122	C₂²⋊C₄	—	178	1	no	—
123	C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	C₂³	—	356	1	no	—
125	C₂³	—	178	1	no	—
126	Q₈	—	712	1	no	—
127	D₄	—	356	1	no	—
128	D₄	—	356	1	no	—
129	D₄	—	356	1	no	—
130	D₄	—	356	1	no	—
131	D₄	—	178	1	no	—
132	D₄	—	178	1	no	—
133	D₄	—	712	1	no	—
134	C₂×C₄	—	356	1	no	—
135	C₂×C₄	—	356	1	no	—
136	C₂×C₄	—	356	1	no	—
137	C₂×C₄	—	178	1	no	—
138	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
140	C₂×C₄	#13083575	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	356	1	no	—
146	C₂²	—	356	1	no	—
147	C₂²	—	356	1	no	—
148	C₂²	—	356	1	no	—
149	C₂²	—	712	1	no	—
150	C₂²	—	712	1	no	—
151	C₂²	—	712	1	no	—
152	C₂²	#15667987	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	356	1	no	—
155	C₄	—	712	1	no	—
156	C₄	#15668034	1	1	yes	characteristic
157	C₄	#15668036	1	1	yes	characteristic
158	C₄	#15668040	1	1	yes	characteristic
159	C₄	#15668044	1	1	yes	characteristic
160	C₂	—	1424	1	no	—
161	C₂	—	712	1	no	—
162	C₂	#19151814	1	1	yes	characteristic
163	C₂	#19151845	1	1	yes	characteristic
164	C₂	#19151869	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#335×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	#19151794	C₂
hypercenter	#335	D₈₉
hypocenter	C₈₉	#1697
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
1. #19151945	C₂	semidirect
2. #19151889	C₂	semidirect
4. #19151677	C₂	semidirect
5. #19151346	C₂	semidirect
6. #19151341	C₂	semidirect
7. #335×C₈₉	C₂	semidirect
83. C₈₉	#1697	semidirect
88. #335	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151945	2
#19151889	2
#19151794	2
#19151677	2
#19151346	2
#19151341	2
#335×C₈₉	2
#1697	89