Finite Group Databasebeta

← #22578086

Subgroups — #22578086

Order 8768.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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#17984083C211yesmaximal, characteristic, complemented
2#17984082C211yesmaximal, characteristic, complemented
3#17984080C211yesmaximal, characteristic, complemented
4#17984078C211yesmaximal, characteristic, complemented
5#17984078C211yesmaximal, characteristic, complemented
6#17984072C211yesmaximal, characteristic
7#17984071C211yesmaximal, characteristic
8#17984066C211yesmaximal, characteristic, complemented
9#17984065C211yesmaximal, characteristic, complemented
10C4×Dic274C211yesmaximal, characteristic, complemented
11#17984048C211yesmaximal, characteristic
12#17984045C211yesmaximal, characteristic, complemented
13#17984044C211yesmaximal, characteristic, complemented
14C2×#14555489C211yesmaximal, characteristic, complemented
15#117×C137C211yesmaximal, characteristic, complemented
16C2×Dic274C2211yescharacteristic
17C2×C22⋊D137C2211yescharacteristic
18C22×Dic137C2211yescharacteristic
19C22×Dic137C2211yescharacteristic
20C2×C4×D137C2211yescharacteristic
21C4⋊Dic137C2211yescharacteristic
22C4⋊Dic137C2211yescharacteristic
23C4⋊Dic137C2211yescharacteristic
24#14555489C2211yescharacteristic
25#14555489C2211yescharacteristic
26#14555489C2211yescharacteristic
27#14555489C2211yescharacteristic
28#14555489C2211yescharacteristic
29#14555489C2211yescharacteristic
30#14555489C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33#14555489C2211yescharacteristic
34#14555489C2211yescharacteristic
35C22⋊Dic137C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic
39#14555487C2211yescharacteristic
40#14555487C2211yescharacteristic
41C4×Dic137C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C22×C4×C137C2211yescharacteristic
45C4⋊C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C22⋊C4×C137C2211yescharacteristic
48C22⋊C4×C137C2211yescharacteristic
49C42×C137C2211yescharacteristic
50C42×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52Dic274—22no—
53C22⋊D137—22no—
54C2×Dic137—22no—
55C2×Dic137—22no—
56C2×Dic137C2311yescharacteristic
57C2×Dic137C2311yescharacteristic
58C2×Dic137C2311yescharacteristic
59C2×Dic137C2311yescharacteristic
60C2×Dic137C2311yescharacteristic
61C2×Dic137C2311yescharacteristic
62C2×Dic137C2311yescharacteristic
63C4×D137—22no—
64C23×C137C2311yescharacteristic
65C2×C4×C137—22no—
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#4112yes—
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—21no—
91Dic137#4112yes—
92C2×C137—22no—
93C2×C137C2×#4111yescharacteristic
94C2×C137C2×#4111yescharacteristic
95C2×C1372−1+411yescharacteristic
96D137—41no—
97C137#55211yescharacteristic, complemented, Sylow
98#552—1371nomaximal, Sylow
99#126—1371no—
100#126—1371no—
101#125—1371no—
102#125—1371no—
103#125—1371no—
104#123—1371no—
105#123—1371no—
106C22⋊Q8—1371no—
107C22⋊Q8—1371no—
108C4×Q8—1371no—
109C4×D4—1371no—
110#117—1371no—
111#117—1371no—
112#117D13711yescharacteristic, complemented
113C2×C4⋊C4—1371no—
114C2×Q8—1371no—
115C2×D4—1371no—
116C22×C4—1371no—
117C22×C4—1371no—
118C22×C4—1371no—
119C22×C4C2×D13711yescharacteristic
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⋊C4—1371no—
130C4⋊C4—1371no—
131C4⋊C4—1371no—
132C4⋊C4—1371no—
133C4⋊C4—1371no—
134C4⋊C4C2×D13711yescharacteristic
135C4⋊C4C2×D13711yescharacteristic
136C22⋊C4—1371no—
137C22⋊C4—1371no—
138C22⋊C4—1371no—
139C22⋊C4—1371no—
140C22⋊C4—1371no—
141C22⋊C4—1371no—
142C22⋊C4C2×D13711yescharacteristic
143C22⋊C4C2×D13711yescharacteristic
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42C2×D13711yescharacteristic
148C42C2×D13711yescharacteristic
149C23—1371no—
150C23C22×D13711yescharacteristic
151Q8—2742no—
152D4—2742no—
153C2×C4—22no—
154C2×C4—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—2741no—
173C22—2741no—
174C22—2741no—
175C22#1455551012yes—
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—2741no—
189C4—1372no—
190C2—22no—
191C2—5481no—
192C2#41×D13711yescharacteristic
193C2C2×#1455551011yescharacteristic
194C2#1798413511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#117×C137C2
hypercenter#117D137
hypocenterC137#552
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #17984083C2semidirect
2. #17984082C2semidirect
3. #17984080C2semidirect
4. #17984078C2semidirect
5. #17984078C2semidirect
8. #17984066C2semidirect
9. #17984065C2semidirect
10. C4×Dic274C2semidirect
12. #17984045C2semidirect
13. #17984044C2semidirect
14. C2×#14555489C2semidirect
15. #117×C137C2semidirect
97. C137#552semidirect
112. #117D137semidirect