Finite Group Databasebeta

Subgroups — #36353024

Order 22144.

144 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 51 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	#26011843	C₂	1	1	yes	maximal, characteristic
2	#26011783	C₂	1	1	yes	maximal, characteristic
3	#26011777	C₂	1	1	yes	maximal, characteristic
4	#26011615	C₂	1	1	yes	maximal, characteristic
5	#26011243	C₂	1	1	yes	maximal, characteristic
6	#26011238	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	#19149679	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₄₆	C₂²	1	1	yes	characteristic
10	#19149612	—	2	1	no	—
11	#19149611	—	2	1	no	—
12	C₄⋊#13063436	C₂²	1	1	yes	characteristic
13	#19149608	—	2	1	no	—
14	#19149604	—	2	1	no	—
15	#19149602	—	2	1	no	—
16	#19149602	—	2	1	no	—
17	#19149602	—	2	1	no	—
18	C₄×#13063436	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₇₃	—	2	1	no	—
20	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₄₆	—	2	1	no	—
24	C₂×Dic₃₄₆	D₄	1	1	yes	characteristic
25	C₂×#13063436	—	2	1	no	—
26	C₂×#13063436	D₄	1	1	yes	characteristic
27	C₂×#13063436	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₇₃	—	2	1	no	—
29	C₄⋊Dic₁₇₃	—	2	1	no	—
30	C₄⋊Dic₁₇₃	D₄	1	1	yes	characteristic
31	#14583805	—	2	1	no	—
32	C₄×Dic₁₇₃	—	2	1	no	—
33	C₂×Q₈×C₁₇₃	—	2	1	no	—
34	C₂×C₈×C₁₇₃	—	2	1	no	—
35	C₂×C₈×C₁₇₃	—	2	1	no	—
36	C₄⋊C₄×C₁₇₃	—	2	1	no	—
37	C₄⋊C₄×C₁₇₃	—	2	1	no	—
38	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
39	Dic₃₄₆	—	2	1	no	—
40	Dic₃₄₆	—	4	1	no	—
41	Dic₃₄₆	—	4	1	no	—
42	Dic₃₄₆	Q₁₆	1	2	yes	—
43	C₂×Dic₁₇₃	—	2	1	no	—
44	C₂×Dic₁₇₃	—	2	1	no	—
45	C₂×Dic₁₇₃	—	4	1	no	—
46	#13063436	—	2	1	no	—
47	#13063436	—	2	1	no	—
48	#13063436	—	4	1	no	—
49	Q₈×C₁₇₃	—	8	1	no	—
50	C₂×C₄×C₁₇₃	—	4	1	no	—
51	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
54	C₈×C₁₇₃	—	4	1	no	—
55	C₈×C₁₇₃	—	4	1	no	—
56	C₂²×C₁₇₃	#121	1	1	yes	characteristic
57	C₄×C₁₇₃	—	2	1	no	—
58	C₄×C₁₇₃	—	8	1	no	—
59	C₄×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₇₃	#135	1	1	yes	characteristic
61	C₄×C₁₇₃	#135	1	1	yes	characteristic
62	C₄×C₁₇₃	#137	1	1	yes	characteristic
63	Dic₁₇₃	—	2	2	no	—
64	Dic₁₇₃	—	4	1	no	—
65	Dic₁₇₃	—	8	1	no	—
66	C₂×C₁₇₃	#464	1	1	yes	characteristic
67	C₂×C₁₇₃	#467	1	1	yes	characteristic
68	C₂×C₁₇₃	#471	1	1	yes	characteristic
69	C₁₇₃	#1684	1	1	yes	characteristic, complemented, Sylow
70	#1684	—	173	1	no	maximal, Sylow
71	#487	—	173	1	no	—
72	#477	—	173	1	no	—
73	#475	—	173	1	no	—
74	C₈×Q₈	—	173	1	no	—
75	#335	—	173	1	no	—
76	#333	D₁₇₃	1	1	yes	characteristic
77	#332	—	173	1	no	—
78	C₄⋊Q₈	—	173	1	no	—
79	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
80	C₄×Q₈	—	173	1	no	—
81	#107	—	346	1	no	—
82	#106	—	346	1	no	—
83	C₄⋊C₈	—	346	1	no	—
84	C₄⋊C₈	—	173	1	no	—
85	C₄⋊C₈	C₂×D₁₇₃	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₁₇₃	1	1	yes	characteristic
87	#103	—	346	1	no	—
88	#103	—	346	1	no	—
89	#103	—	346	1	no	—
90	#103	—	346	1	no	—
91	C₄×C₈	—	346	1	no	—
92	C₄×C₈	—	173	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	346	1	no	—
95	C₂×Q₈	—	173	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	346	1	no	—
99	C₂×C₈	—	173	1	no	—
100	C₂×C₈	—	173	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	346	1	no	—
104	C₄⋊C₄	—	346	1	no	—
105	C₄⋊C₄	—	346	1	no	—
106	C₄⋊C₄	—	173	1	no	—
107	C₄²	—	346	1	no	—
108	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
109	Q₈	—	8	1	no	—
110	Q₈	—	346	1	no	—
111	Q₈	—	173	2	no	—
112	Q₈	—	692	1	no	—
113	Q₈	—	692	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	346	1	no	—
116	C₂×C₄	—	346	1	no	—
117	C₂×C₄	—	692	1	no	—
118	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
119	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
120	C₂×C₄	#14583826	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	346	1	no	—
124	C₈	—	346	1	no	—
125	C₈	—	692	1	no	—
126	C₂²	#19149652	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	346	2	no	—
130	C₄	—	1384	1	no	—
131	C₄	—	692	1	no	—
132	C₄	Q₁₆×D₁₇₃	1	1	yes	characteristic
133	C₄	#19149700	1	1	yes	characteristic
134	C₄	#19149700	1	1	yes	characteristic
135	C₄	#19149711	1	1	yes	characteristic
136	C₂	#26011716	1	1	yes	characteristic
137	C₂	#26011731	1	1	yes	characteristic
138	C₂	#26011753	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149652
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#333×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#121
hypercenter	#333	D₁₇₃
hypocenter	C₁₇₃	#1684
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
69. C₁₇₃	#1684	semidirect

Maximal subgroups

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

subgroup	index
#26011843	2
#26011783	2
#26011777	2
#26011615	2
#26011243	2
#26011238	2
#333×C₁₇₃	2
#1684	173