Finite Group Databasebeta

Subgroups — #36355474

Order 22144.

266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×D₁₃₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19149620	C₂	1	2	yes	maximal, complemented
3	#26011291	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14583801	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₉₂	—	2	1	no	—
7	C₂×D₁₃₈₄	—	2	1	no	—
8	C₂×D₁₃₈₄	—	2	4	no	—
9	C₂×D₁₃₈₄	C₂²	1	2	yes	—
10	#19149620	—	2	4	no	—
11	C₂×#14583801	C₂²	1	2	yes	complemented
12	C₂×D₈×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	#121×C₁₇₃	—	2	1	no	—
15	#107×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
16	#102×C₁₇₃	—	2	1	no	—
17	C₂³×D₁₇₃	—	4	1	no	—
18	C₂×D₆₉₂	—	2	4	no	—
19	C₂×D₆₉₂	—	2	2	no	—
20	C₂×D₆₉₂	—	4	2	no	—
21	C₂×D₆₉₂	—	4	1	no	—
22	D₁₃₈₄	—	2	4	no	—
23	D₁₃₈₄	—	2	2	no	—
24	D₁₃₈₄	D₄	1	4	yes	complemented
25	#14583801	—	2	2	no	—
26	C₂×D₄×C₁₇₃	—	2	1	no	—
27	C₂×D₄×C₁₇₃	—	4	1	no	—
28	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
29	D₈×C₁₇₃	—	2	2	no	—
30	C₂×C₈×C₁₇₃	—	2	2	no	—
31	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇₃	—	2	1	no	—
34	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
35	C₂²×D₁₇₃	—	4	4	no	—
36	C₂²×D₁₇₃	—	4	2	no	—
37	C₂²×D₁₇₃	—	8	2	no	—
38	C₂²×D₁₇₃	—	8	2	no	—
39	D₆₉₂	—	2	4	no	—
40	D₆₉₂	—	4	4	no	—
41	D₆₉₂	—	4	2	no	—
42	C₂³×C₁₇₃	—	4	1	no	—
43	C₂³×C₁₇₃	D₈	1	1	yes	characteristic
44	D₄×C₁₇₃	—	4	1	no	—
45	D₄×C₁₇₃	—	4	1	no	—
46	D₄×C₁₇₃	—	8	2	no	—
47	C₂×C₄×C₁₇₃	—	2	1	no	—
48	C₂×C₄×C₁₇₃	—	2	1	no	—
49	C₂×C₄×C₁₇₃	—	4	1	no	—
50	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₇₃	—	2	1	no	—
53	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
54	C₂²×C₁₇₃	—	2	1	no	—
55	C₂²×C₁₇₃	—	2	1	no	—
56	C₂²×C₁₇₃	—	8	1	no	—
57	C₂²×C₁₇₃	—	8	1	no	—
58	C₂²×C₁₇₃	—	8	1	no	—
59	C₂²×C₁₇₃	D₁₆	1	2	yes	complemented
60	C₂²×C₁₇₃	C₂×D₈	1	1	yes	characteristic
61	C₂×D₁₇₃	—	4	4	no	—
62	C₂×D₁₇₃	—	8	8	no	—
63	C₂×D₁₇₃	—	8	2	no	—
64	C₂×D₁₇₃	—	8	2	no	—
65	C₄×C₁₇₃	—	2	1	no	—
66	C₄×C₁₇₃	—	8	1	no	—
67	C₄×C₁₇₃	#120	1	1	yes	characteristic
68	C₄×C₁₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂×C₁₇₃	—	2	2	no	—
70	C₂×C₁₇₃	—	16	1	no	—
71	C₂×C₁₇₃	#447	1	1	yes	characteristic
72	C₂×C₁₇₃	C₂×D₁₆	1	1	yes	characteristic
73	C₂×C₁₇₃	#509	1	1	yes	characteristic
74	D₁₇₃	—	8	4	no	—
75	C₁₇₃	#2184	1	1	yes	characteristic, complemented, Sylow
76	#2184	—	173	1	no	maximal, Sylow
77	C₂²×D₈	—	173	1	no	—
78	C₂×D₁₆	—	173	2	no	—
79	#466	D₁₇₃	1	1	yes	characteristic, complemented
80	#357	—	173	2	no	—
81	C₂²⋊C₁₆	—	173	1	no	—
82	C₂²×D₄	—	346	1	no	—
83	C₂×D₈	—	346	4	no	—
84	C₂×D₈	—	346	1	no	—
85	C₂×D₈	—	173	2	no	—
86	C₂×D₈	C₂×D₁₇₃	1	1	yes	characteristic, complemented
87	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
88	#121	—	2	1	no	—
89	D₁₆	—	346	4	no	—
90	C₂×C₁₆	—	173	2	no	—
91	#107	C₂×D₁₇₃	1	1	yes	characteristic, complemented
92	#102	—	2	1	no	—
93	C₂⁴	—	692	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	346	4	no	—
97	C₂×D₄	—	346	2	no	—
98	C₂×D₄	—	692	2	no	—
99	C₂×D₄	—	692	1	no	—
100	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
101	D₈	—	2	2	no	—
102	D₈	—	346	4	no	—
103	D₈	—	346	2	no	—
104	D₈	—	173	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	4	1	no	—
110	C₁₆	—	346	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	1384	2	no	—
113	C₂³	—	1384	2	no	—
114	C₂³	—	692	4	no	—
115	C₂³	—	692	2	no	—
116	C₂³	#14583811	1	1	yes	characteristic
117	D₄	—	4	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	8	2	no	—
120	D₄	—	346	4	no	—
121	D₄	—	692	4	no	—
122	D₄	—	692	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#14583811	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	D₄×D₁₇₃	1	2	yes	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	1384	8	no	—
136	C₂²	—	1384	2	no	—
137	C₂²	—	1384	2	no	—
138	C₂²	—	692	4	no	—
139	C₂²	#19149620	1	2	yes	complemented
140	C₂²	C₂×#14583811	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	#19149647	1	1	yes	characteristic
144	C₄	#19149704	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	16	1	no	—
147	C₂	—	1384	4	no	—
148	C₂	#26011624	1	1	yes	characteristic
149	C₂	C₂×#19149620	1	1	yes	characteristic
150	C₂	#26011926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14583811
commutator, cosocle	C₂×C₈×C₁₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₃
Fitting	#466×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×D₈
hypercenter	#466	D₁₇₃
hypocenter	C₁₇₃	#2184
LCS	G ▷ C₂×C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₃₈₄	C₂	semidirect
2. C₂×#19149620	C₂	semidirect
3. #26011291	C₂	semidirect
4. C₂²⋊#14583801	C₂	semidirect
5. #466×C₁₇₃	C₂	semidirect
11. C₂×#14583801	C₂²	semidirect
12. C₂×D₈×C₁₇₃	C₂²	semidirect
15. #107×C₁₇₃	C₂²	semidirect
24. D₁₃₈₄	D₄	semidirect
59. C₂²×C₁₇₃	D₁₆	semidirect
75. C₁₇₃	#2184	semidirect
79. #466	D₁₇₃	semidirect
86. C₂×D₈	C₂×D₁₇₃	semidirect
91. #107	C₂×D₁₇₃	semidirect
139. C₂²	#19149620	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₃₈₄	2
C₂×#19149620	2
#26011291	2
C₂²⋊#14583801	2
#466×C₁₇₃	2
#2184	173