Finite Group Databasebeta

Subgroups — #36369965

Order 22144.

144 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 46 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	#26012535	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012533	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012419	C₂	1	1	yes	maximal, characteristic, complemented
4	#26012406	C₂	1	1	yes	maximal, characteristic, complemented
5	#26012405	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012378	C₂	1	1	yes	maximal, characteristic, complemented
7	#26011922	C₂	1	1	yes	maximal, characteristic
8	#19149767	—	2	1	no	—
9	#19149766	—	2	1	no	—
10	D₁₇₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19149750	C₂²	1	1	yes	characteristic
12	#19149750	C₂²	1	1	yes	characteristic
13	#19149744	—	2	1	no	—
14	#19149743	—	2	1	no	—
15	C₈×#10621924	C₂²	1	1	yes	characteristic
16	#19149702	C₂²	1	1	yes	characteristic
17	#19149698	C₂²	1	1	yes	characteristic
18	#19149625	C₄	1	1	yes	characteristic, complemented
19	#19149624	C₄	1	1	yes	characteristic, complemented
20	#19149620	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₇₃	C₄	1	1	yes	characteristic, complemented
23	#14583836	—	2	1	no	—
24	#14583836	—	2	1	no	—
25	#14583836	—	4	1	no	—
26	#14583835	—	4	1	no	—
27	D₁₇₃⋊C₈	—	4	1	no	—
28	D₁₇₃⋊C₈	D₄	1	1	yes	characteristic
29	C₂×#13063443	—	4	1	no	—
30	C₄×#10621924	D₄	1	1	yes	characteristic
31	#14583829	—	2	1	no	—
32	#14583827	—	2	1	no	—
33	#14583825	—	2	1	no	—
34	Dic₆₉₂	C₂×C₄	1	1	yes	characteristic
35	#14583814	—	2	1	no	—
36	#14583813	—	2	1	no	—
37	D₁₃₈₄	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
39	#14583801	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
41	D₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#10621924	—	2	1	no	—
44	#13063443	—	4	1	no	—
45	#13063443	—	4	1	no	—
46	#13063443	D₈	1	1	yes	characteristic
47	#13063443	D₈	1	1	yes	characteristic
48	Dic₃₄₆	—	2	1	no	—
49	D₆₉₂	—	2	1	no	—
50	D₆₉₂	—	4	1	no	—
51	C₂²⋊D₁₇₃	—	4	1	no	—
52	C₂×Dic₁₇₃	—	4	1	no	—
53	C₄×D₁₇₃	—	4	1	no	—
54	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
55	#13063436	#41	1	1	yes	characteristic
56	Q₈×C₁₇₃	—	2	1	no	—
57	D₄×C₁₇₃	—	2	1	no	—
58	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₁₇₃	—	4	1	no	—
60	C₂×D₁₇₃	—	4	1	no	—
61	C₂×D₁₇₃	#135	1	1	yes	characteristic
62	#10621924	—	4	1	no	—
63	C₄×C₁₇₃	—	4	1	no	—
64	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
65	Dic₁₇₃	—	4	1	no	—
66	Dic₁₇₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₇₃	—	8	1	no	—
68	C₂×C₁₇₃	C₄×D₈	1	1	yes	characteristic
69	D₁₇₃	—	2	1	no	—
70	D₁₇₃	—	8	1	no	—
71	C₁₇₃	#2179	1	1	yes	characteristic, complemented, Sylow
72	#2179	—	173	1	no	maximal, Sylow
73	#508	—	173	1	no	—
74	#443	—	173	1	no	—
75	#443	—	173	1	no	—
76	#368	—	173	1	no	—
77	#359	—	173	1	no	—
78	#359	—	173	1	no	—
79	#346	—	173	1	no	—
80	#135	—	173	1	no	—
81	#135	—	173	1	no	—
82	#131	—	346	1	no	—
83	#131	—	346	1	no	—
84	Q₃₂	—	173	1	no	—
85	SD₃₂	—	173	1	no	—
86	SD₃₂	#10621924	1	1	yes	characteristic, complemented
87	D₁₆	—	173	1	no	—
88	C₂×C₁₆	—	173	1	no	—
89	C₂×C₁₆	—	173	1	no	—
90	#108	—	173	1	no	—
91	#108	—	173	1	no	—
92	#104	—	346	1	no	—
93	#104	—	346	1	no	—
94	C₄×C₈	—	173	1	no	—
95	#41	—	346	1	no	—
96	#41	—	346	1	no	—
97	Q₁₆	—	173	1	no	—
98	Q₁₆	C₂×#10621924	1	1	yes	characteristic
99	SD₁₆	—	346	1	no	—
100	SD₁₆	—	346	1	no	—
101	D₈	—	173	1	no	—
102	D₈	C₂×#10621924	1	1	yes	characteristic
103	M₁₆	—	346	1	no	—
104	M₁₆	—	346	1	no	—
105	M₁₆	—	692	1	no	—
106	M₁₆	—	692	1	no	—
107	C₂×C₈	—	173	1	no	—
108	C₂×C₈	—	173	1	no	—
109	C₂×C₈	—	692	1	no	—
110	C₂×C₈	—	692	1	no	—
111	C₄²	—	173	1	no	—
112	C₁₆	—	346	1	no	—
113	C₁₆	—	173	1	no	—
114	C₁₆	C₂×#10621924	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	346	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	346	1	no	—
119	D₄	—	692	1	no	—
120	D₄	—	692	1	no	—
121	C₂×C₄	—	346	1	no	—
122	C₂×C₄	—	173	1	no	—
123	C₂×C₄	—	692	1	no	—
124	C₂×C₄	—	692	1	no	—
125	C₈	—	173	1	no	—
126	C₈	—	173	1	no	—
127	C₈	—	173	1	no	—
128	C₈	—	692	1	no	—
129	C₈	—	692	1	no	—
130	C₈	C₂²×#10621924	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	173	1	no	—
133	C₂²	—	692	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	173	1	no	—
136	C₄	—	692	1	no	—
137	C₄	—	692	1	no	—
138	C₄	D₄×#10621924	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	346	1	no	—
141	C₂	—	1384	1	no	—
142	C₂	D₈×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#10621924
commutator	C₈×C₁₇₃	C₂²×C₄
Frattini	C₈	C₂²×#10621924
Fitting	SD₃₂×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₄×D₈
prime radical	#26011922	C₂
cosocle	C₈×D₁₇₃	C₂³
hypercenter	SD₃₂	#10621924
hypocenter	C₁₇₃	#2179
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #26012535	C₂	semidirect
2. #26012533	C₂	semidirect
3. #26012419	C₂	semidirect
4. #26012406	C₂	semidirect
5. #26012405	C₂	semidirect
6. #26012378	C₂	semidirect
18. #19149625	C₄	semidirect
19. #19149624	C₄	semidirect
20. #19149620	C₄	semidirect
22. SD₃₂×C₁₇₃	C₄	semidirect
71. C₁₇₃	#2179	semidirect
86. SD₃₂	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012535	2
#26012533	2
#26012419	2
#26012406	2
#26012405	2
#26012378	2
#26011922	2
#2179	173