Finite Group Databasebeta

Subgroups — #37535621

Order 23168.

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

normal subgroups only

This group has 154 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	#26041717	C₂	1	1	yes	maximal, characteristic
2	#26041402	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041392	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041102	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041093	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041090	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#19153838	C₂²	1	1	yes	characteristic
9	C₂×D₁₄₄₈	C₂²	1	1	yes	characteristic
10	#19153822	—	2	1	no	—
11	C₄×D₇₂₄	—	2	1	no	—
12	#19153761	C₄	1	2	yes	complemented
13	D₂₈₉₆	C₄	1	2	yes	complemented
14	#19153752	C₂²	1	1	yes	characteristic
15	#19153748	C₂²	1	1	yes	characteristic
16	#19153744	—	2	1	no	—
17	#19153738	—	2	1	no	—
18	M₃₂×C₁₈₁	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₈₁	C₂²	1	1	yes	characteristic
20	#97×C₁₈₁	C₂²	1	1	yes	characteristic
21	#15668449	—	2	1	no	—
22	C₂×D₇₂₄	—	2	1	no	—
23	C₂×C₄×D₁₈₁	—	4	1	no	—
24	Dic₇₂₄	C₂×C₄	1	1	yes	characteristic, complemented
25	#15668438	—	2	1	no	—
26	D₁₄₄₈	—	2	1	no	—
27	D₁₄₄₈	C₂×C₄	1	2	yes	—
28	D₁₄₄₈	C₂×C₄	1	1	yes	characteristic, complemented
29	#15668433	—	2	1	no	—
30	#15668432	—	4	1	no	—
31	C₈×D₁₈₁	—	4	1	no	—
32	C₄⋊Dic₁₈₁	—	2	1	no	—
33	#15668426	—	4	1	no	—
34	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
36	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
37	C₁₆×C₁₈₁	—	2	1	no	—
38	C₁₆×C₁₈₁	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₈₁	—	4	1	no	—
40	Dic₃₆₂	—	2	1	no	—
41	D₇₂₄	—	2	2	no	—
42	D₇₂₄	—	2	1	no	—
43	D₇₂₄	—	4	1	no	—
44	C₂²⋊D₁₈₁	—	4	1	no	—
45	C₂×Dic₁₈₁	—	4	1	no	—
46	C₄×D₁₈₁	—	4	1	no	—
47	C₄×D₁₈₁	—	8	2	no	—
48	#13089678	—	4	1	no	—
49	C₂×C₄×C₁₈₁	—	2	1	no	—
50	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₈₁	—	2	1	no	—
52	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₈₁	#41	1	1	yes	characteristic
54	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₈₁	—	4	1	no	—
56	C₂×D₁₈₁	—	4	2	no	—
57	C₂×D₁₈₁	—	8	1	no	—
58	C₂×D₁₈₁	—	8	1	no	—
59	C₄×C₁₈₁	—	4	1	no	—
60	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₈₁	—	4	1	no	—
63	Dic₁₈₁	—	8	1	no	—
64	C₂×C₁₈₁	—	2	1	no	—
65	C₂×C₁₈₁	#442	1	1	yes	characteristic
66	D₁₈₁	—	8	1	no	—
67	D₁₈₁	—	8	2	no	—
68	C₁₈₁	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	181	1	no	maximal, Sylow
70	#509	—	181	1	no	—
71	#444	—	181	1	no	—
72	#442	—	181	1	no	—
73	#365	—	181	1	no	—
74	#361	—	181	1	no	—
75	#360	—	181	1	no	—
76	#347	D₁₈₁	1	1	yes	characteristic, complemented
77	#135	—	181	1	no	—
78	C₂×D₈	—	181	1	no	—
79	#131	—	362	1	no	—
80	C₄×D₄	—	362	1	no	—
81	SD₃₂	—	181	2	no	—
82	D₁₆	—	181	2	no	—
83	M₃₂	C₂×D₁₈₁	1	1	yes	characteristic
84	M₃₂	C₂×D₁₈₁	1	1	yes	characteristic
85	#108	—	181	1	no	—
86	#106	—	181	1	no	—
87	#104	—	362	1	no	—
88	#102	—	362	1	no	—
89	#97	C₂×D₁₈₁	1	1	yes	characteristic
90	#41	—	362	1	no	—
91	C₂×D₄	—	362	1	no	—
92	C₂²×C₄	—	724	1	no	—
93	Q₁₆	—	181	1	no	—
94	SD₁₆	—	362	1	no	—
95	D₈	—	362	1	no	—
96	D₈	—	181	1	no	—
97	D₈	—	181	2	no	—
98	M₁₆	—	362	1	no	—
99	M₁₆	—	724	1	no	—
100	C₂×C₈	—	724	1	no	—
101	C₂×C₈	D₇₂₄	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
103	C₄⋊C₄	—	362	1	no	—
104	C₂²⋊C₄	—	724	1	no	—
105	C₄²	D₇₂₄	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₁₈₁	1	2	yes	complemented
108	C₂³	—	724	1	no	—
109	Q₈	—	362	1	no	—
110	D₄	—	362	2	no	—
111	D₄	—	362	1	no	—
112	D₄	—	724	1	no	—
113	D₄	—	724	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	1448	2	no	—
116	C₂×C₄	—	724	1	no	—
117	C₂×C₄	—	724	1	no	—
118	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	724	1	no	—
121	C₈	C₂×C₄×D₁₈₁	1	1	yes	characteristic
122	C₈	#15668449	1	1	yes	characteristic
123	C₂²	—	1448	1	no	—
124	C₂²	—	1448	1	no	—
125	C₂²	—	724	1	no	—
126	C₂²	—	724	2	no	—
127	C₂²	#19153844	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	1448	1	no	—
130	C₄	—	724	1	no	—
131	C₄	C₄×D₇₂₄	1	1	yes	characteristic
132	C₄	#19153844	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1448	2	no	—
135	C₂	—	1448	1	no	—
136	C₂	#26041392	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26041392
commutator	C₈×C₁₈₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#347×C₁₈₁	C₂
socle	C₂×C₁₈₁	#442
prime radical	#26041717	C₂
cosocle	C₂×C₈×C₁₈₁	C₂³
hypercenter	#347	D₁₈₁
hypocenter	C₁₈₁	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #26041402	C₂	semidirect
3. #26041392	C₂	semidirect
4. #26041102	C₂	semidirect
5. #26041093	C₂	semidirect
6. #26041090	C₂	semidirect
7. #347×C₁₈₁	C₂	semidirect
12. #19153761	C₄	semidirect
13. D₂₈₉₆	C₄	semidirect
24. Dic₇₂₄	C₂×C₄	semidirect
28. D₁₄₄₈	C₂×C₄	semidirect
38. C₁₆×C₁₈₁	C₂×C₄	semidirect
68. C₁₈₁	AGL(1,ℤ/16)	semidirect
76. #347	D₁₈₁	semidirect
107. C₁₆	C₄×D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041717	2
#26041402	2
#26041392	2
#26041102	2
#26041093	2
#26041090	2
#347×C₁₈₁	2
AGL(1,ℤ/16)	181