Finite Group Databasebeta

Subgroups — #36369683

Order 22144.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#26012506	C₂	1	2	yes	maximal
2	C₂×#19149750	C₂	1	1	yes	maximal, characteristic
3	#26012454	C₂	1	1	yes	maximal, characteristic
4	#26012374	C₂	1	2	yes	maximal
5	#108×D₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14583836	C₂²	1	2	yes	—
7	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
8	#19149758	C₂²	1	1	yes	characteristic
9	#19149750	—	2	1	no	—
10	#19149750	—	2	1	no	—
11	#19149750	—	2	2	no	—
12	#19149749	—	2	2	no	—
13	#19149733	—	2	1	no	—
14	C₈×#10621924	—	2	1	no	—
15	M₁₆×D₁₇₃	C₂²	1	2	yes	—
16	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
17	#19149615	C₄	1	1	yes	characteristic, complemented
18	#19149614	C₄	1	2	yes	complemented
19	#108×C₁₇₃	C₄	1	1	yes	characteristic, complemented
20	#14583836	—	2	2	no	—
21	#14583836	—	2	2	no	—
22	#14583836	D₄	1	1	yes	characteristic
23	#14583836	D₄	1	1	yes	characteristic
24	#14583835	—	2	1	no	—
25	D₁₇₃⋊C₈	—	2	1	no	—
26	C₂×#13063443	—	2	2	no	—
27	C₄⋊#10621924	—	2	1	no	—
28	C₂²⋊#10621924	Q₈	1	2	yes	—
29	C₄×#10621924	—	2	1	no	—
30	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
31	#14583810	C₂×C₄	1	2	yes	—
32	#14583809	—	2	2	no	—
33	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
34	C₈×D₁₇₃	—	2	2	no	—
35	C₈×D₁₇₃	D₄	1	1	yes	characteristic
36	C₈×D₁₇₃	D₄	1	1	yes	characteristic
37	C₈×D₁₇₃	Q₈	1	1	yes	characteristic
38	C₈×D₁₇₃	Q₈	1	1	yes	characteristic
39	M₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#10621924	—	2	2	no	—
42	#13063443	—	2	1	no	—
43	#13063443	—	2	1	no	—
44	#13063443	—	4	2	no	—
45	C₂²×D₁₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×Dic₁₇₃	#41	1	1	yes	characteristic
47	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
48	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
49	C₄×D₁₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₄×D₁₇₃	#41	1	1	yes	characteristic
51	#13063436	—	2	2	no	—
52	#13063436	C₄⋊C₄	1	1	yes	characteristic
53	#13063436	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₁₇₃	—	2	2	no	—
56	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₇₃	C₄×Q₈	1	1	yes	characteristic
59	C₂×D₁₇₃	—	4	1	no	—
60	C₂×D₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×D₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
62	#10621924	—	4	2	no	—
63	C₄×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
65	Dic₁₇₃	#121	1	1	yes	characteristic
66	Dic₁₇₃	#125	1	1	yes	characteristic
67	C₂×C₁₇₃	—	2	1	no	—
68	C₂×C₁₇₃	#389	1	1	yes	characteristic
69	D₁₇₃	—	2	1	no	—
70	D₁₇₃	—	2	1	no	—
71	C₁₇₃	#1953	1	1	yes	characteristic, complemented, Sylow
72	#1953	—	173	1	no	maximal, Sylow
73	#430	—	173	2	no	—
74	C₂×#108	—	173	1	no	—
75	C₂×#108	—	173	1	no	—
76	#405	—	173	1	no	—
77	#344	—	173	2	no	—
78	C₂×M₁₆	—	173	2	no	—
79	C₂×M₁₆	—	173	1	no	—
80	C₂×M₁₆	—	173	2	no	—
81	C₂²×C₈	—	173	1	no	—
82	#117	—	173	1	no	—
83	#108	—	346	2	no	—
84	#108	—	346	2	no	—
85	#108	—	346	1	no	—
86	#108	—	346	1	no	—
87	#108	—	173	2	no	—
88	#108	—	173	1	no	—
89	#108	#10621924	1	1	yes	characteristic, complemented
90	#97	—	346	1	no	—
91	C₄×C₈	—	346	1	no	—
92	C₂²×C₄	—	173	1	no	—
93	M₁₆	—	346	2	no	—
94	M₁₆	—	346	1	no	—
95	M₁₆	—	346	2	no	—
96	M₁₆	—	346	2	no	—
97	M₁₆	—	173	1	no	—
98	M₁₆	—	173	1	no	—
99	M₁₆	—	173	2	no	—
100	M₁₆	C₂×#10621924	1	2	yes	—
101	C₂×C₈	—	346	2	no	—
102	C₂×C₈	—	346	1	no	—
103	C₂×C₈	—	346	2	no	—
104	C₂×C₈	—	173	1	no	—
105	C₂×C₈	—	173	1	no	—
106	C₂×C₈	—	173	1	no	—
107	C₂×C₈	—	173	1	no	—
108	C₂×C₈	—	173	1	no	—
109	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
110	C₄⋊C₄	—	346	1	no	—
111	C₂²⋊C₄	—	173	2	no	—
112	C₄²	—	346	1	no	—
113	C₂³	—	173	1	no	—
114	C₂×C₄	—	346	2	no	—
115	C₂×C₄	—	173	1	no	—
116	C₂×C₄	—	173	1	no	—
117	C₂×C₄	—	173	1	no	—
118	C₂×C₄	—	173	1	no	—
119	C₂×C₄	—	173	1	no	—
120	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	346	1	no	—
123	C₈	—	346	1	no	—
124	C₈	—	346	2	no	—
125	C₈	—	173	1	no	—
126	C₈	—	173	1	no	—
127	C₈	—	692	2	no	—
128	C₈	C₄⋊#10621924	1	1	yes	characteristic
129	C₈	C₄⋊#10621924	1	1	yes	characteristic
130	C₂²	—	173	1	no	—
131	C₂²	—	173	1	no	—
132	C₂²	—	692	1	no	—
133	C₂²	Q₈×#10621924	1	1	yes	characteristic
134	C₄	—	173	1	no	—
135	C₄	—	173	1	no	—
136	C₄	—	692	2	no	—
137	C₄	C₂×C₄⋊#10621924	1	1	yes	characteristic
138	C₄	D₄×#10621924	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	346	1	no	—
141	C₂	—	346	1	no	—
142	C₂	#26012442	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012442
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10621924
Fitting	#108×C₁₇₃	C₄
socle	C₂×C₁₇₃	#389
prime radical	C₂²×D₁₇₃	C₂×Q₈
cosocle	C₂×C₄×D₁₇₃	C₂³
hypercenter	#108	#10621924
hypocenter	C₁₇₃	#1953
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
17. #19149615	C₄	semidirect
18. #19149614	C₄	semidirect
19. #108×C₁₇₃	C₄	semidirect
71. C₁₇₃	#1953	semidirect
89. #108	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012506	2
C₂×#19149750	2
#26012454	2
#26012374	2
#108×D₁₇₃	2
#1953	173