Finite Group Databasebeta

Subgroups — #37510883

Order 22912.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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	#26038869	C₂	1	1	yes	maximal, characteristic
2	#26038556	C₂	1	1	yes	maximal, characteristic, complemented
3	#26038555	C₂	1	1	yes	maximal, characteristic, complemented
4	#26038268	C₂	1	1	yes	maximal, characteristic, complemented
5	#26038247	C₂	1	1	yes	maximal, characteristic, complemented
6	#26038214	C₂	1	1	yes	maximal, characteristic, complemented
7	#359×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153135	C₂²	1	1	yes	characteristic
9	#19153121	—	2	1	no	—
10	#19153120	—	2	1	no	—
11	#19153059	C₄	1	1	yes	characteristic, complemented
12	#19153057	C₄	1	1	yes	characteristic, complemented
13	#19153056	C₄	1	1	yes	characteristic, complemented
14	#19153054	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19153049	C₂²	1	1	yes	characteristic
17	#19153040	—	2	1	no	—
18	#19153039	—	2	1	no	—
19	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
20	#135×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
22	#108×C₁₇₉	C₂²	1	1	yes	characteristic
23	#15668259	—	2	1	no	—
24	Dic₇₁₆	C₂×C₄	1	1	yes	characteristic
25	#15668248	—	2	1	no	—
26	D₁₄₃₂	C₂×C₄	1	1	yes	characteristic
27	#15668243	—	2	1	no	—
28	#15668243	—	4	1	no	—
29	#15668242	—	4	1	no	—
30	C₂×C₁₇₉⋊C₈	—	4	1	no	—
31	C₂×C₁₇₉⋊C₈	D₄	1	1	yes	characteristic
32	C₈×D₁₇₉	—	4	1	no	—
33	C₄×Dic₁₇₉	D₄	1	1	yes	characteristic
34	C₁₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₁₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
36	#41×C₁₇₉	—	2	1	no	—
37	Q₁₆×C₁₇₉	C₂×C₄	1	1	yes	characteristic
38	SD₁₆×C₁₇₉	—	2	1	no	—
39	D₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₁₇₉	—	2	1	no	—
41	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
42	C₁₆×C₁₇₉	—	2	1	no	—
43	Dic₃₅₈	—	2	1	no	—
44	D₇₁₆	—	2	1	no	—
45	C₂²⋊D₁₇₉	—	4	1	no	—
46	C₂×Dic₁₇₉	—	2	1	no	—
47	C₄×D₁₇₉	—	4	1	no	—
48	C₁₇₉⋊C₈	—	4	1	no	—
49	C₁₇₉⋊C₈	D₈	1	1	yes	characteristic
50	C₁₇₉⋊C₈	D₈	1	1	yes	characteristic
51	Q₈×C₁₇₉	—	2	1	no	—
52	D₄×C₁₇₉	—	2	1	no	—
53	D₄×C₁₇₉	—	4	1	no	—
54	C₂×C₄×C₁₇₉	—	4	1	no	—
55	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₇₉	—	4	1	no	—
57	C₈×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₇₉	#41	1	1	yes	characteristic
59	C₂²×C₁₇₉	—	4	1	no	—
60	C₂²×C₁₇₉	#135	1	1	yes	characteristic
61	C₂×D₁₇₉	—	4	1	no	—
62	C₄×C₁₇₉	—	4	1	no	—
63	C₄×C₁₇₉	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₇₉	C₂×D₈	1	1	yes	characteristic
65	Dic₁₇₉	—	4	1	no	—
66	Dic₁₇₉	—	4	1	no	—
67	C₂×C₁₇₉	—	2	1	no	—
68	C₂×C₁₇₉	—	8	1	no	—
69	C₂×C₁₇₉	C₄×D₈	1	1	yes	characteristic
70	D₁₇₉	—	8	1	no	—
71	C₁₇₉	#2179	1	1	yes	characteristic, complemented, Sylow
72	#2179	—	179	1	no	maximal, Sylow
73	#508	—	179	1	no	—
74	#443	—	179	1	no	—
75	#443	—	179	1	no	—
76	#368	—	179	1	no	—
77	#359	—	179	1	no	—
78	#359	D₁₇₉	1	1	yes	characteristic, complemented
79	#346	—	179	1	no	—
80	#135	—	179	1	no	—
81	#135	C₂×D₁₇₉	1	1	yes	characteristic
82	#131	—	358	1	no	—
83	#131	—	358	1	no	—
84	Q₃₂	—	179	1	no	—
85	SD₃₂	—	179	1	no	—
86	SD₃₂	—	179	1	no	—
87	D₁₆	—	179	1	no	—
88	C₂×C₁₆	—	179	1	no	—
89	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
90	#108	—	179	1	no	—
91	#108	C₂×D₁₇₉	1	1	yes	characteristic
92	#104	—	358	1	no	—
93	#104	—	358	1	no	—
94	C₄×C₈	—	179	1	no	—
95	#41	—	2	1	no	—
96	#41	—	358	1	no	—
97	Q₁₆	—	179	1	no	—
98	Q₁₆	C₄×D₁₇₉	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	358	1	no	—
101	D₈	—	179	1	no	—
102	D₈	C₄×D₁₇₉	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	358	1	no	—
105	M₁₆	—	716	1	no	—
106	M₁₆	—	716	1	no	—
107	C₂×C₈	—	179	1	no	—
108	C₂×C₈	—	716	1	no	—
109	C₂×C₈	—	716	1	no	—
110	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
111	C₄²	—	179	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	179	1	no	—
114	C₁₆	—	179	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	358	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	358	1	no	—
120	D₄	—	716	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	358	1	no	—
123	C₂×C₄	—	716	1	no	—
124	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	179	1	no	—
127	C₈	—	179	1	no	—
128	C₈	—	716	1	no	—
129	C₈	C₂×C₄×D₁₇₉	1	1	yes	characteristic
130	C₈	#15668258	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	716	1	no	—
133	C₂²	#19153134	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	716	1	no	—
136	C₄	—	716	1	no	—
137	C₄	(C₂²⋊D₁₇₉)⋊C₄	1	1	yes	characteristic
138	C₄	D₈×D₁₇₉	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	1432	1	no	—
142	C₂	#26038519	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₇₉
commutator	C₈×C₁₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	#359×C₁₇₉	C₂
socle	C₂×C₁₇₉	C₄×D₈
prime radical	#26038869	C₂
cosocle	C₂×C₈×C₁₇₉	C₂³
hypercenter	#359	D₁₇₉
hypocenter	C₁₇₉	#2179
LCS	G ▷ C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
2. #26038556	C₂	semidirect
3. #26038555	C₂	semidirect
4. #26038268	C₂	semidirect
5. #26038247	C₂	semidirect
6. #26038214	C₂	semidirect
7. #359×C₁₇₉	C₂	semidirect
11. #19153059	C₄	semidirect
12. #19153057	C₄	semidirect
13. #19153056	C₄	semidirect
14. #19153054	C₄	semidirect
71. C₁₇₉	#2179	semidirect
78. #359	D₁₇₉	semidirect

Maximal subgroups

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

subgroup	index
#26038869	2
#26038556	2
#26038555	2
#26038268	2
#26038247	2
#26038214	2
#359×C₁₇₉	2
#2179	179