Finite Group Databasebeta

← #22578096

Subgroups — #22578096

Order 8768.

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

normal subgroups only

This group has 238 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#17984081C211yesmaximal, characteristic, complemented
3#17984078C211yesmaximal, characteristic, complemented
4#17984070C211yesmaximal, characteristic, complemented
5#17984069C211yesmaximal, characteristic, complemented
6#17984060C211yesmaximal, characteristic, complemented
7#17984060C211yesmaximal, characteristic, complemented
8#17984060C211yesmaximal, characteristic, complemented
9D4×Dic137C211yesmaximal, characteristic, complemented
10D4×Dic137C211yesmaximal, characteristic, complemented
11C4×C22⋊D137C211yesmaximal, characteristic, complemented
12#17984048C211yesmaximal, characteristic, complemented
13#17984044C211yesmaximal, characteristic, complemented
14C2×#14555489C211yesmaximal, characteristic, complemented
15#121×C137C211yesmaximal, characteristic, complemented
16C2×C22⋊D137C2211yescharacteristic
17C2×C22⋊D137C2211yescharacteristic
18C2×C22⋊D137C2211yescharacteristic
19C22×Dic137C2211yescharacteristic
20C22×Dic137C2211yescharacteristic
21C22×Dic137C2211yescharacteristic
22C22×Dic137C2211yescharacteristic
23C2×C4×D137C2211yescharacteristic
24C4⋊Dic137C2211yescharacteristic
25C4⋊Dic137C2211yescharacteristic
26#14555489C2211yescharacteristic
27#14555489C2211yescharacteristic
28#14555489C2211yescharacteristic
29#14555489C2211yescharacteristic
30#14555489C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33C22⋊Dic137C2211yescharacteristic
34C22⋊Dic137C2211yescharacteristic
35C22⋊Dic137C2211yescharacteristic
36C22⋊Dic137C2211yescharacteristic
37C22⋊Dic137C2211yescharacteristic
38#14555487C2211yescharacteristic
39#14555487C2211yescharacteristic
40#14555487C2211yescharacteristic
41C4×Dic137C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C2×D4×C137C2211yescharacteristic
45C2×D4×C137C2211yescharacteristic
46C2×D4×C137C2211yescharacteristic
47C22×C4×C137C2211yescharacteristic
48C4⋊C4×C137C2211yescharacteristic
49C22⋊C4×C137C2211yescharacteristic
50C22⋊C4×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52C22⋊D137—22no—
53C22⋊D137—22no—
54C22⋊D137—41no—
55C2×Dic137—22no—
56C2×Dic137—22no—
57C2×Dic137—41no—
58C2×Dic137—41no—
59C2×Dic137C2311yescharacteristic
60C2×Dic137C2311yescharacteristic
61C2×Dic137C2311yescharacteristic
62C2×Dic137C2311yescharacteristic
63C2×Dic137C2311yescharacteristic
64C2×Dic137C2311yescharacteristic
65C2×Dic137C2311yescharacteristic
66C4×D137—41no—
67C23×C137C2311yescharacteristic
68C23×C137C2311yescharacteristic
69C23×C137C2311yescharacteristic
70D4×C137—22no—
71D4×C137—22no—
72D4×C137—41no—
73C2×C4×C137—41no—
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×C137—21no—
85C22×C137—21no—
86C22×C137#4112yes—
87C22×C137C2411yescharacteristic
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—21no—
91C4×C137—21no—
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—21no—
95Dic137—21no—
96Dic137—21no—
97Dic137—21no—
98Dic137—21no—
99Dic137—21no—
100Dic137—21no—
101Dic137#4112yes—
102C2×C137—22no—
103C2×C137—41no—
104C2×C137—41no—
105C2×C137C2×#4111yescharacteristic
106C2×C137C2×#4111yescharacteristic
107C2×C137H(5,𝔽2)11yescharacteristic
108D137—41no—
109C137#55311yescharacteristic, complemented, Sylow
110#553—1371nomaximal, Sylow
111#126—1371no—
112#126—1371no—
113#125—1371no—
114#123—1371no—
115#123—1371no—
116#121—1371no—
117#121—1371no—
118#121—1371no—
119#121D13711yescharacteristic, complemented
120C4×D4—1371no—
121C4×D4—1371no—
122C4×D4—1371no—
123C4×D4—1371no—
124#117—1371no—
125C2×C4⋊C4—1371no—
126C2×D4—1371no—
127C2×D4—1371no—
128C2×D4—1371no—
129C2×D4C2×D13711yescharacteristic
130C2×D4C2×D13711yescharacteristic
131C2×D4C2×D13711yescharacteristic
132C22×C4—1371no—
133C22×C4—1371no—
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⋊C4—1371no—
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⋊C4C2×D13711yescharacteristic
157C22⋊C4C2×D13711yescharacteristic
158C42—1371no—
159C42—1371no—
160C42—1371no—
161C23—1371no—
162C23C22×D13711yescharacteristic
163C23C22×D13711yescharacteristic
164C23C22×D13711yescharacteristic
165D4—22no—
166D4—22no—
167D4—41no—
168D4—2742no—
169D4—2742no—
170D4—5481no—
171C2×C4—41no—
172C2×C4—2742no—
173C2×C4—2742no—
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×C4—1371no—
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—21no—
195C22—21no—
196C22—2741no—
197C22—2741no—
198C22—2741no—
199C22#1455550912yes—
200C22C23×D13711yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—2741no—
206C4—2741no—
207C4—2741no—
208C4—2741no—
209C4—2741no—
210C4—2741no—
211C4—1372no—
212C2—22no—
213C2—41no—
214C2—41no—
215C2—5481no—
216C2#41×D13711yescharacteristic
217C2C2×#1455550911yescharacteristic
218C2#1798413311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#121×C137C2
hypercenter#121D137
hypocenterC137#553
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #17984083C2semidirect
2. #17984081C2semidirect
3. #17984078C2semidirect
4. #17984070C2semidirect
5. #17984069C2semidirect
6. #17984060C2semidirect
7. #17984060C2semidirect
8. #17984060C2semidirect
9. D4×Dic137C2semidirect
10. D4×Dic137C2semidirect
11. C4×C22⋊D137C2semidirect
12. #17984048C2semidirect
13. #17984044C2semidirect
14. C2×#14555489C2semidirect
15. #121×C137C2semidirect
109. C137#553semidirect
119. #121D137semidirect