Finite Group Databasebeta

Subgroups — #36470879

Order 22784.

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

normal subgroups only

This group has 306 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	C₂×#19151427	C₂	1	4	yes	maximal
2	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
3	#26020424	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₂²	1	2	yes	—
6	C₂×#15667948	—	2	1	no	—
7	C₂×#15667948	C₂²	1	2	yes	—
8	#19151622	—	2	1	no	—
9	#19151427	—	2	8	no	—
10	#19151374	C₄	1	4	yes	—
11	C₂×#108×C₈₉	—	2	1	no	—
12	C₂×#108×C₈₉	C₂²	1	2	yes	—
13	#423×C₈₉	—	2	1	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₂×#13083549	C₂×C₄	1	4	yes	—
17	C₂×#13083549	D₄	1	4	yes	—
18	C₂×#13083549	Q₈	1	4	yes	—
19	#15667948	—	2	4	no	—
20	#15667948	—	4	2	no	—
21	C₄⋊#10623494	—	4	2	no	—
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	#108×C₈₉	—	2	4	no	—
27	#108×C₈₉	—	4	2	no	—
28	C₄⋊C₈×C₈₉	—	4	2	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	Q₈	1	2	yes	—
31	#13083558	—	4	2	no	—
32	#13083558	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	#13083549	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
36	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₈₉	—	4	2	no	—
38	M₁₆×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	—	4	2	no	—
40	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₈₉	D₈	1	2	yes	—
42	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
43	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₄²×C₈₉	D₈	1	2	yes	—
49	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	4	2	no	—
51	C₂×C₄×C₈₉	#107	1	4	yes	—
52	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₈₉	—	8	2	no	—
59	C₈×C₈₉	#107	1	4	yes	—
60	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
62	#10623494	—	8	2	no	—
63	C₂²×C₈₉	—	2	2	no	—
64	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
65	C₂²×C₈₉	#494	1	1	yes	characteristic
66	C₂²×C₈₉	#500	1	1	yes	characteristic
67	C₄×C₈₉	—	2	4	no	—
68	C₄×C₈₉	#384	1	1	yes	characteristic
69	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
70	C₄×C₈₉	#493	1	1	yes	characteristic
71	C₄×C₈₉	#500	1	1	yes	characteristic
72	#12910	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	—	2	1	no	—
74	C₂×C₈₉	—	2	1	no	—
75	C₂×C₈₉	#1845	1	1	yes	characteristic
76	C₂×C₈₉	#2161	1	2	yes	—
77	C₂×#368	—	89	4	no	—
78	C₂×#346	—	89	1	no	—
79	#1848	—	89	1	no	—
80	#1848	D₈₉	1	1	yes	characteristic
81	C₈₉	#12910	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₁₆	—	89	2	no	—
83	C₂×#108	—	2	1	no	—
84	C₂×#108	—	89	2	no	—
85	C₂×#108	—	178	1	no	—
86	C₂×#108	C₂×D₈₉	1	2	yes	—
87	#423	—	2	1	no	—
88	#423	—	178	1	no	—
89	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
90	#368	—	178	8	no	—
91	#346	—	89	4	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	178	2	no	—
94	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂×C₁₆	—	89	4	no	—
98	C₂×C₁₆	—	89	4	no	—
99	C₂×C₁₆	—	89	4	no	—
100	#108	—	2	4	no	—
101	#108	—	4	2	no	—
102	#108	—	356	2	no	—
103	#108	—	178	4	no	—
104	C₄⋊C₈	—	4	2	no	—
105	C₄⋊C₈	—	356	2	no	—
106	C₄×C₈	Dic₁₇₈	1	2	yes	—
107	C₄×C₈	C₄×D₈₉	1	2	yes	—
108	C₂²×C₄	#13083564	1	2	yes	—
109	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	356	2	no	—
113	M₁₆	—	356	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	356	2	no	—
116	C₂×C₈	#13083553	1	2	yes	—
117	C₂×C₈	#13083559	1	2	yes	—
118	C₂×C₈	#13083564	1	2	yes	—
119	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
120	C₂×C₈	D₄×D₈₉	1	2	yes	—
121	C₂×C₈	Q₈×D₈₉	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
123	C₄²	#13083553	1	2	yes	—
124	C₄²	#13083559	1	2	yes	—
125	C₁₆	—	89	8	no	—
126	C₂³	C₂×#13083564	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	#15667945	1	4	yes	—
129	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
130	C₂×C₄	#15668008	1	1	yes	characteristic
131	C₂×C₄	#15668010	1	1	yes	characteristic
132	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
133	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
134	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
135	C₈	—	8	2	no	—
136	C₈	—	712	2	no	—
137	C₈	#15667945	1	4	yes	—
138	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
139	C₈	#15668011	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	C₂×#15667945	1	1	yes	characteristic
142	C₂²	#19151977	1	1	yes	characteristic
143	C₂²	#19152001	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	#19151467	1	1	yes	characteristic
146	C₄	C₂×#15667945	1	1	yes	characteristic
147	C₄	#19151974	1	1	yes	characteristic
148	C₄	#19152001	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#26020411	1	1	yes	characteristic
152	C₂	#26022046	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#12910
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
81. C₈₉	#12910	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151427	2
C₂×#19151374	2
#26020424	2
C₈₉×#1848	2
#12910	89