Finite Group Databasebeta

Subgroups — #36469176

Order 22784.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 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	#26022534	C₂	1	2	yes	maximal
2	#26022527	C₂	1	2	yes	maximal
3	#26019870	C₂	1	1	yes	maximal, characteristic
4	#26019862	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
6	#19152003	—	2	1	no	—
7	#19152003	C₂²	1	1	yes	characteristic
8	#19152003	C₂²	1	1	yes	characteristic
9	#19152003	C₂²	1	2	yes	—
10	#19151978	—	2	1	no	—
11	#19151949	—	2	1	no	—
12	#19151949	—	2	1	no	—
13	#19151426	—	2	4	no	—
14	#19151424	—	2	4	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	4	2	no	—
18	#15668012	—	2	1	no	—
19	#15668012	—	2	1	no	—
20	#15668012	—	2	2	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	2	2	no	—
23	#15667946	—	2	2	no	—
24	#15667946	—	2	2	no	—
25	#15667946	—	4	2	no	—
26	#15667937	—	4	2	no	—
27	#15667937	—	4	2	no	—
28	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇₈	—	4	2	no	—
33	C₂×Dic₁₇₈	—	4	1	no	—
34	C₂×Dic₁₇₈	—	4	1	no	—
35	Dic₃₅₆	—	8	4	no	—
36	C₄⋊Dic₈₉	—	4	1	no	—
37	C₄⋊Dic₈₉	—	4	1	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	1	no	—
40	C₄⋊Dic₈₉	—	4	1	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₂×C₈×C₈₉	D₈	1	2	yes	—
43	C₂×C₈×C₈₉	D₈	1	2	yes	—
44	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
45	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₈₉	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₈₉	—	8	1	no	—
51	C₂×Dic₈₉	—	8	2	no	—
52	C₂×Dic₈₉	—	8	1	no	—
53	Dic₁₇₈	—	8	4	no	—
54	Dic₁₇₈	—	8	2	no	—
55	Dic₁₇₈	—	8	2	no	—
56	C₈×C₈₉	—	2	2	no	—
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₈₉	#493	1	1	yes	characteristic
60	C₄×C₈₉	#494	1	2	yes	—
61	C₄×C₈₉	#509	1	2	yes	—
62	C₄×C₈₉	#510	1	2	yes	—
63	Dic₈₉	—	16	2	no	—
64	Dic₈₉	—	16	1	no	—
65	Dic₈₉	—	16	1	no	—
66	#12524	—	89	1	no	maximal, Sylow
67	C₂×C₈₉	#1713	1	1	yes	characteristic
68	C₂×C₈₉	#2250	1	1	yes	characteristic
69	C₂×C₈₉	#2251	1	1	yes	characteristic
70	#2254	—	89	2	no	—
71	#2252	—	89	2	no	—
72	#1713	—	89	1	no	—
73	#1710	—	89	1	no	—
74	#1311	D₈₉	1	1	yes	characteristic
75	C₈₉	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	89	1	no	—
77	#500	—	89	1	no	—
78	#500	—	89	2	no	—
79	#500	—	178	1	no	—
80	#494	—	178	1	no	—
81	#487	—	178	1	no	—
82	#487	—	178	1	no	—
83	#367	—	178	4	no	—
84	#366	—	178	4	no	—
85	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
86	C₈²	C₂×D₈₉	1	1	yes	characteristic
87	C₂×Q₁₆	—	356	2	no	—
88	C₄⋊Q₈	—	178	1	no	—
89	C₄⋊Q₈	—	178	1	no	—
90	C₄⋊Q₈	—	178	2	no	—
91	C₂×C₁₆	D₃₅₆	1	4	yes	—
92	#107	—	356	2	no	—
93	#107	—	178	2	no	—
94	#107	—	178	2	no	—
95	#107	—	178	2	no	—
96	#107	—	178	2	no	—
97	#103	—	356	2	no	—
98	#103	—	356	2	no	—
99	C₄×C₈	D₃₅₆	1	1	yes	characteristic
100	C₄×C₈	D₃₅₆	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	C₂×Q₈	—	356	1	no	—
103	C₂×Q₈	—	356	1	no	—
104	C₂×Q₈	—	356	2	no	—
105	Q₁₆	—	712	4	no	—
106	C₂×C₈	D₇₁₂	1	2	yes	—
107	C₂×C₈	D₇₁₂	1	2	yes	—
108	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
109	C₄⋊C₄	—	356	1	no	—
110	C₄⋊C₄	—	356	2	no	—
111	C₄⋊C₄	—	356	1	no	—
112	C₄⋊C₄	—	356	1	no	—
113	C₄⋊C₄	—	356	1	no	—
114	C₄⋊C₄	—	356	2	no	—
115	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
116	C₁₆	Dic₃₅₆	1	8	yes	complemented
117	Q₈	—	712	4	no	—
118	Q₈	—	712	2	no	—
119	Q₈	—	712	2	no	—
120	C₂×C₄	—	712	1	no	—
121	C₂×C₄	—	712	2	no	—
122	C₂×C₄	—	712	1	no	—
123	C₂×C₄	#15668009	1	1	yes	characteristic
124	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
125	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₃₅₆	1	4	yes	—
129	C₂²	#19151975	1	1	yes	characteristic
130	C₄	—	1424	2	no	—
131	C₄	—	1424	1	no	—
132	C₄	—	1424	1	no	—
133	C₄	#19151978	1	2	yes	—
134	C₄	#19152036	1	2	yes	—
135	C₄	#19152041	1	2	yes	—
136	C₂	#26019870	1	1	yes	characteristic
137	C₂	#26022518	1	1	yes	characteristic
138	C₂	#26022523	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12524
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
46. C₁₆×C₈₉	Q₁₆	semidirect
75. C₈₉	#12524	semidirect
116. C₁₆	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26022534	2
#26022527	2
#26019870	2
#26019862	2
C₈₉×#1311	2
#12524	89