Finite Group Databasebeta

Subgroups — #26022213

Order 11392.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19152412	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151865	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151415	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151405	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151383	C₂	1	1	yes	maximal, characteristic, complemented
7	#509×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668059	—	2	1	no	—
9	#15668042	—	2	1	no	—
10	#15668041	—	2	2	no	—
11	#15668038	—	2	1	no	—
12	#15668031	—	2	1	no	—
13	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
14	Q₁₆×D₈₉	—	2	1	no	—
15	#15668018	C₂²	1	1	yes	characteristic
16	#15667956	—	2	2	no	—
17	C₂×#13083549	C₂²	1	1	yes	characteristic
18	#15667948	C₂²	1	1	yes	characteristic
19	#15667929	—	2	1	no	—
20	#135×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
22	SD₃₂×C₈₉	—	2	1	no	—
23	D₁₆×C₈₉	—	2	1	no	—
24	M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	#13083574	—	4	1	no	—
26	#13083574	D₄	1	1	yes	characteristic
27	#13083573	—	2	2	no	—
28	#13083573	—	4	2	no	—
29	C₂×Dic₁₇₈	—	2	1	no	—
30	C₂×Dic₁₇₈	—	4	1	no	—
31	Q₈×D₈₉	—	2	2	no	—
32	Dic₃₅₆	—	2	1	no	—
33	Dic₃₅₆	D₄	1	2	yes	—
34	#13083564	—	2	2	no	—
35	#13083562	—	2	2	no	—
36	#13083562	—	4	1	no	—
37	#13083559	—	4	1	no	—
38	#13083558	—	2	1	no	—
39	#13083558	D₄	1	1	yes	characteristic
40	#13083557	—	2	1	no	—
41	C₈×D₈₉	—	2	1	no	—
42	#13083549	—	2	1	no	—
43	#41×C₈₉	—	2	1	no	—
44	C₂×D₄×C₈₉	—	2	1	no	—
45	Q₁₆×C₈₉	D₄	1	1	yes	characteristic
46	SD₁₆×C₈₉	—	2	1	no	—
47	D₈×C₈₉	—	2	1	no	—
48	D₈×C₈₉	—	2	1	no	—
49	D₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₁₆×C₈₉	—	2	1	no	—
52	C₂³×C₈₉	—	4	1	no	—
53	Q₈×C₈₉	—	2	1	no	—
54	D₄×C₈₉	—	2	1	no	—
55	D₄×C₈₉	—	4	1	no	—
56	D₄×C₈₉	—	4	1	no	—
57	D₄×C₈₉	—	4	1	no	—
58	C₂×C₄×C₈₉	—	4	1	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₈₉	—	2	1	no	—
61	C₂²⋊D₈₉	—	4	2	no	—
62	C₂×Dic₈₉	—	4	2	no	—
63	C₂×Dic₈₉	—	4	1	no	—
64	D₃₅₆	D₈	1	1	yes	characteristic
65	C₄×D₈₉	—	2	1	no	—
66	C₄×D₈₉	—	4	2	no	—
67	Dic₁₇₈	—	2	2	no	—
68	Dic₁₇₈	—	4	2	no	—
69	Dic₁₇₈	—	4	1	no	—
70	Dic₁₇₈	D₈	1	1	yes	characteristic
71	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
73	#10623494	—	2	1	no	—
74	#10623494	—	4	1	no	—
75	C₂²×C₈₉	—	4	1	no	—
76	C₂²×C₈₉	—	8	1	no	—
77	C₂²×C₈₉	—	16	1	no	—
78	C₂²×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₈₉	—	2	1	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	—	2	1	no	—
84	Dic₈₉	—	4	2	no	—
85	C₂×C₈₉	—	2	1	no	—
86	C₂×C₈₉	—	8	1	no	—
87	C₂×C₈₉	—	16	1	no	—
88	C₂×C₈₉	#447	1	1	yes	characteristic
89	D₈₉	—	4	1	no	—
90	#2193	—	89	1	no	maximal, Sylow
91	C₈₉	#2193	1	1	yes	characteristic, complemented, Sylow
92	#578	—	89	1	no	—
93	#509	D₈₉	1	1	yes	characteristic, complemented
94	C₂×SD₃₂	—	89	1	no	—
95	#472	—	89	1	no	—
96	#362	—	89	1	no	—
97	#359	—	89	1	no	—
98	#350	—	89	1	no	—
99	2₋¹⁺⁴	—	178	1	no	—
100	#137	—	178	1	no	—
101	#137	—	178	2	no	—
102	AGL(1,ℤ/8)	—	178	1	no	—
103	#135	—	178	1	no	—
104	#135	C₂×D₈₉	1	1	yes	characteristic
105	C₂×Q₁₆	—	89	1	no	—
106	C₂×Q₁₆	—	178	1	no	—
107	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
108	#131	—	89	1	no	—
109	SD₃₂	—	2	1	no	—
110	SD₃₂	—	178	2	no	—
111	D₁₆	—	2	1	no	—
112	M₃₂	C₂×D₈₉	1	1	yes	characteristic
113	C₂×C₁₆	—	89	1	no	—
114	#108	—	89	1	no	—
115	#100	—	178	1	no	—
116	#41	—	2	1	no	—
117	#41	—	89	1	no	—
118	#41	—	356	1	no	—
119	#41	—	356	2	no	—
120	#41	—	178	2	no	—
121	C₂×Q₈	—	356	1	no	—
122	C₂×Q₈	—	178	2	no	—
123	C₂×Q₈	—	178	1	no	—
124	C₂×D₄	—	2	1	no	—
125	Q₁₆	—	89	2	no	—
126	Q₁₆	—	178	2	no	—
127	Q₁₆	—	178	1	no	—
128	Q₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	356	1	no	—
131	SD₁₆	—	178	2	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	356	1	no	—
135	D₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	M₁₆	—	89	1	no	—
137	M₁₆	—	178	1	no	—
138	M₁₆	—	178	1	no	—
139	C₂×C₈	—	178	1	no	—
140	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	178	1	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	89	1	no	—
146	Q₈	—	356	2	no	—
147	Q₈	—	356	1	no	—
148	Q₈	—	178	2	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	89	1	no	—
154	D₄	—	356	2	no	—
155	D₄	—	178	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	356	2	no	—
158	C₂×C₄	—	356	1	no	—
159	C₂×C₄	—	356	2	no	—
160	C₂×C₄	—	178	1	no	—
161	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
162	C₈	—	356	1	no	—
163	C₈	—	178	1	no	—
164	C₈	D₄×D₈₉	1	1	yes	characteristic
165	C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
166	C₂²	—	4	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	178	1	no	—
170	C₂²	#15668036	1	1	yes	characteristic
171	C₄	—	4	1	no	—
172	C₄	—	356	2	no	—
173	C₄	—	178	1	no	—
174	C₄	#15667980	1	1	yes	characteristic
175	C₄	D₈×D₈₉	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	—	16	1	no	—
179	C₂	—	356	1	no	—
180	C₂	#19151730	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151730
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#509×C₈₉	C₂
socle	C₂×C₈₉	#447
hypercenter	#509	D₈₉
hypocenter	C₈₉	#2193
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #19152412	C₂	semidirect
2. C₂×#15667956	C₂	semidirect
3. #19151865	C₂	semidirect
4. #19151415	C₂	semidirect
5. #19151405	C₂	semidirect
6. #19151383	C₂	semidirect
7. #509×C₈₉	C₂	semidirect
91. C₈₉	#2193	semidirect
93. #509	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152412	2
C₂×#15667956	2
#19151865	2
#19151415	2
#19151405	2
#19151383	2
#509×C₈₉	2
#2193	89