Finite Group Databasebeta

Subgroups — #36353022

Order 22144.

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

normal subgroups only

This group has 144 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	#26011842	C₂	1	1	yes	maximal, characteristic
2	#26011787	C₂	1	1	yes	maximal, characteristic
3	#26011777	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₃₄₆	C₂	1	1	yes	maximal, characteristic
5	#26011239	C₂	1	1	yes	maximal, characteristic
6	#26011236	C₂	1	1	yes	maximal, characteristic
7	#335×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	#19149679	C₂²	1	1	yes	characteristic
9	#19149679	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₄₆	C₂²	1	1	yes	characteristic
11	#19149613	—	2	1	no	—
12	#19149611	—	2	1	no	—
13	C₄⋊#13063436	C₂²	1	1	yes	characteristic
14	#19149608	—	2	1	no	—
15	#19149604	—	2	1	no	—
16	#19149604	—	2	1	no	—
17	#19149604	—	2	1	no	—
18	#19149604	—	2	1	no	—
19	C₈×Dic₁₇₃	—	2	1	no	—
20	C₄⋊C₈×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	—	2	1	no	—
25	C₂×Dic₃₄₆	D₄	1	1	yes	characteristic
26	C₂×#13063436	—	2	1	no	—
27	C₄⋊Dic₁₇₃	—	2	1	no	—
28	C₄⋊Dic₁₇₃	—	2	1	no	—
29	C₄⋊Dic₁₇₃	—	2	1	no	—
30	C₄⋊Dic₁₇₃	—	2	1	no	—
31	C₄⋊Dic₁₇₃	D₄	1	1	yes	characteristic
32	#14583805	—	2	1	no	—
33	C₄×Dic₁₇₃	—	2	1	no	—
34	C₂×C₈×C₁₇₃	—	2	1	no	—
35	C₂×C₈×C₁₇₃	—	2	1	no	—
36	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
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	—	8	1	no	—
43	Dic₃₄₆	Q₁₆	1	2	yes	—
44	C₂×Dic₁₇₃	—	2	1	no	—
45	C₂×Dic₁₇₃	—	2	1	no	—
46	C₂×Dic₁₇₃	—	4	1	no	—
47	C₂×Dic₁₇₃	—	4	1	no	—
48	#13063436	—	4	1	no	—
49	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
52	C₈×C₁₇₃	—	2	1	no	—
53	C₈×C₁₇₃	—	2	1	no	—
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	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₁₇₃	#135	1	1	yes	characteristic
60	C₄×C₁₇₃	#135	1	1	yes	characteristic
61	C₄×C₁₇₃	#137	1	1	yes	characteristic
62	Dic₁₇₃	—	2	2	no	—
63	Dic₁₇₃	—	4	1	no	—
64	Dic₁₇₃	—	8	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	D₁₇₃	1	1	yes	characteristic
76	#333	—	173	1	no	—
77	#332	—	173	1	no	—
78	C₄⋊Q₈	—	173	1	no	—
79	C₄⋊Q₈	—	173	1	no	—
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	C₂×D₁₇₃	1	1	yes	characteristic
93	C₂×Q₈	—	346	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	D₆₉₂	1	1	yes	characteristic
100	C₂×C₈	D₆₉₂	1	1	yes	characteristic
101	C₄⋊C₄	—	346	1	no	—
102	C₄⋊C₄	—	346	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	—	346	1	no	—
110	Q₈	—	1384	1	no	—
111	Q₈	—	173	2	no	—
112	Q₈	—	692	1	no	—
113	Q₈	—	692	1	no	—
114	C₂×C₄	—	346	1	no	—
115	C₂×C₄	—	346	1	no	—
116	C₂×C₄	—	692	1	no	—
117	C₂×C₄	—	692	1	no	—
118	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
120	C₂×C₄	#14583827	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	692	1	no	—
126	C₂²	#19149654	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	346	2	no	—
129	C₄	—	1384	1	no	—
130	C₄	—	1384	1	no	—
131	C₄	—	692	1	no	—
132	C₄	Q₁₆×D₁₇₃	1	1	yes	characteristic
133	C₄	#19149698	1	1	yes	characteristic
134	C₄	#19149701	1	1	yes	characteristic
135	C₄	#19149712	1	1	yes	characteristic
136	C₂	#26011714	1	1	yes	characteristic
137	C₂	#26011733	1	1	yes	characteristic
138	C₂	#26011755	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149654
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#335×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#121
hypercenter	#335	D₁₇₃
hypocenter	C₁₇₃	#1684
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

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
#26011842	2
#26011787	2
#26011777	2
C₈×Dic₃₄₆	2
#26011239	2
#26011236	2
#335×C₁₇₃	2
#1684	173