Finite Group Databasebeta

Subgroups — #36543757

Order 22784.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#26030997	C₂	1	3	yes	maximal, complemented
2	#26030627	C₂	1	3	yes	maximal
3	#26019912	C₂	1	6	yes	maximal, complemented
4	#26019864	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152155	C₂²	1	3	yes	—
7	C₂×#15668009	C₂²	1	1	yes	characteristic
8	#19151996	C₂²	1	3	yes	complemented
9	#19151978	C₂²	1	3	yes	complemented
10	#19151963	—	2	6	no	—
11	#19151949	C₂²	1	6	yes	complemented
12	#19151945	C₂²	1	6	yes	—
13	#19151945	C₂²	1	6	yes	—
14	#19151856	—	2	6	no	—
15	#19151706	—	2	6	no	—
16	#19151604	—	2	6	no	—
17	C₄×M₁₆×C₈₉	C₂²	1	3	yes	—
18	#322×C₈₉	C₂²	1	3	yes	complemented
19	C₈²×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₃₅₆	—	2	3	no	—
21	C₂×Dic₃₅₆	—	2	6	no	—
22	#15668012	C₂³	1	6	yes	—
23	#15668009	—	2	3	no	—
24	#15668009	C₂³	1	2	yes	—
25	#15668004	—	2	3	no	—
26	#15667991	—	2	6	no	—
27	C₄×Dic₁₇₈	—	2	6	no	—
28	#15667971	—	2	3	no	—
29	#15667944	—	2	6	no	—
30	#15667937	—	2	12	no	—
31	#15667934	—	2	12	no	—
32	#15667934	—	2	12	no	—
33	C₂×M₁₆×C₈₉	D₄	1	6	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	#97×C₈₉	D₄	1	6	yes	—
36	C₄×C₈×C₈₉	C₂³	1	3	yes	—
37	C₄×C₈×C₈₉	C₂³	1	3	yes	—
38	C₂³×D₈₉	—	4	1	no	—
39	C₂×Dic₁₇₈	—	2	6	no	—
40	C₂×D₃₅₆	—	2	6	no	—
41	C₂×D₃₅₆	—	4	12	no	—
42	C₂×D₃₅₆	—	4	3	no	—
43	Dic₃₅₆	—	4	12	no	—
44	C₄⋊Dic₈₉	—	2	6	no	—
45	C₄⋊Dic₈₉	—	2	6	no	—
46	#13083553	—	4	6	no	—
47	C₂²⋊Dic₈₉	—	4	3	no	—
48	C₄×Dic₈₉	—	4	3	no	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
50	M₁₆×C₈₉	—	4	6	no	—
51	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
52	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
53	C₄²×C₈₉	C₂×D₄	1	3	yes	—
54	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
55	C₂³×C₈₉	#127	1	1	yes	characteristic
56	C₂²×D₈₉	—	4	2	no	—
57	C₂²×D₈₉	—	8	6	no	—
58	C₂×C₄×C₈₉	—	2	6	no	—
59	C₂×C₄×C₈₉	#127	1	3	yes	—
60	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
61	C₂×Dic₈₉	—	4	6	no	—
62	D₃₅₆	—	4	12	no	—
63	D₃₅₆	—	8	6	no	—
64	Dic₁₇₈	—	4	12	no	—
65	C₈×C₈₉	—	2	6	no	—
66	C₈×C₈₉	—	2	6	no	—
67	C₂²×C₈₉	—	2	3	no	—
68	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
69	C₂×D₈₉	—	8	6	no	—
70	C₂×D₈₉	—	16	4	no	—
71	C₄×C₈₉	—	2	3	no	—
72	C₄×C₈₉	#575	1	6	yes	—
73	Dic₈₉	—	8	6	no	—
74	#19753	—	89	1	no	maximal, Sylow
75	C₂×C₈₉	—	4	1	no	—
76	C₂×C₈₉	#3153	1	3	yes	—
77	D₈₉	—	16	2	no	—
78	#3185	—	89	3	no	—
79	#3138	—	89	3	no	—
80	#1720	—	89	6	no	—
81	#1711	—	89	2	no	—
82	#1455	D₈₉	1	1	yes	characteristic, complemented
83	C₈₉	#19753	1	1	yes	characteristic, complemented, Sylow
84	#533	—	89	3	no	—
85	C₂×#127	—	89	1	no	—
86	#498	—	89	3	no	—
87	#494	—	89	3	no	—
88	#490	—	178	6	no	—
89	#487	—	89	6	no	—
90	#486	—	89	6	no	—
91	#486	—	89	6	no	—
92	#470	—	178	6	no	—
93	#441	—	178	6	no	—
94	#418	—	178	6	no	—
95	C₄×M₁₆	C₂×D₈₉	1	3	yes	—
96	#322	C₂×D₈₉	1	3	yes	complemented
97	C₈²	C₂×D₈₉	1	1	yes	characteristic, complemented
98	C₂²×D₄	—	178	3	no	—
99	C₂×Q₁₆	—	178	6	no	—
100	C₂×M₁₆	D₃₅₆	1	6	yes	—
101	C₄⋊Q₈	—	89	6	no	—
102	#127	—	89	2	no	—
103	#127	—	178	3	no	—
104	#125	—	178	3	no	—
105	C₂²⋊Q₈	—	178	6	no	—
106	C₄×Q₈	—	178	6	no	—
107	#117	—	178	3	no	—
108	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
109	#106	—	178	6	no	—
110	#103	—	178	12	no	—
111	#102	—	178	12	no	—
112	#102	—	178	12	no	—
113	#97	D₃₅₆	1	6	yes	—
114	C₄×C₈	C₂²×D₈₉	1	3	yes	—
115	C₄×C₈	C₂²×D₈₉	1	3	yes	—
116	C₂⁴	—	356	1	no	—
117	C₂×Q₈	—	178	6	no	—
118	C₂×D₄	—	356	12	no	—
119	C₂×D₄	—	356	3	no	—
120	C₂×D₄	—	178	6	no	—
121	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
122	Q₁₆	—	356	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
125	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
126	C₄⋊C₄	—	356	6	no	—
127	C₄⋊C₄	—	178	6	no	—
128	C₄⋊C₄	—	178	6	no	—
129	C₂²⋊C₄	—	356	3	no	—
130	C₄²	—	356	3	no	—
131	C₄²	C₂×D₃₅₆	1	3	yes	—
132	C₄²	C₂³×D₈₉	1	1	yes	characteristic
133	C₂³	—	356	2	no	—
134	C₂³	—	712	6	no	—
135	C₂³	#15668009	1	1	yes	characteristic
136	Q₈	—	356	12	no	—
137	D₄	—	356	12	no	—
138	D₄	—	712	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	356	6	no	—
141	C₂×C₄	#15668009	1	3	yes	—
142	C₂×C₄	C₂²×D₃₅₆	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	1424	4	no	—
147	C₂²	—	712	6	no	—
148	C₂²	C₂×#15668009	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	712	6	no	—
151	C₄	#19152398	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	1424	2	no	—
154	C₂	#26030718	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂×#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1455	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂×#15668009	C₂²
hypercenter	#1455	D₈₉
hypocenter	C₈₉	#19753
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
1. #26030997	C₂	semidirect
3. #26019912	C₂	semidirect
4. #26019864	C₂	semidirect
5. C₈₉×#1455	C₂	semidirect
8. #19151996	C₂²	semidirect
9. #19151978	C₂²	semidirect
11. #19151949	C₂²	semidirect
18. #322×C₈₉	C₂²	semidirect
19. C₈²×C₈₉	C₂²	semidirect
82. #1455	D₈₉	semidirect
83. C₈₉	#19753	semidirect
96. #322	C₂×D₈₉	semidirect
97. C₈²	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030997	2
#26030627	2
#26019912	2
#26019864	2
C₈₉×#1455	2
#19753	89