Finite Group Databasebeta

Subgroups — #26021709

Order 11392.

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

normal subgroups only

This group has 204 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	#19151647	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151603	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667929	C₂	1	1	yes	maximal, characteristic
4	#19151369	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151368	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151354	C₂	1	1	yes	maximal, characteristic, complemented
7	#418×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13083558	C₂²	1	1	yes	characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	C₂×#13083558	C₂²	1	1	yes	characteristic
11	#15667971	C₂²	1	1	yes	characteristic
12	#15667948	—	2	2	no	—
13	#15667933	—	2	2	no	—
14	#15667929	—	2	2	no	—
15	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#117×C₈₉	C₂²	1	1	yes	characteristic
18	#102×C₈₉	—	2	2	no	—
19	#13083558	—	2	1	no	—
20	#13083558	—	2	1	no	—
21	#13083558	—	2	1	no	—
22	#13083558	—	2	1	no	—
23	#13083558	—	2	1	no	—
24	#13083558	—	2	1	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₂×#10623494	—	2	1	no	—
27	C₂×#10623494	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₂²⋊Dic₈₉	—	2	1	no	—
30	C₄×Dic₈₉	—	2	1	no	—
31	C₂⁴×C₈₉	—	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	D₄	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	M₁₆×C₈₉	—	2	1	no	—
40	M₁₆×C₈₉	—	2	1	no	—
41	C₂×C₈×C₈₉	D₄	1	2	yes	—
42	C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂²⋊C₄×C₈₉	—	2	1	no	—
44	C₄²×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	—	2	2	no	—
46	C₂³×C₈₉	—	4	1	no	—
47	C₂³×C₈₉	—	4	1	no	—
48	C₂³×C₈₉	—	8	2	no	—
49	C₂³×C₈₉	#41	1	1	yes	characteristic
50	D₄×C₈₉	—	2	2	no	—
51	D₄×C₈₉	—	2	2	no	—
52	D₄×C₈₉	—	4	1	no	—
53	D₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
61	C₂×Dic₈₉	—	4	1	no	—
62	C₈×C₈₉	—	2	2	no	—
63	#10623494	—	4	1	no	—
64	#10623494	—	4	1	no	—
65	#10623494	—	4	1	no	—
66	C₂²×C₈₉	—	4	1	no	—
67	C₂²×C₈₉	—	4	2	no	—
68	C₂²×C₈₉	—	8	2	no	—
69	C₂²×C₈₉	—	8	2	no	—
70	C₂²×C₈₉	—	8	2	no	—
71	C₂²×C₈₉	#123	1	1	yes	characteristic
72	C₂²×C₈₉	#124	1	1	yes	characteristic
73	C₂²×C₈₉	#126	1	1	yes	characteristic
74	C₄×C₈₉	—	8	1	no	—
75	C₄×C₈₉	#121	1	1	yes	characteristic
76	C₄×C₈₉	#123	1	1	yes	characteristic
77	C₄×C₈₉	#123	1	1	yes	characteristic
78	C₄×C₈₉	#126	1	1	yes	characteristic
79	Dic₈₉	—	8	1	no	—
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	—	8	2	no	—
84	C₂×C₈₉	#397	1	1	yes	characteristic
85	#2081	—	89	1	no	maximal, Sylow
86	C₈₉	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	89	1	no	—
88	#418	—	89	1	no	—
89	#418	D₈₉	1	1	yes	characteristic
90	C₂×#100	—	89	1	no	—
91	#344	—	89	1	no	—
92	#344	—	89	1	no	—
93	H(3,ℤ/4)	—	89	1	no	—
94	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
95	C₂×M₁₆	—	89	1	no	—
96	C₂×M₁₆	—	89	1	no	—
97	C₂×M₁₆	—	89	1	no	—
98	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
99	#117	—	89	1	no	—
100	#117	C₂×D₈₉	1	1	yes	characteristic
101	#108	—	178	2	no	—
102	#102	—	2	2	no	—
103	#102	—	178	2	no	—
104	#100	—	178	2	no	—
105	C₂⁴	—	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	—	2	1	no	—
111	C₂×D₄	D₃₅₆	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	178	1	no	—
116	M₁₆	—	178	1	no	—
117	M₁₆	—	178	1	no	—
118	M₁₆	—	178	1	no	—
119	M₁₆	—	178	1	no	—
120	M₁₆	—	178	1	no	—
121	C₂×C₈	—	178	1	no	—
122	C₂×C₈	—	178	1	no	—
123	C₂×C₈	—	178	1	no	—
124	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	178	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	178	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	178	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	8	2	no	—
135	C₂³	#13083573	1	1	yes	characteristic
136	D₄	—	2	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	356	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
144	C₂×C₄	#13083573	1	1	yes	characteristic
145	C₂×C₄	#13083573	1	1	yes	characteristic
146	C₂×C₄	#13083573	1	1	yes	characteristic
147	C₂×C₄	#13083574	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	356	1	no	—
150	C₈	—	356	1	no	—
151	C₈	—	356	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	#15667996	1	1	yes	characteristic
158	C₂²	#15667999	1	1	yes	characteristic
159	C₂²	#15668005	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	712	1	no	—
162	C₄	#15667986	1	1	yes	characteristic
163	C₄	#15667993	1	1	yes	characteristic
164	C₄	#15667996	1	1	yes	characteristic
165	C₄	#15668005	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#19151523	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151523
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#418×C₈₉	C₂
socle	C₂×C₈₉	#397
prime radical	C₂²×D₄×C₈₉	C₂²
hypercenter	#418	D₈₉
hypocenter	C₈₉	#2081
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
1. #19151647	C₂	semidirect
4. #19151369	C₂	semidirect
5. #19151368	C₂	semidirect
6. #19151354	C₂	semidirect
86. C₈₉	#2081	semidirect

Maximal subgroups

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

subgroup	index
#19151647	2
#19151603	2
C₂×#15667929	2
#19151369	2
#19151368	2
#19151354	2
#418×C₈₉	2
#2081	89