Finite Group Databasebeta

← #21456824

Subgroups — #21456824

Order 7808.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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#16840319C211yesmaximal, characteristic, complemented
2C2×#14519765C211yesmaximal, characteristic
3#16840156C212yesmaximal, complemented
4#16840122C212yesmaximal, complemented
5#16839985C214yesmaximal, complemented
6#16839920C212yesmaximal, complemented
7#16839915C212yesmaximal
8C61×#432C211yesmaximal, characteristic, complemented
9C22×Dic122C2211yescharacteristic
10C22×D244C2211yescharacteristic
11C2×Dic244C2214yescomplemented
12C2×#11758190C2214yes—
13#14519777C2211yescharacteristic, complemented
14#14519777C2211yescharacteristic, complemented
15#14519769—21no—
16#14519765—21no—
17#14519765—21no—
18#14519765C2212yes—
19#14519765C2212yes—
20#14519756—21no—
21#14519756C2212yes—
22C4×Dic122—21no—
23C4×Dic122C2212yescomplemented
24#14519736—21no—
25C2×#11758198—21no—
26C2×#11758198—21no—
27#14519702C2214yescomplemented
28#14519699C2214yes—
29C2×C42×C61C2211yescharacteristic
30C4⋊C8×C61C2212yescomplemented
31C22⋊C8×C61C2212yes—
32#97×C61C2212yescomplemented
33C23×D61—21no—
34C22×C4×C61D411yescharacteristic
35C22×C4×C61D411yescharacteristic
36C22×C4×C61C2311yescharacteristic
37C22×Dic61—21no—
38C2×D244—21no—
39C2×D244—42no—
40C2×D244C2312yes—
41C2×Dic122—21no—
42C2×Dic122—21no—
43C2×Dic122—42no—
44C2×Dic122C2312yes—
45C2×Dic122C2312yes—
46C2×C8×C61C2314yes—
47C42×C61D411yescharacteristic
48C42×C61D411yescharacteristic
49C42×C61C2311yescharacteristic
50C42×C61C2311yescharacteristic
51C22⋊Dic61—22no—
52#11758198—22no—
53#11758198—22no—
54#11758198—22no—
55#11758198—22no—
56C4⋊Dic61—21no—
57C4⋊Dic61—21no—
58C4⋊Dic61—21no—
59C4⋊Dic61C2312yes—
60#11758196—22no—
61#11758196—22no—
62C4×Dic61—22no—
63Dic244—44no—
64#11758190—44no—
65C23×C61C2×D411yescharacteristic
66C22×D61—22no—
67C22×D61—42no—
68C22×D61—42no—
69C22×D61—42no—
70C2×C4×C61—22no—
71C2×C4×C61—22no—
72C2×C4×C61—41no—
73C2×C4×C61C2×D411yescharacteristic
74C2×C4×C61C2×D411yescharacteristic
75C2×C4×C61C2×D411yescharacteristic
76C2×C4×C61C2×D411yescharacteristic
77C2×C4×C61C2×D411yescharacteristic
78C2×C4×C61C2411yescharacteristic
79C2×Dic61—22no—
80C2×Dic61—22no—
81C2×Dic61—22no—
82C2×Dic61—42no—
83D244—24no—
84D244—42no—
85Dic122—24no—
86Dic122—24no—
87Dic122—42no—
88Dic122—42no—
89C8×C61—24no—
90C22×C61—21no—
91C22×C61—21no—
92C22×C61#13712yes—
93C22×C61C22×D411yescharacteristic
94C2×D61—42no—
95C2×D61—42no—
96C2×D61—42no—
97C2×D61—84no—
98C4×C61—21no—
99C4×C61—21no—
100C4×C61—21no—
101C4×C61—21no—
102C4×C61—21no—
103C4×C61H(5,𝔽2)12yes—
104Dic61—42no—
105Dic61—42no—
106Dic61—42no—
107#3215—611nomaximal, Sylow
108C2×C61—22no—
109C2×C61#53511yescharacteristic
110C2×C61C2×#13711yescharacteristic
111C2×C61#57511yescharacteristic
112D61—82no—
113#533—611no—
114C2×#124—611no—
115#497—612no—
116#489—612no—
117#464—614no—
118#451—612no—
119#450—612no—
120#432D6111yescharacteristic, complemented
121C61#321511yescharacteristic, complemented, Sylow
122C22×Q8—611no—
123C22×D4—611no—
124C2×Q16—614no—
125C2×SD16—614no—
126C4⋊Q8—611no—
127C4⋊Q8—611no—
128#125—1221no—
129#124—612no—
130#124—612no—
131#124—1221no—
132#124—1221no—
133C22⋊Q8—612no—
134C22⋊Q8—1221no—
135C4×Q8—612no—
136C4×Q8—1221no—
137#117—1221no—
138C2×C22⋊C4—1221no—
139C2×C22⋊C4—1221no—
140C2×C42C2×D6111yescharacteristic
141C4⋊C8C2×D6112yescomplemented
142#103—614no—
143#102—614no—
144C22⋊C8C2×D6112yes—
145#97C2×D6112yescomplemented
146C24—1221no—
147C2×Q8—612no—
148C2×Q8—612no—
149C2×Q8—1221no—
150C2×Q8—1221no—
151C2×Q8—2442no—
152C2×D4—612no—
153C2×D4—1221no—
154C2×D4—2442no—
155C22×C4—1221no—
156C22×C4D24411yescharacteristic
157C22×C4D24411yescharacteristic
158C22×C4C22×D6111yescharacteristic
159Q16—2444no—
160SD16—2444no—
161C2×C8C22×D6114yes—
162C4⋊C4—612no—
163C4⋊C4—1222no—
164C4⋊C4—1222no—
165C4⋊C4—1221no—
166C4⋊C4—1221no—
167C4⋊C4—1221no—
168C22⋊C4—1222no—
169C22⋊C4—1222no—
170C22⋊C4—1222no—
171C22⋊C4—1222no—
172C22⋊C4—1222no—
173C42—1222no—
174C42D24411yescharacteristic
175C42D24411yescharacteristic
176C42C22×D6111yescharacteristic
177C42C22×D6111yescharacteristic
178C23—1222no—
179C23—2442no—
180C23—2442no—
181C23—2442no—
182C23C2×D24411yescharacteristic
183Q8—1224no—
184Q8—1224no—
185Q8—2442no—
186Q8—2442no—
187D4—1224no—
188D4—2442no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—1222no—
193C2×C4—1222no—
194C2×C4—1222no—
195C2×C4—2442no—
196C2×C4C2×D24411yescharacteristic
197C2×C4C2×D24411yescharacteristic
198C2×C4C2×D24411yescharacteristic
199C2×C4C2×D24411yescharacteristic
200C2×C4C2×D24411yescharacteristic
201C2×C4C23×D6111yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—4884no—
206C22—2442no—
207C22—2442no—
208C22—2442no—
209C22#1451981012yes—
210C22C22×D24411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—2442no—
217C4—2442no—
218C4—2442no—
219C4#1451982312yes—
220C2—22no—
221C2—4882no—
222C2#1684033111yescharacteristic
223C2C2×#1451981011yescharacteristic
224C2#1684056211yescharacteristic
subgroupG/H
centerC22C22×D244
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#432C2
socleC22×C61C22×D4
hypercenter#432D61
hypocenterC61#3215
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #16840319C2semidirect
3. #16840156C2semidirect
4. #16840122C2semidirect
5. #16839985C2semidirect
6. #16839920C2semidirect
8. C61×#432C2semidirect
11. C2×Dic244C22semidirect
13. #14519777C22semidirect
14. #14519777C22semidirect
23. C4×Dic122C22semidirect
27. #14519702C22semidirect
30. C4⋊C8×C61C22semidirect
32. #97×C61C22semidirect
120. #432D61semidirect
121. C61#3215semidirect
141. C4⋊C8C2×D61semidirect
145. #97C2×D61semidirect