Finite Group Databasebeta

Subgroups — #36354352

Order 22144.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#26011540	C₂	1	1	yes	maximal, characteristic
2	#26011539	C₂	1	1	yes	maximal, characteristic
3	C₂×#19149614	C₂	1	1	yes	maximal, characteristic
4	C₂×#19149614	C₂	1	1	yes	maximal, characteristic
5	#26011447	C₂	1	1	yes	maximal, characteristic
6	#26011261	C₂	1	1	yes	maximal, characteristic
7	#344×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14583810	C₂²	1	1	yes	characteristic
9	C₂×#14583810	C₂²	1	1	yes	characteristic
10	C₂×#14583810	C₂²	1	1	yes	characteristic
11	C₂²×#13063436	C₂²	1	1	yes	characteristic
12	#19149615	—	2	2	no	—
13	#19149614	—	2	2	no	—
14	#19149614	—	2	1	no	—
15	#19149614	—	2	1	no	—
16	#19149614	C₄	1	4	yes	complemented
17	#19149590	—	2	1	no	—
18	C₄×#13063436	—	2	1	no	—
19	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
21	#117×C₁₇₃	C₂²	1	1	yes	characteristic
22	#14583810	—	2	1	no	—
23	#14583810	—	2	1	no	—
24	#14583810	—	2	1	no	—
25	#14583810	—	2	1	no	—
26	#14583810	C₂×C₄	1	2	yes	—
27	#14583810	D₄	1	2	yes	—
28	C₂×#13063436	—	2	1	no	—
29	C₂×#13063436	—	2	1	no	—
30	C₂×#13063436	—	2	1	no	—
31	C₂×#13063436	C₂×C₄	1	2	yes	—
32	C₂×#13063436	D₄	1	1	yes	characteristic
33	C₂×#13063436	D₄	1	1	yes	characteristic
34	C₂×#13063436	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₇₃	—	2	1	no	—
37	M₁₆×C₁₇₃	—	2	1	no	—
38	M₁₆×C₁₇₃	—	2	1	no	—
39	M₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₇₃	—	2	1	no	—
41	C₂×C₈×C₁₇₃	—	2	1	no	—
42	C₄⋊C₄×C₁₇₃	—	2	1	no	—
43	C₂²⋊C₄×C₁₇₃	Q₈	1	2	yes	—
44	C₄²×C₁₇₃	—	2	1	no	—
45	#13063436	—	2	2	no	—
46	#13063436	—	2	2	no	—
47	#13063436	—	4	1	no	—
48	#13063436	C₄⋊C₄	1	2	yes	—
49	#13063436	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	—	2	2	no	—
52	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
58	C₈×C₁₇₃	—	2	2	no	—
59	C₈×C₁₇₃	—	4	1	no	—
60	C₂²×C₁₇₃	—	4	1	no	—
61	C₂²×C₁₇₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₁₇₃	—	4	2	no	—
65	C₄×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
67	C₄×C₁₇₃	#121	1	1	yes	characteristic
68	C₄×C₁₇₃	#125	1	1	yes	characteristic
69	C₂×C₁₇₃	—	2	1	no	—
70	C₂×C₁₇₃	—	2	1	no	—
71	C₂×C₁₇₃	—	2	1	no	—
72	C₂×C₁₇₃	#389	1	1	yes	characteristic
73	C₁₇₃	#1953	1	1	yes	characteristic, complemented, Sylow
74	#1953	—	173	1	no	maximal, Sylow
75	#430	—	173	1	no	—
76	#430	—	173	1	no	—
77	C₂×#108	—	173	1	no	—
78	C₂×#108	—	173	1	no	—
79	#405	—	173	1	no	—
80	#344	—	173	1	no	—
81	#344	D₁₇₃	1	1	yes	characteristic
82	C₂×M₁₆	—	173	1	no	—
83	C₂×M₁₆	—	173	1	no	—
84	C₂×M₁₆	—	173	1	no	—
85	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
87	C₂²×C₈	—	173	1	no	—
88	#117	C₂×D₁₇₃	1	1	yes	characteristic
89	#108	—	346	2	no	—
90	#108	—	346	2	no	—
91	#108	—	346	1	no	—
92	#108	—	346	1	no	—
93	#108	—	173	4	no	—
94	#97	—	346	1	no	—
95	C₄×C₈	—	346	1	no	—
96	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	346	1	no	—
101	M₁₆	—	346	1	no	—
102	M₁₆	—	346	1	no	—
103	M₁₆	—	346	1	no	—
104	M₁₆	—	173	2	no	—
105	M₁₆	—	173	2	no	—
106	M₁₆	C₄×D₁₇₃	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	346	1	no	—
110	C₂×C₈	—	346	1	no	—
111	C₂×C₈	—	346	1	no	—
112	C₂×C₈	—	173	2	no	—
113	C₂×C₈	—	173	1	no	—
114	C₂×C₈	—	173	1	no	—
115	C₂×C₈	—	173	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₃₄₆	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₃₄₆	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
122	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₇₃	1	1	yes	characteristic
125	C₂×C₄	#14583825	1	1	yes	characteristic
126	C₂×C₄	#14583826	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	346	2	no	—
130	C₈	—	346	2	no	—
131	C₈	—	173	2	no	—
132	C₈	—	173	2	no	—
133	C₈	—	692	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₃₄₆	1	1	yes	characteristic
136	C₂²	#19149656	1	1	yes	characteristic
137	C₂²	#19149678	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
140	C₄	#19149639	1	1	yes	characteristic
141	C₄	#19149652	1	1	yes	characteristic
142	C₄	#19149669	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#26011388	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149652
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	#344×C₁₇₃	C₂
socle	C₂×C₁₇₃	#389
prime radical	C₂³×C₁₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	#344	D₁₇₃
hypocenter	C₁₇₃	#1953
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
16. #19149614	C₄	semidirect
73. C₁₇₃	#1953	semidirect

Maximal subgroups

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

subgroup	index
#26011540	2
#26011539	2
C₂×#19149614	2
#26011447	2
#26011261	2
#344×C₁₇₃	2
#1953	173