Finite Group Databasebeta

← #22578120

Subgroups — #22578120

Order 8768.

212 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 95 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, complemented
3#17984083C211yesmaximal, characteristic, complemented
4#17984081C211yesmaximal, characteristic, complemented
5#17984074C211yesmaximal, characteristic
6#17984074C211yesmaximal, characteristic
7#17984066C211yesmaximal, characteristic, complemented
8#17984065C211yesmaximal, characteristic, complemented
9C4×Dic274C211yesmaximal, characteristic, complemented
10#17984053C211yesmaximal, characteristic, complemented
11#17984053C211yesmaximal, characteristic, complemented
12#17984048C211yesmaximal, characteristic
13#17984045C211yesmaximal, characteristic, complemented
14#17984044C211yesmaximal, characteristic, complemented
15#126×C137C211yesmaximal, characteristic, complemented
16C2×Dic274C2211yescharacteristic
17C2×Dic274C2211yescharacteristic
18C2×Dic274C2211yescharacteristic
19C2×C22⋊D137C2211yescharacteristic
20C22×Dic137C2211yescharacteristic
21C2×C4×D137C2211yescharacteristic
22C4⋊Dic137C2211yescharacteristic
23C4⋊Dic137C2211yescharacteristic
24#14555489C2211yescharacteristic
25#14555489C2211yescharacteristic
26#14555489C2211yescharacteristic
27#14555489C2211yescharacteristic
28#14555489C2211yescharacteristic
29#14555489C2211yescharacteristic
30#14555489C2211yescharacteristic
31C22⋊Dic137C2211yescharacteristic
32C22⋊Dic137C2211yescharacteristic
33#14555487C2211yescharacteristic
34#14555487C2211yescharacteristic
35#14555487C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38C4×Dic137C2211yescharacteristic
39C4×Dic137C2211yescharacteristic
40C4×Dic137C2211yescharacteristic
41C4×Dic137C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C4⋊C4×C137C2211yescharacteristic
45C4⋊C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C22⋊C4×C137C2211yescharacteristic
48C22⋊C4×C137C2211yescharacteristic
49C22⋊C4×C137C2211yescharacteristic
50C42×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52Dic274—22no—
53Dic274—22no—
54Dic274—22no—
55C22⋊D137—22no—
56C2×Dic137—22no—
57C2×Dic137C2311yescharacteristic
58C2×Dic137C2311yescharacteristic
59C2×Dic137C2311yescharacteristic
60C2×Dic137C2311yescharacteristic
61C2×Dic137C2311yescharacteristic
62C2×Dic137C2311yescharacteristic
63C2×Dic137C2311yescharacteristic
64C4×D137—22no—
65C23×C137C2311yescharacteristic
66C2×C4×C137C2311yescharacteristic
67C2×C4×C137C2311yescharacteristic
68C2×C4×C137C2311yescharacteristic
69C2×C4×C137C2311yescharacteristic
70C2×C4×C137C2311yescharacteristic
71C2×C4×C137C2311yescharacteristic
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—21no—
75C22×C137C2411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—21no—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—21no—
83C4×C137—21no—
84C4×C137—21no—
85Dic137—21no—
86Dic137—21no—
87Dic137—21no—
88Dic137—21no—
89Dic137—21no—
90Dic137#4112yes—
91Dic137#4112yes—
92C2×C137—41no—
93C2×C137C2×#4111yescharacteristic
94C2×C137C2×#4111yescharacteristic
95C2×C1372−1+411yescharacteristic
96D137—41no—
97C137#55611yescharacteristic, complemented, Sylow
98#556—1371nomaximal, Sylow
99C4⋊Q8—1371no—
100#126—1371no—
101#126—1371no—
102#126—1371no—
103#126D13711yescharacteristic, complemented
104#124—1371no—
105#124—1371no—
106C22⋊Q8—1371no—
107C22⋊Q8—1371no—
108C4×Q8—1371no—
109C4×Q8—1371no—
110C4×Q8—1371no—
111C4×D4—1371no—
112#117—1371no—
113#117—1371no—
114C2×Q8—1371no—
115C2×Q8—1371no—
116C2×Q8—1371no—
117C2×D4—1371no—
118C22×C4—1371no—
119C22×C4—1371no—
120C4⋊C4—1371no—
121C4⋊C4—1371no—
122C4⋊C4—1371no—
123C4⋊C4—1371no—
124C4⋊C4—1371no—
125C4⋊C4—1371no—
126C4⋊C4—1371no—
127C4⋊C4—1371no—
128C4⋊C4—1371no—
129C4⋊C4C2×D13711yescharacteristic
130C4⋊C4C2×D13711yescharacteristic
131C4⋊C4C2×D13711yescharacteristic
132C22⋊C4—1371no—
133C22⋊C4—1371no—
134C22⋊C4—1371no—
135C22⋊C4—1371no—
136C22⋊C4—1371no—
137C22⋊C4—1371no—
138C22⋊C4—1371no—
139C22⋊C4C2×D13711yescharacteristic
140C22⋊C4C2×D13711yescharacteristic
141C22⋊C4C2×D13711yescharacteristic
142C42—1371no—
143C42—1371no—
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42—1371no—
148C42C2×D13711yescharacteristic
149C23—1371no—
150C23C22×D13711yescharacteristic
151Q8—2742no—
152Q8—2742no—
153Q8—2742no—
154D4—2742no—
155C2×C4—2742no—
156C2×C4—2742no—
157C2×C4—1371no—
158C2×C4—1371no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1371no—
163C2×C4—1371no—
164C2×C4C22×D13711yescharacteristic
165C2×C4C22×D13711yescharacteristic
166C2×C4C22×D13711yescharacteristic
167C2×C4C22×D13711yescharacteristic
168C2×C4C22×D13711yescharacteristic
169C2×C4C22×D13711yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—2741no—
174C22—2741no—
175C22—2741no—
176C22C23×D13711yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—2741no—
184C4—2741no—
185C4—2741no—
186C4—2741no—
187C4—2741no—
188C4—1372no—
189C4—1372no—
190C2—41no—
191C2—5481no—
192C2#41×D13711yescharacteristic
193C2#41×D13711yescharacteristic
194C2#1798413511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#126×C137C2
hypercenter#126D137
hypocenterC137#556
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #17984087C2semidirect
2. #17984083C2semidirect
3. #17984083C2semidirect
4. #17984081C2semidirect
7. #17984066C2semidirect
8. #17984065C2semidirect
9. C4×Dic274C2semidirect
10. #17984053C2semidirect
11. #17984053C2semidirect
13. #17984045C2semidirect
14. #17984044C2semidirect
15. #126×C137C2semidirect
97. C137#556semidirect
103. #126D137semidirect