Finite Group Databasebeta

Subgroups — #26019917

Order 11392.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19151974	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151974	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151970	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151945	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151944	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151320	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083562	—	2	2	no	—
9	C₂×#13083559	—	2	2	no	—
10	C₂×#13083559	—	2	2	no	—
11	#15668012	C₂²	1	1	yes	characteristic
12	#15668009	C₂²	1	1	yes	characteristic
13	#15667934	—	2	2	no	—
14	#15667933	—	2	2	no	—
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₈₉	—	2	2	no	—
18	#127×C₈₉	C₂²	1	1	yes	characteristic
19	#127×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×D₃₅₆	—	2	1	no	—
23	C₂×D₃₅₆	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	#13083562	—	4	4	no	—
26	#13083559	—	4	4	no	—
27	#13083559	—	4	4	no	—
28	C₂×#10623494	D₄	1	2	yes	—
29	C₂×#10623494	D₄	1	2	yes	—
30	C₄⋊Dic₈₉	—	2	1	no	—
31	C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×D₄×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	1	no	—
34	C₂×D₄×C₈₉	—	2	1	no	—
35	C₂×D₄×C₈₉	—	2	1	no	—
36	C₂×D₄×C₈₉	—	2	1	no	—
37	C₂×D₄×C₈₉	—	2	1	no	—
38	D₈×C₈₉	—	4	4	no	—
39	C₂×C₈×C₈₉	D₄	1	2	yes	—
40	C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₂³×C₈₉	—	4	1	no	—
42	C₂³×C₈₉	—	4	1	no	—
43	C₂²×D₈₉	—	4	1	no	—
44	D₄×C₈₉	—	4	2	no	—
45	D₄×C₈₉	—	4	2	no	—
46	D₄×C₈₉	—	4	2	no	—
47	D₄×C₈₉	—	4	2	no	—
48	D₄×C₈₉	—	4	2	no	—
49	D₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₈₉	—	4	1	no	—
54	D₃₅₆	—	4	2	no	—
55	D₃₅₆	—	4	2	no	—
56	D₃₅₆	—	4	2	no	—
57	Dic₁₇₈	—	4	2	no	—
58	C₈×C₈₉	—	2	2	no	—
59	#10623494	—	2	2	no	—
60	#10623494	D₈	1	4	yes	complemented
61	C₂²×C₈₉	—	8	1	no	—
62	C₂²×C₈₉	—	8	1	no	—
63	C₂²×C₈₉	—	8	1	no	—
64	C₂²×C₈₉	—	8	1	no	—
65	C₂²×C₈₉	—	8	1	no	—
66	C₂²×C₈₉	—	8	1	no	—
67	C₂²×C₈₉	#127	1	1	yes	characteristic
68	C₂×D₈₉	—	8	1	no	—
69	C₂×D₈₉	—	8	1	no	—
70	C₂×D₈₉	—	8	1	no	—
71	C₄×C₈₉	C₂×D₈	1	2	yes	—
72	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₈₉	—	8	1	no	—
75	C₂×C₈₉	—	16	1	no	—
76	C₂×C₈₉	—	16	1	no	—
77	C₂×C₈₉	#493	1	1	yes	characteristic
78	C₂×C₈₉	#496	1	1	yes	characteristic
79	C₂×C₈₉	#496	1	1	yes	characteristic
80	D₈₉	—	16	1	no	—
81	#1721	—	89	1	no	maximal, Sylow
82	C₈₉	#1721	1	1	yes	characteristic, complemented, Sylow
83	#493	—	89	1	no	—
84	#493	—	89	1	no	—
85	#493	D₈₉	1	1	yes	characteristic, complemented
86	#492	—	89	1	no	—
87	#486	—	89	1	no	—
88	#486	—	89	1	no	—
89	#322	—	89	1	no	—
90	C₂×SD₁₆	—	178	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	178	2	no	—
93	C₂×D₈	—	178	2	no	—
94	C₄⋊Q₈	—	89	1	no	—
95	#127	—	89	1	no	—
96	#127	C₂×D₈₉	1	1	yes	characteristic
97	#127	C₂×D₈₉	1	1	yes	characteristic
98	#102	—	178	2	no	—
99	#102	—	178	2	no	—
100	C₄×C₈	—	89	1	no	—
101	C₄×C₈	—	89	1	no	—
102	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
103	C₂×Q₈	—	178	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	178	1	no	—
111	C₂×D₄	—	178	1	no	—
112	C₂×D₄	—	178	1	no	—
113	SD₁₆	—	356	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	356	4	no	—
116	D₈	—	356	4	no	—
117	C₂×C₈	—	89	2	no	—
118	C₂×C₈	—	89	2	no	—
119	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
120	C₄⋊C₄	—	178	1	no	—
121	C₄⋊C₄	—	178	1	no	—
122	C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	356	1	no	—
126	Q₈	—	356	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	356	2	no	—
134	D₄	—	356	2	no	—
135	D₄	—	356	2	no	—
136	C₂×C₄	—	356	1	no	—
137	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	89	4	no	—
142	C₈	—	178	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	712	1	no	—
150	C₂²	—	712	1	no	—
151	C₂²	—	712	1	no	—
152	C₂²	#15668008	1	1	yes	characteristic
153	C₄	—	712	1	no	—
154	C₄	D₈×D₈₉	1	2	yes	—
155	C₄	#15668036	1	2	yes	—
156	C₄	#15668038	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	1424	1	no	—
160	C₂	#19151973	1	1	yes	characteristic
161	C₂	#19151985	1	1	yes	characteristic
162	C₂	#19151986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#493×C₈₉	C₂
socle	C₂²×C₈₉	#127
hypercenter	#493	D₈₉
hypocenter	C₈₉	#1721
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #19151974	C₂	semidirect
2. #19151974	C₂	semidirect
3. #19151970	C₂	semidirect
4. #19151945	C₂	semidirect
5. #19151944	C₂	semidirect
6. #19151320	C₂	semidirect
7. #493×C₈₉	C₂	semidirect
60. #10623494	D₈	semidirect
82. C₈₉	#1721	semidirect
85. #493	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151974	2
#19151970	2
#19151945	2
#19151944	2
#19151320	2
#493×C₈₉	2
#1721	89