Finite Group Databasebeta

Subgroups — #26022228

Order 11392.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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	#19152415	C₂	1	1	yes	maximal, characteristic
2	#19152039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151413	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151404	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151382	C₂	1	1	yes	maximal, characteristic, complemented
7	#473×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668059	—	2	1	no	—
9	#15668045	—	2	2	no	—
10	#15668045	—	2	1	no	—
11	#15668034	—	2	1	no	—
12	#15668034	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₅₆	—	2	1	no	—
14	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
15	#15667958	—	2	2	no	—
16	#15667958	—	2	1	no	—
17	#15667955	—	2	1	no	—
18	#15667952	C₂²	1	1	yes	characteristic
19	C₂×#13083549	C₂²	1	1	yes	characteristic
20	#137×C₈₉	—	2	1	no	—
21	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#131×C₈₉	C₂²	1	1	yes	characteristic
23	#108×C₈₉	C₂²	1	1	yes	characteristic
24	#101×C₈₉	—	2	1	no	—
25	#13083574	—	2	2	no	—
26	#13083574	—	2	1	no	—
27	#13083573	—	4	2	no	—
28	#13083573	—	4	1	no	—
29	C₂×Dic₁₇₈	—	2	2	no	—
30	C₂×Dic₁₇₈	—	2	1	no	—
31	Q₈×D₈₉	—	4	1	no	—
32	Dic₃₅₆	—	2	2	no	—
33	Dic₃₅₆	—	2	1	no	—
34	Dic₃₅₆	D₄	1	1	yes	characteristic
35	Dic₃₅₆	D₄	1	2	yes	—
36	#13083563	—	2	2	no	—
37	#13083563	—	2	1	no	—
38	D₇₁₂	D₄	1	1	yes	characteristic
39	#13083549	—	2	1	no	—
40	#13083549	—	2	1	no	—
41	#41×C₈₉	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₈₉	—	2	1	no	—
43	Q₁₆×C₈₉	—	2	1	no	—
44	Q₁₆×C₈₉	—	2	1	no	—
45	Q₁₆×C₈₉	—	4	1	no	—
46	SD₁₆×C₈₉	—	4	1	no	—
47	M₁₆×C₈₉	—	2	1	no	—
48	M₁₆×C₈₉	—	2	1	no	—
49	M₁₆×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	2	1	no	—
51	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	Q₈×C₈₉	—	4	1	no	—
53	Q₈×C₈₉	—	4	1	no	—
54	Q₈×C₈₉	D₈	1	1	yes	characteristic
55	D₄×C₈₉	—	2	1	no	—
56	D₄×C₈₉	D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	4	1	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₈₉	—	4	2	no	—
61	C₂²⋊D₈₉	—	4	1	no	—
62	C₂×Dic₈₉	—	4	2	no	—
63	C₂×Dic₈₉	—	4	1	no	—
64	D₃₅₆	—	2	1	no	—
65	C₄×D₈₉	—	4	2	no	—
66	C₄×D₈₉	—	4	1	no	—
67	Dic₁₇₈	—	2	2	no	—
68	Dic₁₇₈	—	2	1	no	—
69	Dic₁₇₈	—	4	2	no	—
70	Dic₁₇₈	—	4	1	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	—	4	1	no	—
73	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₈₉	—	4	1	no	—
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	—	8	1	no	—
80	C₄×C₈₉	#120	1	1	yes	characteristic
81	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
82	Dic₈₉	—	4	1	no	—
83	Dic₈₉	—	4	2	no	—
84	C₂×C₈₉	—	2	1	no	—
85	C₂×C₈₉	—	4	1	no	—
86	C₂×C₈₉	#447	1	1	yes	characteristic
87	D₈₉	—	8	1	no	—
88	#2195	—	89	1	no	maximal, Sylow
89	C₈₉	#2195	1	1	yes	characteristic, complemented, Sylow
90	#578	—	89	1	no	—
91	#510	—	89	1	no	—
92	C₂×Q₃₂	—	89	1	no	—
93	#473	D₈₉	1	1	yes	characteristic, complemented
94	#362	—	89	1	no	—
95	#359	—	89	1	no	—
96	#350	—	89	1	no	—
97	2₋¹⁺⁴	—	178	1	no	—
98	#137	—	2	1	no	—
99	#137	—	178	2	no	—
100	#137	—	178	1	no	—
101	#135	—	89	1	no	—
102	#135	—	178	1	no	—
103	C₂×Q₁₆	—	89	1	no	—
104	C₂×Q₁₆	—	178	1	no	—
105	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
106	#131	C₂×D₈₉	1	1	yes	characteristic
107	Q₃₂	—	178	2	no	—
108	Q₃₂	—	178	1	no	—
109	SD₃₂	—	178	1	no	—
110	M₃₂	—	89	1	no	—
111	C₂×C₁₆	—	89	1	no	—
112	#108	C₂×D₈₉	1	1	yes	characteristic
113	#101	—	2	1	no	—
114	#41	—	356	2	no	—
115	#41	—	356	1	no	—
116	#41	—	178	2	no	—
117	#41	—	178	1	no	—
118	#41	C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	356	1	no	—
121	C₂×Q₈	—	178	2	no	—
122	C₂×Q₈	—	178	1	no	—
123	Q₁₆	—	2	1	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	4	1	no	—
126	Q₁₆	—	89	2	no	—
127	Q₁₆	—	89	1	no	—
128	Q₁₆	—	178	2	no	—
129	Q₁₆	—	178	1	no	—
130	SD₁₆	—	4	1	no	—
131	SD₁₆	—	178	2	no	—
132	SD₁₆	—	178	1	no	—
133	D₈	—	89	1	no	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
139	C₁₆	—	178	1	no	—
140	C₁₆	—	178	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	356	2	no	—
144	Q₈	—	356	1	no	—
145	Q₈	—	178	2	no	—
146	Q₈	—	178	1	no	—
147	Q₈	#13083559	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	356	2	no	—
150	D₄	—	356	1	no	—
151	D₄	—	178	1	no	—
152	D₄	#13083559	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	356	2	no	—
156	C₂×C₄	—	356	2	no	—
157	C₂×C₄	—	356	1	no	—
158	C₂×C₄	—	356	1	no	—
159	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	4	1	no	—
162	C₈	D₄×D₈₉	1	1	yes	characteristic
163	C₈	D₄×D₈₉	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	356	1	no	—
166	C₂²	#15668037	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	8	1	no	—
169	C₄	—	356	2	no	—
170	C₄	—	356	1	no	—
171	C₄	#15667980	1	1	yes	characteristic
172	C₄	C₂×#13083559	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	712	1	no	—
176	C₂	#19151731	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151731
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#473×C₈₉	C₂
socle	C₂×C₈₉	#447
prime radical	#19152415	C₂
hypercenter	#473	D₈₉
hypocenter	C₈₉	#2195
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #473

Complements

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

N	H ≅ G/N	type
2. #19152039	C₂	semidirect
3. C₂×#15667958	C₂	semidirect
4. #19151413	C₂	semidirect
5. #19151404	C₂	semidirect
6. #19151382	C₂	semidirect
7. #473×C₈₉	C₂	semidirect
89. C₈₉	#2195	semidirect
93. #473	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152415	2
#19152039	2
C₂×#15667958	2
#19151413	2
#19151404	2
#19151382	2
#473×C₈₉	2
#2195	89