Finite Group Databasebeta

← #22578012

Subgroups — #22578012

Order 8768.

192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17984087C211yesmaximal, characteristic, complemented
2#17984083C211yesmaximal, characteristic
3#17984083C211yesmaximal, characteristic
4#17984083C211yesmaximal, characteristic
5#17984082C211yesmaximal, characteristic
6#17984079C211yesmaximal, characteristic, complemented
7#17984078C211yesmaximal, characteristic, complemented
8#17984078C211yesmaximal, characteristic, complemented
9#17984078C211yesmaximal, characteristic, complemented
10#17984066C211yesmaximal, characteristic
11#17984066C211yesmaximal, characteristic
12C4×Dic274C211yesmaximal, characteristic, complemented
13#17984053C211yesmaximal, characteristic, complemented
14#17984046C211yesmaximal, characteristic
15#125×C137C211yesmaximal, characteristic, complemented
16C2×Dic274C2211yescharacteristic
17C2×Dic274C2211yescharacteristic
18C2×C4×D137C2211yescharacteristic
19C4⋊Dic137C2211yescharacteristic
20C4⋊Dic137C2211yescharacteristic
21C4⋊Dic137C2211yescharacteristic
22C4⋊Dic137C2211yescharacteristic
23#14555489C2211yescharacteristic
24#14555489C2211yescharacteristic
25#14555489C2211yescharacteristic
26#14555489C2211yescharacteristic
27#14555489C2211yescharacteristic
28#14555489C2211yescharacteristic
29#14555489C2211yescharacteristic
30#14555489C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33#14555487C2211yescharacteristic
34#14555487C2211yescharacteristic
35#14555487C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic
39C4×Dic137C2211yescharacteristic
40C4×Dic137C2211yescharacteristic
41C4×Dic137C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C4⋊C4×C137C2211yescharacteristic
45C4⋊C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C4⋊C4×C137C2211yescharacteristic
48C4⋊C4×C137C2211yescharacteristic
49C4⋊C4×C137C2211yescharacteristic
50C42×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52Dic274—22no—
53Dic274—22no—
54C2×Dic137C2311yescharacteristic
55C2×Dic137C2311yescharacteristic
56C2×Dic137C2311yescharacteristic
57C2×Dic137C2311yescharacteristic
58C2×Dic137C2311yescharacteristic
59C2×Dic137C2311yescharacteristic
60C2×Dic137C2311yescharacteristic
61C4×D137—22no—
62C2×C4×C137C2311yescharacteristic
63C2×C4×C137C2311yescharacteristic
64C2×C4×C137C2311yescharacteristic
65C2×C4×C137C2311yescharacteristic
66C2×C4×C137C2311yescharacteristic
67C2×C4×C137C2311yescharacteristic
68C2×C4×C137C2311yescharacteristic
69C22×C137C2411yescharacteristic
70C2×D137—21no—
71C2×D137—21no—
72C2×D137—21no—
73C4×C137—21no—
74C4×C137—21no—
75C4×C137—21no—
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—21no—
79C4×C137—21no—
80Dic137—21no—
81Dic137—21no—
82Dic137—21no—
83Dic137—21no—
84Dic137—21no—
85Dic137—21no—
86Dic137#4112yes—
87C2×C137C2×#4111yescharacteristic
88C2×C1372−1+411yescharacteristic
89C2×C1372−1+411yescharacteristic
90D137—41no—
91C137#54111yescharacteristic, complemented, Sylow
92#541—1371nomaximal, Sylow
93C4⋊Q8—1371no—
94#126—1371no—
95#126—1371no—
96#126—1371no—
97#126—1371no—
98#125—1371no—
99#125—1371no—
100#125—1371no—
101#125—1371no—
102#125D13711yescharacteristic, complemented
103C22⋊Q8—1371no—
104C22⋊Q8—1371no—
105C4×Q8—1371no—
106C4×Q8—1371no—
107#117—1371no—
108C2×Q8—1371no—
109C2×Q8—1371no—
110C22×C4—1371no—
111C4⋊C4—1371no—
112C4⋊C4—1371no—
113C4⋊C4—1371no—
114C4⋊C4—1371no—
115C4⋊C4—1371no—
116C4⋊C4—1371no—
117C4⋊C4—1371no—
118C4⋊C4—1371no—
119C4⋊C4—1371no—
120C4⋊C4—1371no—
121C4⋊C4—1371no—
122C4⋊C4—1371no—
123C4⋊C4—1371no—
124C4⋊C4—1371no—
125C4⋊C4C2×D13711yescharacteristic
126C4⋊C4C2×D13711yescharacteristic
127C4⋊C4C2×D13711yescharacteristic
128C4⋊C4C2×D13711yescharacteristic
129C4⋊C4C2×D13711yescharacteristic
130C4⋊C4C2×D13711yescharacteristic
131C22⋊C4—1371no—
132C22⋊C4—1371no—
133C22⋊C4—1371no—
134C22⋊C4—1371no—
135C22⋊C4—1371no—
136C22⋊C4—1371no—
137C42—1371no—
138C42—1371no—
139C42—1371no—
140C42—1371no—
141C42—1371no—
142C42C2×D13711yescharacteristic
143C23—1371no—
144Q8—2742no—
145Q8—2742no—
146C2×C4—2742no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4—1371no—
152C2×C4—1371no—
153C2×C4—1371no—
154C2×C4C22×D13711yescharacteristic
155C2×C4C22×D13711yescharacteristic
156C2×C4C22×D13711yescharacteristic
157C2×C4C22×D13711yescharacteristic
158C2×C4C22×D13711yescharacteristic
159C2×C4C22×D13711yescharacteristic
160C2×C4C22×D13711yescharacteristic
161C22—2741no—
162C22—2741no—
163C22—2741no—
164C22C23×D13711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—2741no—
173C4—2741no—
174C4—2741no—
175C4—2741no—
176C4—2741no—
177C4—2741no—
178C4—1372no—
179C2—5481no—
180C2#41×D13711yescharacteristic
181C2#1798413511yescharacteristic
182C2#1798413611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#125×C137C2
hypercenter#125D137
hypocenterC137#541
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #17984087C2semidirect
6. #17984079C2semidirect
7. #17984078C2semidirect
8. #17984078C2semidirect
9. #17984078C2semidirect
12. C4×Dic274C2semidirect
13. #17984053C2semidirect
15. #125×C137C2semidirect
91. C137#541semidirect
102. #125D137semidirect