Finite Group Databasebeta

← #22578150

Subgroups — #22578150

Order 8768.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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#17984079C211yesmaximal, characteristic, complemented
2#17984074C211yesmaximal, characteristic, complemented
3#17984074C211yesmaximal, characteristic, complemented
4#17984073C211yesmaximal, characteristic, complemented
5#17984072C211yesmaximal, characteristic, complemented
6#17984070C211yesmaximal, characteristic, complemented
7#17984068C211yesmaximal, characteristic, complemented
8#17984067C211yesmaximal, characteristic, complemented
9#17984066C211yesmaximal, characteristic, complemented
10#17984065C211yesmaximal, characteristic, complemented
11#17984062C211yesmaximal, characteristic, complemented
12#17984061C211yesmaximal, characteristic, complemented
13#17984060C211yesmaximal, characteristic, complemented
14#17984059C211yesmaximal, characteristic, complemented
15#123×C137C211yesmaximal, characteristic, complemented
16C2×Dic274C2211yescharacteristic
17C2×Dic274C2211yescharacteristic
18C2×D548C2211yescharacteristic
19C2×C22⋊D137C2211yescharacteristic
20C2×C22⋊D137C2211yescharacteristic
21C2×C22⋊D137C2211yescharacteristic
22C2×C22⋊D137C2211yescharacteristic
23C22×Dic137C2211yescharacteristic
24C2×C4×D137C2211yescharacteristic
25C2×C4×D137C2211yescharacteristic
26C4⋊Dic137C2211yescharacteristic
27C4⋊Dic137C2211yescharacteristic
28C4⋊Dic137C2211yescharacteristic
29C4⋊Dic137C2211yescharacteristic
30#14555489C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33#14555489C2211yescharacteristic
34C22⋊Dic137C2211yescharacteristic
35C22⋊Dic137C2211yescharacteristic
36C22⋊Dic137C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic
39#14555487C2211yescharacteristic
40#14555487C2211yescharacteristic
41#14555487C2211yescharacteristic
42#14555487C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C2×D4×C137C2211yescharacteristic
45C22×C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C4⋊C4×C137C2211yescharacteristic
48C22⋊C4×C137C2211yescharacteristic
49C22⋊C4×C137C2211yescharacteristic
50C22⋊C4×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52C22×D137C2311yescharacteristic
53Dic274—41no—
54Dic274—41no—
55D548—41no—
56C22⋊D137—41no—
57C22⋊D137—41no—
58C22⋊D137—41no—
59C22⋊D137—41no—
60C2×Dic137—41no—
61C2×Dic137C2311yescharacteristic
62C2×Dic137C2311yescharacteristic
63C2×Dic137C2311yescharacteristic
64C2×Dic137C2311yescharacteristic
65C2×Dic137C2311yescharacteristic
66C2×Dic137C2311yescharacteristic
67C4×D137—41no—
68C4×D137—41no—
69C23×C137C2311yescharacteristic
70C23×C137C2311yescharacteristic
71D4×C137—41no—
72C2×C4×C137—41no—
73C2×C4×C137C2311yescharacteristic
74C2×C4×C137C2311yescharacteristic
75C2×C4×C137C2311yescharacteristic
76C2×C4×C137C2311yescharacteristic
77C2×C4×C137C2311yescharacteristic
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—21no—
83C22×C137—21no—
84C22×C137C2411yescharacteristic
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—21no—
91C4×C137—21no—
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—21no—
95C4×C137—21no—
96Dic137—21no—
97Dic137—21no—
98Dic137—21no—
99Dic137—21no—
100Dic137—21no—
101Dic137—21no—
102C2×C137—41no—
103C2×C137—41no—
104C2×C137H(5,𝔽2)11yescharacteristic
105C2×C137H(5,𝔽2)11yescharacteristic
106C2×C1372−1+411yescharacteristic
107D137—41no—
108D137—41no—
109C137#56211yescharacteristic, complemented, Sylow
110#562—1371nomaximal, Sylow
111#125—1371no—
112#124—1371no—
113#124—1371no—
114#123—1371no—
115#123—1371no—
116#123—1371no—
117#123D13711yescharacteristic, complemented
118C22⋊Q8—1371no—
119C22⋊Q8—1371no—
120C22⋊Q8—1371no—
121C22⋊Q8—1371no—
122#121—1371no—
123#121—1371no—
124#121—1371no—
125#121—1371no—
126C2×Q8—1371no—
127C2×Q8—1371no—
128C2×D4—1371no—
129C2×D4—1371no—
130C2×D4—1371no—
131C2×D4—1371no—
132C2×D4—1371no—
133C2×D4C2×D13711yescharacteristic
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4C2×D13711yescharacteristic
138C4⋊C4—1371no—
139C4⋊C4—1371no—
140C4⋊C4—1371no—
141C4⋊C4—1371no—
142C4⋊C4—1371no—
143C4⋊C4—1371no—
144C4⋊C4—1371no—
145C4⋊C4—1371no—
146C4⋊C4C2×D13711yescharacteristic
147C4⋊C4C2×D13711yescharacteristic
148C22⋊C4—1371no—
149C22⋊C4—1371no—
150C22⋊C4—1371no—
151C22⋊C4—1371no—
152C22⋊C4—1371no—
153C22⋊C4—1371no—
154C22⋊C4—1371no—
155C22⋊C4—1371no—
156C22⋊C4—1371no—
157C22⋊C4C2×D13711yescharacteristic
158C22⋊C4C2×D13711yescharacteristic
159C22⋊C4C2×D13711yescharacteristic
160C42—1371no—
161C23—1371no—
162C23—1371no—
163C23C22×D13711yescharacteristic
164C23C22×D13711yescharacteristic
165Q8—5481no—
166Q8—5481no—
167D4—41no—
168D4—5481no—
169D4—5481no—
170D4—5481no—
171D4—5481no—
172D4—5481no—
173C2×C4—41no—
174C2×C4—5481no—
175C2×C4—5481no—
176C2×C4—5481no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4C22×D13711yescharacteristic
184C2×C4C22×D13711yescharacteristic
185C2×C4C22×D13711yescharacteristic
186C2×C4C22×D13711yescharacteristic
187C2×C4C22×D13711yescharacteristic
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—2741no—
195C22—2741no—
196C22—2741no—
197C22—2741no—
198C22—2741no—
199C22—2741no—
200C22C23×D13711yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—2741no—
207C4—2741no—
208C4—2741no—
209C4—2741no—
210C4—2741no—
211C4—2741no—
212C2—41no—
213C2—41no—
214C2—5481no—
215C2—5481no—
216C2#1798413311yescharacteristic
217C2#1798413411yescharacteristic
218C2#1798413511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#123×C137C2
hypercenter#123D137
hypocenterC137#562
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #17984079C2semidirect
2. #17984074C2semidirect
3. #17984074C2semidirect
4. #17984073C2semidirect
5. #17984072C2semidirect
6. #17984070C2semidirect
7. #17984068C2semidirect
8. #17984067C2semidirect
9. #17984066C2semidirect
10. #17984065C2semidirect
11. #17984062C2semidirect
12. #17984061C2semidirect
13. #17984060C2semidirect
14. #17984059C2semidirect
15. #123×C137C2semidirect
109. C137#562semidirect
117. #123D137semidirect