Finite Group Databasebeta

Subgroups — #36470820

Order 22784.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 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×#19151374	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic
3	#26018345	C₂	1	4	yes	maximal
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	#19151374	C₄	1	4	yes	—
10	C₂×#108×C₈₉	—	2	1	no	—
11	C₂×#108×C₈₉	C₂²	1	2	yes	—
12	#423×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#13083558	—	2	2	no	—
15	C₂×#13083549	—	2	2	no	—
16	C₂×#13083549	—	2	2	no	—
17	C₂×#13083549	C₂×C₄	1	4	yes	—
18	#15667948	—	2	4	no	—
19	#15667948	—	4	2	no	—
20	C₄⋊#10623494	—	4	2	no	—
21	C₂×M₁₆×C₈₉	—	2	2	no	—
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	#108×C₈₉	—	2	4	no	—
26	#108×C₈₉	—	4	2	no	—
27	C₄⋊C₈×C₈₉	—	4	2	no	—
28	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₈₉	Q₈	1	2	yes	—
30	#13083558	—	4	2	no	—
31	#13083558	—	4	2	no	—
32	C₂×#10623494	—	4	2	no	—
33	#13083549	—	2	4	no	—
34	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₈₉	—	4	2	no	—
37	M₁₆×C₈₉	—	4	2	no	—
38	C₂×C₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₈₉	#41	1	2	yes	—
45	C₂×C₈×C₈₉	#41	1	2	yes	—
46	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₉	D₈	1	2	yes	—
48	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	#107	1	4	yes	—
51	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₈₉	—	8	2	no	—
58	C₈×C₈₉	#106	1	4	yes	—
59	C₈×C₈₉	#117	1	2	yes	—
60	C₈×C₈₉	#125	1	2	yes	—
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
64	C₂²×C₈₉	#487	1	1	yes	characteristic
65	C₂²×C₈₉	#487	1	1	yes	characteristic
66	C₄×C₈₉	—	2	4	no	—
67	C₄×C₈₉	#382	1	1	yes	characteristic
68	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
69	C₄×C₈₉	#486	1	1	yes	characteristic
70	C₄×C₈₉	#486	1	1	yes	characteristic
71	#12892	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	—	2	1	no	—
73	C₂×C₈₉	—	2	1	no	—
74	C₂×C₈₉	#1834	1	1	yes	characteristic
75	C₂×C₈₉	#2161	1	2	yes	—
76	C₂×#346	—	89	1	no	—
77	#1848	—	89	1	no	—
78	#1848	D₈₉	1	1	yes	characteristic
79	#1381	—	89	4	no	—
80	C₈₉	#12892	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₁₆	—	89	2	no	—
82	C₂×#108	—	2	1	no	—
83	C₂×#108	—	89	2	no	—
84	C₂×#108	—	178	1	no	—
85	C₂×#108	C₂×D₈₉	1	2	yes	—
86	#423	—	2	1	no	—
87	#423	—	178	1	no	—
88	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	#346	—	89	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	178	2	no	—
92	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
95	C₂×C₁₆	—	89	4	no	—
96	C₂×C₁₆	—	178	2	no	—
97	C₂×C₁₆	—	178	2	no	—
98	#108	—	2	4	no	—
99	#108	—	4	2	no	—
100	#108	—	356	2	no	—
101	#108	—	178	4	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	356	2	no	—
104	C₄×C₈	Dic₁₇₈	1	2	yes	—
105	C₄×C₈	C₄×D₈₉	1	2	yes	—
106	C₂²×C₄	#13083564	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	356	2	no	—
111	M₁₆	—	356	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	356	2	no	—
114	C₂×C₈	#13083553	1	2	yes	—
115	C₂×C₈	#13083561	1	2	yes	—
116	C₂×C₈	#13083562	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
119	C₂×C₈	#13083573	1	2	yes	—
120	C₂×C₈	#13083575	1	2	yes	—
121	C₄²	#13083553	1	2	yes	—
122	C₄²	#13083559	1	2	yes	—
123	C₁₆	—	178	4	no	—
124	C₂³	C₂×#13083564	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#15667945	1	4	yes	—
127	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
128	C₂×C₄	#15667999	1	1	yes	characteristic
129	C₂×C₄	#15668001	1	1	yes	characteristic
130	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
131	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
132	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	712	2	no	—
135	C₈	#15667943	1	4	yes	—
136	C₈	#15667970	1	2	yes	—
137	C₈	#15668003	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#15667945	1	1	yes	characteristic
140	C₂²	#19151948	1	1	yes	characteristic
141	C₂²	#19151950	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	#19151463	1	1	yes	characteristic
144	C₄	C₂×#15667943	1	1	yes	characteristic
145	C₄	#19151944	1	1	yes	characteristic
146	C₄	#19151946	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#26020379	1	1	yes	characteristic
150	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	#12892
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
80. C₈₉	#12892	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151374	2
#26020424	2
#26018345	2
C₈₉×#1848	2
#12892	89