Finite Group Databasebeta

Subgroups — #26022230

Order 11392.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152038	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151873	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151414	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151405	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151382	C₂	1	1	yes	maximal, characteristic, complemented
7	#578×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668042	—	2	1	no	—
9	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	#15667958	—	2	2	no	—
11	#15667958	—	2	1	no	—
12	#15667956	—	2	1	no	—
13	#15667952	C₂²	1	1	yes	characteristic
14	C₂×#13083549	C₂²	1	1	yes	characteristic
15	#15667948	C₂²	1	1	yes	characteristic
16	#15667930	—	2	1	no	—
17	2₋¹⁺⁴×C₈₉	—	2	1	no	—
18	#137×C₈₉	—	2	2	no	—
19	#137×C₈₉	—	2	1	no	—
20	#135×C₈₉	—	2	1	no	—
21	#135×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₈₉	—	2	1	no	—
23	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	#131×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	2	1	no	—
26	Dic₃₅₆	—	2	1	no	—
27	Dic₃₅₆	—	2	1	no	—
28	#13083564	—	4	1	no	—
29	#13083562	—	4	1	no	—
30	#13083558	—	2	1	no	—
31	#13083549	—	2	1	no	—
32	#13083549	—	2	1	no	—
33	#41×C₈₉	—	2	2	no	—
34	#41×C₈₉	—	2	1	no	—
35	#41×C₈₉	—	4	2	no	—
36	#41×C₈₉	—	4	1	no	—
37	#41×C₈₉	D₄	1	1	yes	characteristic
38	C₂×Q₈×C₈₉	—	2	2	no	—
39	C₂×Q₈×C₈₉	—	2	1	no	—
40	C₂×Q₈×C₈₉	—	4	1	no	—
41	Q₁₆×C₈₉	—	2	2	no	—
42	Q₁₆×C₈₉	—	2	1	no	—
43	Q₁₆×C₈₉	D₄	1	1	yes	characteristic
44	Q₁₆×C₈₉	D₄	1	2	yes	—
45	SD₁₆×C₈₉	—	2	2	no	—
46	SD₁₆×C₈₉	—	2	1	no	—
47	D₈×C₈₉	D₄	1	1	yes	characteristic
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	—	2	2	no	—
53	Q₈×C₈₉	—	2	1	no	—
54	Q₈×C₈₉	—	4	2	no	—
55	Q₈×C₈₉	—	4	1	no	—
56	Q₈×C₈₉	D₈	1	1	yes	characteristic
57	D₄×C₈₉	—	2	1	no	—
58	D₄×C₈₉	—	2	1	no	—
59	D₄×C₈₉	—	4	2	no	—
60	D₄×C₈₉	—	4	1	no	—
61	D₄×C₈₉	D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	4	2	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	1	no	—
66	C₂×C₄×C₈₉	—	4	1	no	—
67	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₈₉	—	4	1	no	—
69	Dic₁₇₈	—	4	1	no	—
70	Dic₁₇₈	—	4	1	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
74	#10623494	—	4	1	no	—
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	—	4	1	no	—
77	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	—	4	2	no	—
80	C₄×C₈₉	—	4	1	no	—
81	C₄×C₈₉	#120	1	1	yes	characteristic
82	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
83	Dic₈₉	—	8	1	no	—
84	C₂×C₈₉	—	2	1	no	—
85	C₂×C₈₉	—	4	1	no	—
86	C₂×C₈₉	—	8	1	no	—
87	C₂×C₈₉	#447	1	1	yes	characteristic
88	#2195	—	89	1	no	maximal, Sylow
89	C₈₉	#2195	1	1	yes	characteristic, complemented, Sylow
90	#578	D₈₉	1	1	yes	characteristic
91	#510	—	89	1	no	—
92	C₂×Q₃₂	—	89	1	no	—
93	#473	—	89	1	no	—
94	#362	—	89	1	no	—
95	#359	—	89	1	no	—
96	#350	—	89	1	no	—
97	2₋¹⁺⁴	—	2	1	no	—
98	#137	—	2	2	no	—
99	#137	—	2	1	no	—
100	#137	—	178	1	no	—
101	#135	—	2	1	no	—
102	#135	C₂×D₈₉	1	1	yes	characteristic
103	C₂×Q₁₆	—	2	1	no	—
104	C₂×Q₁₆	—	89	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	—	89	1	no	—
113	#101	—	178	1	no	—
114	#41	—	2	2	no	—
115	#41	—	2	1	no	—
116	#41	—	4	2	no	—
117	#41	—	4	1	no	—
118	#41	C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	178	1	no	—
123	Q₁₆	—	2	2	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	356	1	no	—
126	Q₁₆	—	178	1	no	—
127	Q₁₆	—	178	1	no	—
128	Q₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
129	Q₁₆	C₂²⋊D₈₉	1	2	yes	—
130	SD₁₆	—	2	2	no	—
131	SD₁₆	—	2	1	no	—
132	SD₁₆	—	356	1	no	—
133	D₈	C₂²⋊D₈₉	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	178	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	—	2	2	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	356	1	no	—
146	Q₈	—	356	1	no	—
147	Q₈	#13083559	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	1	no	—
152	D₄	#13083559	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	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	—	356	1	no	—
162	C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#15668038	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	712	1	no	—
171	C₄	#15667981	1	1	yes	characteristic
172	C₄	C₂×#13083559	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	8	1	no	—
176	C₂	#19151732	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151732
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting, prime radical	#578×C₈₉	C₂
socle	C₂×C₈₉	#447
hypercenter	#578	D₈₉
hypocenter	C₈₉	#2195
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #19152038	C₂	semidirect
2. C₂×#15667958	C₂	semidirect
3. #19151873	C₂	semidirect
4. #19151414	C₂	semidirect
5. #19151405	C₂	semidirect
6. #19151382	C₂	semidirect
89. C₈₉	#2195	semidirect

Maximal subgroups

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

subgroup	index
#19152038	2
C₂×#15667958	2
#19151873	2
#19151414	2
#19151405	2
#19151382	2
#578×C₈₉	2
#2195	89