Finite Group Databasebeta

Subgroups — #36466105

Order 22784.

200 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26019015	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018318	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic
5	#19151655	—	2	1	no	—
6	#19151651	—	2	1	no	—
7	#19151351	—	2	1	no	—
8	#19151351	—	2	1	no	—
9	#19151349	C₂²	1	2	yes	—
10	#19151349	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	—
13	C₂²⋊C₁₆×C₈₉	C₄	1	2	yes	—
14	#346×C₈₉	C₂²	1	2	yes	—
15	C₄⋊#10623494	—	2	2	no	—
16	C₄⋊#10623494	—	2	2	no	—
17	C₂²⋊#10623494	—	4	1	no	—
18	C₂²⋊#10623494	—	4	1	no	—
19	#15667923	—	4	1	no	—
20	C₄×#10623494	—	4	1	no	—
21	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×#10623494	—	4	2	no	—
30	C₂×#10623494	—	4	2	no	—
31	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	4	1	no	—
36	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
42	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	4	no	—
45	C₂³×C₈₉	#106	1	1	yes	characteristic
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	4	1	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₈₉	—	2	1	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	#131	1	2	yes	—
60	#10623494	—	8	2	no	—
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	—	2	1	no	—
63	C₂²×C₈₉	—	2	1	no	—
64	C₂²×C₈₉	—	2	1	no	—
65	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	#424	1	1	yes	characteristic
72	C₄×C₈₉	#430	1	1	yes	characteristic
73	#12031	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	—	4	1	no	—
75	C₂×C₈₉	#1570	1	1	yes	characteristic
76	C₂×C₈₉	#2155	1	1	yes	characteristic
77	C₂×C₈₉	#2155	1	1	yes	characteristic
78	#2163	D₈₉	1	1	yes	characteristic
79	#1568	—	89	1	no	—
80	#1564	—	89	1	no	—
81	#1368	—	89	4	no	—
82	C₈₉	#12031	1	1	yes	characteristic, complemented, Sylow
83	#432	—	178	1	no	—
84	#431	—	178	1	no	—
85	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₄⋊C₁₆	Dic₈₉	1	2	yes	—
87	C₂²⋊C₁₆	Dic₈₉	1	2	yes	—
88	#346	C₂×D₈₉	1	2	yes	—
89	#335	—	178	1	no	—
90	#335	—	178	1	no	—
91	#334	—	89	2	no	—
92	#334	—	89	2	no	—
93	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
95	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
97	C₄⋊C₈	—	178	2	no	—
98	C₄⋊C₈	—	178	2	no	—
99	C₂²⋊C₈	—	356	1	no	—
100	C₂²⋊C₈	—	356	1	no	—
101	#97	—	356	1	no	—
102	C₄×C₈	—	356	1	no	—
103	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
104	C₄×C₈	D₃₅₆	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
107	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
108	C₂²×C₄	#13083563	1	1	yes	characteristic
109	C₂²×C₄	#13083563	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	356	2	no	—
113	C₂×C₈	—	356	2	no	—
114	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
117	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
118	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
119	C₄²	#13083563	1	1	yes	characteristic
120	C₄²	#13083563	1	1	yes	characteristic
121	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₂³	#15667944	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#15667944	1	1	yes	characteristic
129	C₂×C₄	#15667944	1	1	yes	characteristic
130	C₂×C₄	#15667944	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
132	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
133	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	712	2	no	—
138	C₈	—	712	2	no	—
139	C₈	#15668019	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#15667944	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#19151625	1	1	yes	characteristic
150	C₄	#19151647	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#26019023	1	1	yes	characteristic
153	C₂	#26022025	1	1	yes	characteristic
154	C₂	#26022025	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667944
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2163	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×C₈₉	#106
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#2163	D₈₉
hypocenter	C₈₉	#12031
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #26018318	C₂	semidirect
82. C₈₉	#12031	semidirect

Maximal subgroups

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

subgroup	index
#26019015	2
#26019000	2
#26018318	2
C₈₉×#2163	2
#12031	89