Finite Group Databasebeta

Subgroups — #36444870

Order 22784.

354 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 155 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	#26019939	C₂	1	1	yes	maximal, characteristic
2	#26018536	C₂	1	2	yes	maximal
3	#26018082	C₂	1	2	yes	maximal
4	#26018081	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1274	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×#15667923	C₄	1	2	yes	—
8	C₂×#15667923	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₄	1	2	yes	—
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	#19151320	C₄	1	4	yes	complemented
12	C₈×#10623494	C₄	1	4	yes	complemented
13	#431×C₈₉	C₂²	1	2	yes	—
14	#376×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×#10623494	C₂×C₄	1	2	yes	—
16	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
18	C₄⋊#10623494	—	2	2	no	—
19	C₂²⋊#10623494	—	2	2	no	—
20	#15667923	—	2	1	no	—
21	#15667923	—	2	1	no	—
22	#15667923	C₈	1	8	yes	complemented
23	C₄×#10623494	—	2	4	no	—
24	C₄×#10623494	C₂×C₄	1	4	yes	—
25	C₄×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	2	no	—
30	C₂²⋊C₈×C₈₉	—	2	2	no	—
31	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
32	#95×C₈₉	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	C₄²	1	2	yes	—
37	C₂×#10623494	C₄²	1	1	yes	characteristic
38	C₂×#10623494	C₄²	1	1	yes	characteristic
39	C₂×#10623494	C₂×C₈	1	4	yes	—
40	C₂×#10623494	C₂×C₈	1	4	yes	—
41	C₂×#10623494	M₁₆	1	4	yes	—
42	C₂²×C₄×C₈₉	—	2	2	no	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	#41	1	2	yes	—
45	C₂×C₈×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	—	2	2	no	—
47	C₄²×C₈₉	C₂×C₈	1	4	yes	—
48	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
49	C₄²×C₈₉	#41	1	1	yes	characteristic
50	C₄²×C₈₉	#41	1	1	yes	characteristic
51	C₂³×C₈₉	#117	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	—	4	2	no	—
54	C₂×C₄×C₈₉	—	4	4	no	—
55	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#117	1	2	yes	—
57	C₂×C₄×C₈₉	#117	1	2	yes	—
58	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
61	C₈×C₈₉	—	4	4	no	—
62	#10623494	—	2	4	no	—
63	#10623494	—	4	4	no	—
64	#10623494	C₄×C₈	1	4	yes	—
65	#10623494	#97	1	2	yes	—
66	#10623494	#97	1	2	yes	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	#376	1	1	yes	characteristic
69	C₂²×C₈₉	#431	1	2	yes	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	—	4	4	no	—
73	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
75	C₄×C₈₉	#431	1	2	yes	—
76	#8138	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	—	2	2	no	—
78	C₂×C₈₉	#1744	1	1	yes	characteristic
79	C₂×C₈₉	#1747	1	2	yes	—
80	#1726	—	89	1	no	—
81	#1450	—	89	2	no	—
82	#1275	—	89	2	no	—
83	#1274	—	89	1	no	—
84	#1274	D₈₉	1	1	yes	characteristic
85	C₈₉	#8138	1	1	yes	characteristic, complemented, Sylow
86	#431	—	89	2	no	—
87	#431	C₂×D₈₉	1	2	yes	—
88	C₂×#97	—	89	2	no	—
89	C₂×#97	—	89	1	no	—
90	C₂×C₄×C₈	—	89	1	no	—
91	C₂×C₄×C₈	—	89	2	no	—
92	#376	C₂×D₈₉	1	1	yes	characteristic
93	#322	—	89	4	no	—
94	C₈²	—	89	4	no	—
95	C₂²×C₈	—	89	1	no	—
96	C₂²×C₈	—	89	1	no	—
97	C₂²×C₈	—	89	2	no	—
98	C₂×C₄²	C₄×D₈₉	1	2	yes	—
99	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	178	2	no	—
102	C₂²⋊C₈	—	2	2	no	—
103	C₂²⋊C₈	—	178	2	no	—
104	#97	—	89	8	no	—
105	#97	—	178	1	no	—
106	#97	—	178	1	no	—
107	C₄×C₈	—	89	2	no	—
108	C₄×C₈	—	89	2	no	—
109	C₄×C₈	—	89	4	no	—
110	C₄×C₈	—	178	4	no	—
111	C₄×C₈	C₄×D₈₉	1	4	yes	—
112	#95	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₂²×C₄	#13083573	1	2	yes	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	89	4	no	—
118	C₂×C₈	—	89	4	no	—
119	C₂×C₈	—	89	2	no	—
120	C₂×C₈	—	89	1	no	—
121	C₂×C₈	—	89	1	no	—
122	C₂×C₈	—	89	4	no	—
123	C₂×C₈	—	178	4	no	—
124	C₂×C₈	—	178	2	no	—
125	C₂×C₈	—	178	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₈×D₈₉	1	4	yes	—
128	C₄²	C₂×C₄×D₈₉	1	2	yes	—
129	C₄²	#13083573	1	1	yes	characteristic
130	C₄²	#13083575	1	1	yes	characteristic
131	C₂³	#15667968	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
136	C₂×C₄	#15667968	1	2	yes	—
137	C₂×C₄	#15667970	1	2	yes	—
138	C₂×C₄	#15667970	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
140	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
141	C₈	—	4	4	no	—
142	C₈	—	89	4	no	—
143	C₈	—	89	2	no	—
144	C₈	—	89	2	no	—
145	C₈	—	356	4	no	—
146	C₈	—	178	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#19151443	1	1	yes	characteristic
149	C₂²	#19151648	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
154	C₄	#97×D₈₉	1	1	yes	characteristic
155	C₄	#19151650	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#26019996	1	1	yes	characteristic
158	C₂	#26020006	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1274	C₂
socle	C₂²×C₈₉	#376
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1274	D₈₉
hypocenter	C₈₉	#8138
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #1274

Complements

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

N	H ≅ G/N	type
11. #19151320	C₄	semidirect
12. C₈×#10623494	C₄	semidirect
22. #15667923	C₈	semidirect
85. C₈₉	#8138	semidirect

Maximal subgroups

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

subgroup	index
#26019939	2
#26018536	2
#26018082	2
#26018081	2
C₈₉×#1274	2
#8138	89