Finite Group Databasebeta

Subgroups — #36355508

Order 22144.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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	#26012291	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011927	C₂	1	2	yes	maximal, complemented
3	#26011300	C₂	1	2	yes	maximal, complemented
4	#26011273	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14583826	—	2	1	no	—
7	#19149712	—	2	2	no	—
8	#19149707	—	2	2	no	—
9	#19149701	—	2	1	no	—
10	#19149701	C₂²	1	2	yes	—
11	#19149623	—	2	2	no	—
12	#19149620	—	2	2	no	—
13	#19149619	C₂²	1	2	yes	complemented
14	C₂×D₈×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
16	#121×C₁₇₃	—	2	1	no	—
17	#106×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
18	#102×C₁₇₃	—	2	1	no	—
19	#14583826	—	2	2	no	—
20	#14583826	—	2	2	no	—
21	#14583826	—	4	1	no	—
22	#14583826	—	4	1	no	—
23	C₂×Dic₃₄₆	—	2	1	no	—
24	C₂×D₆₉₂	—	2	1	no	—
25	C₂×C₂²⋊D₁₇₃	—	4	1	no	—
26	C₂×C₄×D₁₇₃	—	4	1	no	—
27	Dic₆₉₂	—	2	1	no	—
28	Dic₆₉₂	D₄	1	2	yes	complemented
29	#14583815	—	2	2	no	—
30	#14583815	—	2	2	no	—
31	D₁₃₈₄	—	2	1	no	—
32	D₁₃₈₄	D₄	1	2	yes	complemented
33	#14583801	—	2	2	no	—
34	C₂×D₄×C₁₇₃	—	2	1	no	—
35	C₂×D₄×C₁₇₃	—	4	1	no	—
36	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
37	D₈×C₁₇₃	—	2	2	no	—
38	M₁₆×C₁₇₃	—	2	2	no	—
39	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₇₃	—	2	1	no	—
42	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
43	C₂²×D₁₇₃	—	4	1	no	—
44	Dic₃₄₆	—	2	2	no	—
45	Dic₃₄₆	—	4	1	no	—
46	D₆₉₂	—	2	2	no	—
47	D₆₉₂	—	4	1	no	—
48	C₂²⋊D₁₇₃	—	4	2	no	—
49	C₂²⋊D₁₇₃	—	4	2	no	—
50	C₂×Dic₁₇₃	—	4	1	no	—
51	C₄×D₁₇₃	—	4	2	no	—
52	C₄×D₁₇₃	—	4	2	no	—
53	C₂³×C₁₇₃	—	4	1	no	—
54	C₂³×C₁₇₃	D₈	1	1	yes	characteristic
55	D₄×C₁₇₃	—	4	1	no	—
56	D₄×C₁₇₃	—	4	1	no	—
57	D₄×C₁₇₃	—	8	2	no	—
58	C₂×C₄×C₁₇₃	—	2	1	no	—
59	C₂×C₄×C₁₇₃	—	2	1	no	—
60	C₂×C₄×C₁₇₃	—	4	1	no	—
61	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₇₃	—	2	1	no	—
64	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
65	C₂²×C₁₇₃	—	2	1	no	—
66	C₂²×C₁₇₃	—	4	1	no	—
67	C₂²×C₁₇₃	—	8	1	no	—
68	C₂²×C₁₇₃	—	8	1	no	—
69	C₂²×C₁₇₃	—	8	1	no	—
70	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂×D₁₇₃	—	4	2	no	—
72	C₂×D₁₇₃	—	8	2	no	—
73	C₄×C₁₇₃	—	2	1	no	—
74	C₄×C₁₇₃	—	8	1	no	—
75	C₄×C₁₇₃	#120	1	1	yes	characteristic
76	C₄×C₁₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₁₇₃	—	4	2	no	—
78	C₂×C₁₇₃	—	2	1	no	—
79	C₂×C₁₇₃	—	4	1	no	—
80	C₂×C₁₇₃	—	16	1	no	—
81	C₂×C₁₇₃	#447	1	1	yes	characteristic
82	D₁₇₃	—	8	2	no	—
83	C₁₇₃	#2190	1	1	yes	characteristic, complemented, Sylow
84	#2190	—	173	1	no	maximal, Sylow
85	#575	—	173	1	no	—
86	#509	—	173	2	no	—
87	#469	D₁₇₃	1	1	yes	characteristic, complemented
88	#360	—	173	2	no	—
89	#349	—	173	1	no	—
90	C₂×#41	—	346	1	no	—
91	#137	—	346	2	no	—
92	AGL(1,ℤ/8)	—	346	2	no	—
93	#135	—	346	1	no	—
94	#135	—	173	2	no	—
95	C₂×D₈	C₂×D₁₇₃	1	1	yes	characteristic, complemented
96	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
97	#121	—	2	1	no	—
98	SD₃₂	—	346	2	no	—
99	D₁₆	—	346	2	no	—
100	M₃₂	—	173	2	no	—
101	#106	C₂×D₁₇₃	1	1	yes	characteristic, complemented
102	#102	—	2	1	no	—
103	#41	—	346	2	no	—
104	#41	—	346	2	no	—
105	#41	—	692	1	no	—
106	#41	—	692	1	no	—
107	C₂×Q₈	—	346	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	346	1	no	—
111	C₂×D₄	—	692	1	no	—
112	C₂²×C₄	—	692	1	no	—
113	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
114	Q₁₆	—	346	1	no	—
115	Q₁₆	—	173	2	no	—
116	SD₁₆	—	346	2	no	—
117	SD₁₆	—	346	2	no	—
118	D₈	—	2	2	no	—
119	D₈	—	346	1	no	—
120	D₈	—	173	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₁₆	—	346	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	692	1	no	—
129	C₂³	#14583811	1	1	yes	characteristic
130	Q₈	—	346	2	no	—
131	Q₈	—	692	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	8	2	no	—
135	D₄	—	346	2	no	—
136	D₄	—	692	1	no	—
137	D₄	—	692	2	no	—
138	D₄	—	692	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	692	2	no	—
143	C₂×C₄	—	692	1	no	—
144	C₂×C₄	—	692	2	no	—
145	C₂×C₄	#14583811	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₁₇₃	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	1384	2	no	—
155	C₂²	—	692	2	no	—
156	C₂²	C₂×#14583811	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	8	1	no	—
159	C₄	—	692	2	no	—
160	C₄	#19149647	1	1	yes	characteristic
161	C₄	#19149704	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	16	1	no	—
165	C₂	—	1384	2	no	—
166	C₂	#26011624	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011624
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#469×C₁₇₃	C₂
socle	C₂×C₁₇₃	#447
hypercenter	#469	D₁₇₃
hypocenter	C₁₇₃	#2190
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #26012291	C₂	semidirect
2. #26011927	C₂	semidirect
3. #26011300	C₂	semidirect
4. #26011273	C₂	semidirect
5. #469×C₁₇₃	C₂	semidirect
13. #19149619	C₂²	semidirect
14. C₂×D₈×C₁₇₃	C₂²	semidirect
17. #106×C₁₇₃	C₂²	semidirect
28. Dic₆₉₂	D₄	semidirect
32. D₁₃₈₄	D₄	semidirect
83. C₁₇₃	#2190	semidirect
87. #469	D₁₇₃	semidirect
95. C₂×D₈	C₂×D₁₇₃	semidirect
101. #106	C₂×D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26012291	2
#26011927	2
#26011300	2
#26011273	2
#469×C₁₇₃	2
#2190	173