Finite Group Databasebeta

Subgroups — #26030783

Order 11392.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19152226	C₂	1	2	yes	maximal
2	#19152007	C₂	1	4	yes	maximal, complemented
3	#19152000	C₂	1	2	yes	maximal, complemented
4	#19151996	C₂	1	2	yes	maximal, complemented
5	#19151640	C₂	1	2	yes	maximal
6	#19151636	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667944	C₂	1	1	yes	maximal, characteristic
8	#405×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#15668012	C₂²	1	4	yes	—
10	#15668011	—	2	2	no	—
11	#15668004	C₂²	1	4	yes	—
12	#15668003	—	2	2	no	—
13	#15667989	—	2	4	no	—
14	#15667971	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	#15667946	C₂²	1	2	yes	—
17	#15667946	C₂²	1	4	yes	—
18	#15667944	C₂²	1	2	yes	—
19	#15667944	C₂²	1	2	yes	—
20	#15667944	C₂²	1	1	yes	characteristic
21	#15667944	C₂²	1	1	yes	characteristic
22	#15667944	C₂²	1	4	yes	—
23	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#117×C₈₉	C₂²	1	1	yes	characteristic
26	#97×C₈₉	C₂²	1	2	yes	—
27	C₄×C₈×C₈₉	C₂²	1	2	yes	—
28	C₂×Dic₁₇₈	—	2	4	no	—
29	C₂²×Dic₈₉	—	2	2	no	—
30	C₄⋊Dic₈₉	—	2	4	no	—
31	C₄⋊Dic₈₉	—	2	2	no	—
32	C₄⋊Dic₈₉	C₂³	1	2	yes	—
33	C₄⋊Dic₈₉	C₂³	1	2	yes	—
34	C₄⋊Dic₈₉	C₂³	1	4	yes	—
35	#13083553	—	2	4	no	—
36	#13083553	—	2	4	no	—
37	C₂²⋊Dic₈₉	—	2	2	no	—
38	C₄×Dic₈₉	—	2	2	no	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₈₉	Q₈	1	4	yes	complemented
41	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
42	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
46	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
47	C₄²×C₈₉	C₂³	1	2	yes	—
48	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₈₉	—	2	2	no	—
55	C₂×Dic₈₉	—	2	2	no	—
56	C₂×Dic₈₉	—	2	4	no	—
57	C₂×Dic₈₉	—	8	2	no	—
58	Dic₁₇₈	—	8	4	no	—
59	C₈×C₈₉	C₂×Q₈	1	2	yes	—
60	C₈×C₈₉	C₂×Q₈	1	2	yes	—
61	C₈×C₈₉	C₂×Q₈	1	4	yes	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₈₉	—	2	4	no	—
66	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
67	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
69	Dic₈₉	—	4	2	no	—
70	Dic₈₉	—	4	4	no	—
71	Dic₈₉	—	4	2	no	—
72	C₂×C₈₉	—	2	2	no	—
73	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₈₉	#577	1	2	yes	—
75	#3163	—	89	1	no	maximal, Sylow
76	C₈₉	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	89	2	no	—
78	#501	—	89	4	no	—
79	#499	—	89	2	no	—
80	#498	—	89	2	no	—
81	#428	—	89	2	no	—
82	#427	—	89	1	no	—
83	C₂×#106	—	89	1	no	—
84	#405	D₈₉	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
87	C₄⋊Q₈	—	89	4	no	—
88	C₄⋊Q₈	—	178	2	no	—
89	#125	—	89	4	no	—
90	#125	—	178	2	no	—
91	C₂²⋊Q₈	—	178	4	no	—
92	#117	—	89	2	no	—
93	#117	C₂×D₈₉	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	89	2	no	—
95	#107	—	89	2	no	—
96	#107	—	89	4	no	—
97	#106	—	89	2	no	—
98	#106	—	89	2	no	—
99	#106	—	89	1	no	—
100	#106	—	89	1	no	—
101	#106	—	89	4	no	—
102	#97	C₂×D₈₉	1	2	yes	—
103	C₄×C₈	C₂×D₈₉	1	2	yes	—
104	C₂×Q₈	—	178	4	no	—
105	C₂²×C₄	—	178	2	no	—
106	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
107	M₁₆	Dic₁₇₈	1	4	yes	complemented
108	C₂×C₈	Dic₁₇₈	1	4	yes	complemented
109	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₈₉	1	2	yes	—
111	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	C₄⋊C₄	—	89	4	no	—
113	C₄⋊C₄	—	89	2	no	—
114	C₄⋊C₄	—	89	2	no	—
115	C₄⋊C₄	—	178	2	no	—
116	C₄⋊C₄	—	178	4	no	—
117	C₄⋊C₄	—	178	4	no	—
118	C₄⋊C₄	—	178	4	no	—
119	C₄⋊C₄	D₃₅₆	1	2	yes	—
120	C₂²⋊C₄	—	178	2	no	—
121	C₂²⋊C₄	D₃₅₆	1	2	yes	—
122	C₄²	—	178	2	no	—
123	C₄²	C₂²×D₈₉	1	2	yes	—
124	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
125	Q₈	—	712	4	no	—
126	C₂×C₄	—	178	2	no	—
127	C₂×C₄	—	178	2	no	—
128	C₂×C₄	—	178	4	no	—
129	C₂×C₄	—	712	2	no	—
130	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
131	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
133	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
134	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
135	C₈	C₂×Dic₁₇₈	1	2	yes	—
136	C₈	C₂×Dic₁₇₈	1	2	yes	—
137	C₈	C₂×Dic₁₇₈	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#15668012	1	2	yes	—
140	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	356	4	no	—
143	C₄	—	356	2	no	—
144	C₄	—	356	2	no	—
145	C₄	#15668012	1	2	yes	—
146	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
147	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#15668012	1	1	yes	characteristic
150	C₂	#19152410	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#405	D₈₉
hypocenter	C₈₉	#3163
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19152007	C₂	semidirect
3. #19152000	C₂	semidirect
4. #19151996	C₂	semidirect
40. M₁₆×C₈₉	Q₈	semidirect
41. C₂×C₈×C₈₉	Q₈	semidirect
76. C₈₉	#3163	semidirect
107. M₁₆	Dic₁₇₈	semidirect
108. C₂×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#19152226	2
#19152007	2
#19152000	2
#19151996	2
#19151640	2
#19151636	2
C₂×#15667944	2
#405×C₈₉	2
#3163	89