Finite Group Databasebeta

Subgroups — #36366711

Order 22144.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#26012308	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012296	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011934	C₂	1	3	yes	maximal, complemented
4	#26011929	C₂	1	3	yes	maximal, complemented
5	#26011923	C₂	1	3	yes	maximal, complemented
6	C₂×#19149624	C₂	1	3	yes	maximal, complemented
7	#504×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19149726	—	2	1	no	—
9	#19149725	—	2	1	no	—
10	#19149712	—	2	3	no	—
11	#19149707	—	2	3	no	—
12	#19149701	C₂²	1	3	yes	—
13	#19149701	C₂²	1	3	yes	complemented
14	C₂×Dic₆₉₂	C₂²	1	3	yes	complemented
15	C₂×D₁₃₈₄	C₂²	1	3	yes	—
16	Dic₁₃₈₄	C₂²	1	3	yes	complemented
17	#19149624	C₂²	1	3	yes	complemented
18	#19149624	C₂²	1	6	yes	complemented
19	#19149624	C₂²	1	1	yes	characteristic, complemented
20	D₂₇₆₈	C₂²	1	3	yes	complemented
21	#131×C₁₇₃	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₇₃	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₇₃	C₂²	1	3	yes	complemented
24	#14583827	—	4	1	no	—
25	#14583826	—	2	3	no	—
26	#14583826	—	2	3	no	—
27	#14583825	—	4	3	no	—
28	C₂×Dic₃₄₆	—	2	3	no	—
29	Q₈×D₁₇₃	—	4	1	no	—
30	C₂×D₆₉₂	—	2	3	no	—
31	D₄×D₁₇₃	—	4	3	no	—
32	Dic₆₉₂	—	2	3	no	—
33	Dic₆₉₂	C₂³	1	3	yes	—
34	Dic₆₉₂	C₂³	1	1	yes	characteristic
35	#14583815	—	2	3	no	—
36	#14583815	—	2	3	no	—
37	D₁₃₈₄	—	2	3	no	—
38	D₁₃₈₄	C₂³	1	1	yes	characteristic
39	D₁₃₈₄	C₂³	1	3	yes	—
40	#41×C₁₇₃	D₄	1	1	yes	characteristic
41	M₁₆×C₁₇₃	D₄	1	3	yes	—
42	C₂×C₈×C₁₇₃	C₂³	1	3	yes	—
43	C₁₆×C₁₇₃	C₂³	1	3	yes	—
44	C₁₆×C₁₇₃	C₂³	1	1	yes	characteristic
45	C₂²×D₁₇₃	—	4	3	no	—
46	Dic₃₄₆	—	2	1	no	—
47	Dic₃₄₆	—	2	3	no	—
48	Dic₃₄₆	—	4	3	no	—
49	D₆₉₂	—	2	3	no	—
50	D₆₉₂	—	2	1	no	—
51	D₆₉₂	—	4	3	no	—
52	C₂²⋊D₁₇₃	—	4	3	no	—
53	C₂²⋊D₁₇₃	—	4	3	no	—
54	C₂×Dic₁₇₃	—	4	3	no	—
55	C₄×D₁₇₃	—	4	3	no	—
56	C₄×D₁₇₃	—	4	3	no	—
57	Q₈×C₁₇₃	D₈	1	1	yes	characteristic
58	D₄×C₁₇₃	D₈	1	3	yes	—
59	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
60	C₈×C₁₇₃	C₂×D₄	1	3	yes	—
61	C₈×C₁₇₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₇₃	C₂×D₈	1	3	yes	—
63	C₂×D₁₇₃	—	4	1	no	—
64	C₂×D₁₇₃	—	4	3	no	—
65	C₂×D₁₇₃	—	16	3	no	—
66	C₄×C₁₇₃	C₂×D₈	1	3	yes	—
67	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₇₃	—	4	1	no	—
69	Dic₁₇₃	—	4	3	no	—
70	C₂×C₁₇₃	—	2	3	no	—
71	C₂×C₁₇₃	C₂²×D₈	1	1	yes	characteristic
72	D₁₇₃	—	8	1	no	—
73	D₁₇₃	—	8	3	no	—
74	C₁₇₃	#3416	1	1	yes	characteristic, complemented, Sylow
75	#3416	—	173	1	no	maximal, Sylow
76	#578	—	173	1	no	—
77	#576	—	173	1	no	—
78	#510	—	173	3	no	—
79	#509	—	173	3	no	—
80	#508	—	173	3	no	—
81	C₂×SD₃₂	—	173	3	no	—
82	#504	D₁₇₃	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	346	1	no	—
84	H(5,𝔽₂)	—	346	1	no	—
85	#137	—	346	3	no	—
86	AGL(1,ℤ/8)	—	346	3	no	—
87	#135	—	173	3	no	—
88	#135	—	173	3	no	—
89	C₂×Q₁₆	—	173	3	no	—
90	C₂×D₈	—	173	3	no	—
91	#131	C₂×D₁₇₃	1	1	yes	characteristic
92	Q₃₂	—	173	3	no	—
93	SD₃₂	—	173	3	no	—
94	SD₃₂	—	173	6	no	—
95	SD₃₂	—	173	1	no	—
96	D₁₆	—	173	3	no	—
97	M₃₂	C₂×D₁₇₃	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₇₃	1	3	yes	complemented
99	#41	—	346	3	no	—
100	#41	—	346	3	no	—
101	#41	—	692	1	no	—
102	#41	—	692	3	no	—
103	#41	D₆₉₂	1	1	yes	characteristic
104	C₂×Q₈	—	346	3	no	—
105	C₂×Q₈	—	692	1	no	—
106	C₂×D₄	—	346	3	no	—
107	C₂×D₄	—	692	3	no	—
108	Q₁₆	—	346	3	no	—
109	Q₁₆	—	173	3	no	—
110	Q₁₆	—	173	1	no	—
111	SD₁₆	—	346	3	no	—
112	SD₁₆	—	346	3	no	—
113	D₈	—	346	3	no	—
114	D₈	—	173	1	no	—
115	D₈	—	173	3	no	—
116	M₁₆	D₆₉₂	1	3	yes	—
117	C₂×C₈	C₂²×D₁₇₃	1	3	yes	—
118	C₁₆	C₂²×D₁₇₃	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₇₃	1	3	yes	—
120	C₂³	—	692	3	no	—
121	Q₈	—	346	1	no	—
122	Q₈	—	346	3	no	—
123	Q₈	—	692	3	no	—
124	Q₈	D₁₃₈₄	1	1	yes	characteristic
125	D₄	—	346	3	no	—
126	D₄	—	346	1	no	—
127	D₄	—	692	3	no	—
128	D₄	—	692	3	no	—
129	D₄	—	692	3	no	—
130	D₄	D₁₃₈₄	1	3	yes	—
131	C₂×C₄	—	692	3	no	—
132	C₂×C₄	—	692	3	no	—
133	C₂×C₄	—	692	3	no	—
134	C₂×C₄	C₂×D₆₉₂	1	3	yes	—
135	C₈	C₂×D₆₉₂	1	3	yes	—
136	C₈	C₂³×D₁₇₃	1	1	yes	characteristic
137	C₂²	—	692	1	no	—
138	C₂²	—	692	3	no	—
139	C₂²	—	2768	3	no	—
140	C₂²	C₂×D₁₃₈₄	1	3	yes	—
141	C₄	—	692	1	no	—
142	C₄	—	692	3	no	—
143	C₄	C₂×D₁₃₈₄	1	3	yes	—
144	C₄	C₂²×D₆₉₂	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	1384	1	no	—
147	C₂	—	1384	3	no	—
148	C₂	C₂²×D₁₃₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₃₈₄
commutator, cosocle	C₈×C₁₇₃	C₂⁴
Frattini	C₈	C₂³×D₁₇₃
Fitting	#504×C₁₇₃	C₂
socle	C₂×C₁₇₃	C₂²×D₈
hypercenter	#504	D₁₇₃
hypocenter	C₁₇₃	#3416
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #26012308	C₂	semidirect
2. #26012296	C₂	semidirect
3. #26011934	C₂	semidirect
4. #26011929	C₂	semidirect
5. #26011923	C₂	semidirect
6. C₂×#19149624	C₂	semidirect
7. #504×C₁₇₃	C₂	semidirect
13. #19149701	C₂²	semidirect
14. C₂×Dic₆₉₂	C₂²	semidirect
16. Dic₁₃₈₄	C₂²	semidirect
17. #19149624	C₂²	semidirect
18. #19149624	C₂²	semidirect
19. #19149624	C₂²	semidirect
20. D₂₇₆₈	C₂²	semidirect
22. M₃₂×C₁₇₃	C₂²	semidirect
23. C₂×C₁₆×C₁₇₃	C₂²	semidirect
74. C₁₇₃	#3416	semidirect
82. #504	D₁₇₃	semidirect
97. M₃₂	C₂×D₁₇₃	semidirect
98. C₂×C₁₆	C₂×D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26012308	2
#26012296	2
#26011934	2
#26011929	2
#26011923	2
C₂×#19149624	2
#504×C₁₇₃	2
#3416	173