Finite Group Databasebeta

← #22577999

Subgroups — #22577999

Order 8768.

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

normal subgroups only

This group has 218 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#17984083C211yesmaximal, characteristic, complemented
2#17984081C211yesmaximal, characteristic, complemented
3#17984078C211yesmaximal, characteristic, complemented
4#17984078C211yesmaximal, characteristic, complemented
5#17984073C211yesmaximal, characteristic, complemented
6#17984071C211yesmaximal, characteristic, complemented
7#17984070C211yesmaximal, characteristic, complemented
8#17984070C211yesmaximal, characteristic, complemented
9#17984064C211yesmaximal, characteristic, complemented
10#17984064C211yesmaximal, characteristic, complemented
11#17984061C211yesmaximal, characteristic
12#17984050C211yesmaximal, characteristic, complemented
13C4×C22⋊D137C211yesmaximal, characteristic, complemented
14C4⋊C4×D137C211yesmaximal, characteristic, complemented
15C22⋊Q8×C137C211yesmaximal, characteristic, complemented
16C2×D548C2211yescharacteristic
17C2×C22⋊D137C2211yescharacteristic
18C2×C22⋊D137C2211yescharacteristic
19C2×C4×D137C2211yescharacteristic
20C2×C4×D137C2211yescharacteristic
21C2×C4×D137C2211yescharacteristic
22C2×C4×D137C2211yescharacteristic
23C4⋊Dic137C2211yescharacteristic
24C4⋊Dic137C2211yescharacteristic
25C4⋊Dic137C2211yescharacteristic
26#14555489C2211yescharacteristic
27#14555489C2211yescharacteristic
28#14555489C2211yescharacteristic
29#14555489C2211yescharacteristic
30#14555489C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33#14555489C2211yescharacteristic
34C22⋊Dic137C2211yescharacteristic
35C22⋊Dic137C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic
39#14555487C2211yescharacteristic
40#14555487C2211yescharacteristic
41#14555487C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C2×Q8×C137C2211yescharacteristic
45C22×C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C4⋊C4×C137C2211yescharacteristic
48C4⋊C4×C137C2211yescharacteristic
49C22⋊C4×C137C2211yescharacteristic
50C22⋊C4×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52C22×D137C2311yescharacteristic
53D548—22no—
54C22⋊D137—22no—
55C22⋊D137—41no—
56C2×Dic137C2311yescharacteristic
57C2×Dic137C2311yescharacteristic
58C2×Dic137C2311yescharacteristic
59C2×Dic137C2311yescharacteristic
60C2×Dic137C2311yescharacteristic
61C2×Dic137C2311yescharacteristic
62C4×D137—22no—
63C4×D137—41no—
64C4×D137—41no—
65C4×D137—41no—
66C23×C137C2311yescharacteristic
67Q8×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137C2311yescharacteristic
70C2×C4×C137C2311yescharacteristic
71C2×C4×C137C2311yescharacteristic
72C2×C4×C137C2311yescharacteristic
73C2×C4×C137C2311yescharacteristic
74C2×C4×C137C2311yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—21no—
78C22×C137C2411yescharacteristic
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—21no—
84C2×D137#4112yes—
85C4×C137—21no—
86C4×C137—21no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—21no—
90C4×C137—21no—
91Dic137—21no—
92Dic137—21no—
93Dic137—21no—
94Dic137—21no—
95Dic137—21no—
96Dic137—21no—
97C2×C137—41no—
98C2×C137C2×#4111yescharacteristic
99C2×C137H(5,𝔽2)11yescharacteristic
100C2×C1372−1+411yescharacteristic
101D137—22no—
102D137—41no—
103C137#53911yescharacteristic, complemented, Sylow
104#539—1371nomaximal, Sylow
105#126—1371no—
106#126—1371no—
107#125—1371no—
108#125—1371no—
109#123—1371no—
110#123—1371no—
111#123—1371no—
112#123—1371no—
113C22⋊Q8—1371no—
114C22⋊Q8—1371no—
115C22⋊Q8D13711yescharacteristic, complemented
116#121—1371no—
117C4×D4—1371no—
118C4×D4—1371no—
119C2×C4⋊C4—1371no—
120C2×Q8C2×D13711yescharacteristic
121C2×D4—1371no—
122C2×D4—1371no—
123C2×D4—1371no—
124C22×C4—1371no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4—1371no—
128C22×C4C2×D13711yescharacteristic
129C4⋊C4—1371no—
130C4⋊C4—1371no—
131C4⋊C4—1371no—
132C4⋊C4—1371no—
133C4⋊C4—1371no—
134C4⋊C4—1371no—
135C4⋊C4—1371no—
136C4⋊C4—1371no—
137C4⋊C4—1371no—
138C4⋊C4—1371no—
139C4⋊C4—1371no—
140C4⋊C4C2×D13711yescharacteristic
141C4⋊C4C2×D13711yescharacteristic
142C4⋊C4C2×D13711yescharacteristic
143C22⋊C4—1371no—
144C22⋊C4—1371no—
145C22⋊C4—1371no—
146C22⋊C4—1371no—
147C22⋊C4—1371no—
148C22⋊C4—1371no—
149C22⋊C4—1371no—
150C22⋊C4—1371no—
151C22⋊C4C2×D13711yescharacteristic
152C22⋊C4C2×D13711yescharacteristic
153C42—1371no—
154C42—1371no—
155C23—1371no—
156C23—1371no—
157C23C22×D13711yescharacteristic
158Q8—41no—
159D4—2742no—
160D4—2742no—
161D4—5481no—
162C2×C4—41no—
163C2×C4—2742no—
164C2×C4—5481no—
165C2×C4—5481no—
166C2×C4—5481no—
167C2×C4—1371no—
168C2×C4—1371no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4C22×D13711yescharacteristic
174C2×C4C22×D13711yescharacteristic
175C2×C4C22×D13711yescharacteristic
176C2×C4C22×D13711yescharacteristic
177C2×C4C22×D13711yescharacteristic
178C2×C4C22×D13711yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—2741no—
183C22—2741no—
184C22—2741no—
185C22—2741no—
186C22—2741no—
187C22—1372no—
188C22C23×D13711yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—2741no—
196C4—2741no—
197C4—2741no—
198C4—2741no—
199C4—2741no—
200C4—2741no—
201C2—41no—
202C2—2742no—
203C2—5481no—
204C2#41×D13711yescharacteristic
205C2#1798413311yescharacteristic
206C2#1798413611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
FittingC22⋊Q8×C137C2
hypercenterC22⋊Q8D137
hypocenterC137#539
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #17984083C2semidirect
2. #17984081C2semidirect
3. #17984078C2semidirect
4. #17984078C2semidirect
5. #17984073C2semidirect
6. #17984071C2semidirect
7. #17984070C2semidirect
8. #17984070C2semidirect
9. #17984064C2semidirect
10. #17984064C2semidirect
12. #17984050C2semidirect
13. C4×C22⋊D137C2semidirect
14. C4⋊C4×D137C2semidirect
15. C22⋊Q8×C137C2semidirect
103. C137#539semidirect
115. C22⋊Q8D137semidirect