Finite Group Databasebeta

← #21450978

Subgroups — #21450978

Order 7808.

672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#11758198C211yesmaximal, characteristic
2#16839688C212yesmaximal, complemented
3#16839673C212yesmaximal, complemented
4#16839652C214yesmaximal, complemented
5#16839623C212yesmaximal, complemented
6#16839617C212yesmaximal
7C4×C22⋊Dic61C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
9C24×D61C2211yescharacteristic
10C23×Dic61C2211yescharacteristic
11C2×C22⋊Dic61C2212yes—
12C2×#11758198—24no—
13C2×#11758198C2212yes—
14C2×#11758198C2214yes—
15C2×#11758198C2212yes—
16C2×#11758198C2214yes—
17C2×C4×Dic61C2212yescomplemented
18#14519684C2212yescomplemented
19#14519684C2214yescomplemented
20#14519684C2214yescomplemented
21C22×Q8×C61C2211yescharacteristic, complemented
22C23×C4×C61C2211yescharacteristic
23C22⋊Q8×C61—24no—
24C2×C4⋊C4×C61C2211yescharacteristic, complemented
25C2×C4⋊C4×C61C2212yescomplemented
26C2×C22⋊C4×C61C2212yes—
27C24×C61C2311yescharacteristic
28C23×D61—22no—
29C23×D61—24no—
30C23×D61—24no—
31C23×D61—42no—
32C23×D61C2312yes—
33C2×Q8×C61—22no—
34C2×Q8×C61—42no—
35C22×C4×C61—42no—
36C22×C4×C61D414yes—
37C22×C4×C61C2312yes—
38C22×C4×C61C2312yes—
39C22×C4×C61C2311yescharacteristic
40C22×C4×C61C2311yescharacteristic
41C22×Dic61—22no—
42C22×Dic61—24no—
43C22×Dic61C2312yes—
44C22×Dic61C2314yes—
45C4⋊C4×C61—22no—
46C4⋊C4×C61—24no—
47C22⋊C4×C61—24no—
48C22⋊Dic61—24no—
49#11758198—28no—
50#11758198—24no—
51#11758198—24no—
52#11758198—44no—
53C4×Dic61—42no—
54C23×C61—22no—
55C23×C61—22no—
56C23×C61C2×D412yes—
57C23×C61#4114yes—
58C23×C61C2411yescharacteristic
59C22×D61—28no—
60C22×D61—22no—
61C22×D61—22no—
62C22×D61—24no—
63C22×D61—24no—
64C22×D61—416no—
65C22×D61—44no—
66C22×D61—44no—
67C22×D61#4114yes—
68Q8×C61—44no—
69C2×C4×C61—22no—
70C2×C4×C61—22no—
71C2×C4×C61—22no—
72C2×C4×C61—22no—
73C2×C4×C61—24no—
74C2×C4×C61—24no—
75C2×C4×C61—44no—
76C2×C4×C61C2×D412yes—
77C2×C4×C61C2×D412yes—
78C2×Dic61—22no—
79C2×Dic61—24no—
80C2×Dic61—24no—
81C2×Dic61—24no—
82C2×Dic61—24no—
83C2×Dic61—44no—
84C22×C61—22no—
85C22×C61—22no—
86C22×C61—28no—
87C22×C61#12414yes—
88C22×C61C22×D411yescharacteristic
89C22×C61C2×#4112yes—
90C22×C61C2×#4112yes—
91C22×C61H(5,𝔽2)12yes—
92C2×D61—24no—
93C2×D61—28no—
94C2×D61—416no—
95C2×D61—42no—
96C2×D61—42no—
97C2×D61—44no—
98C4×C61—22no—
99C4×C61—22no—
100C4×C61—42no—
101C4×C61—42no—
102Dic61—42no—
103Dic61—44no—
104#2561—611nomaximal, Sylow
105C2×C61—24no—
106C2×C61#52511yescharacteristic
107C2×C61C2×#12411yescharacteristic
108C2×C61#53411yescharacteristic
109C2×C61#55114yes—
110D61—44no—
111C2×C22⋊Q8D6111yescharacteristic, complemented
112C22×C22⋊C4—611no—
113#397—612no—
114#393—612no—
115#388—614no—
116#381—612no—
117#379—612no—
118C4×C22⋊C4—611no—
119C61#256111yescharacteristic, complemented, Sylow
120C25—611no—
121C22×Q8C2×D6111yescharacteristic, complemented
122C23×C4—611no—
123C23×C4C2×D6111yescharacteristic
124C22⋊Q8—24no—
125C2×C4⋊C4C2×D6112yescomplemented
126C2×C4⋊C4C2×D6111yescharacteristic, complemented
127C2×C22⋊C4—612no—
128C2×C22⋊C4—612no—
129C2×C22⋊C4—612no—
130C2×C22⋊C4—614no—
131C2×C22⋊C4—614no—
132C2×C22⋊C4—1224no—
133C2×C22⋊C4C2×D6112yes—
134C2×C42—612no—
135#95—612no—
136#95—614no—
137#95—614no—
138C24—612no—
139C24—1224no—
140C24—1222no—
141C24—1224no—
142C24—2442no—
143C24C22×D6111yescharacteristic
144C2×Q8—22no—
145C2×Q8—42no—
146C22×C4—42no—
147C22×C4—612no—
148C22×C4—614no—
149C22×C4—1224no—
150C22×C4—1222no—
151C22×C4C22⋊D6114yes—
152C22×C4C22×D6112yes—
153C22×C4C22×D6112yes—
154C22×C4C22×D6111yescharacteristic
155C22×C4C22×D6111yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—1228no—
160C22⋊C4—1224no—
161C22⋊C4—1224no—
162C22⋊C4—1224no—
163C22⋊C4—2444no—
164C42—2442no—
165C23—22no—
166C23—22no—
167C23—614no—
168C23—1228no—
169C23—1222no—
170C23—1222no—
171C23—1224no—
172C23—1224no—
173C23—24416no—
174C23—2444no—
175C23—2444no—
176C23#1175822614yes—
177C23C2×C22⋊D6112yes—
178C23C23×D6111yescharacteristic
179Q8—44no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—44no—
187C2×C4—1222no—
188C2×C4—1224no—
189C2×C4—1224no—
190C2×C4—1224no—
191C2×C4—1224no—
192C2×C4—2444no—
193C2×C4C2×C22⋊D6112yes—
194C2×C4C2×C22⋊D6112yes—
195C22—22no—
196C22—22no—
197C22—28no—
198C22—1224no—
199C22—1228no—
200C22—24416no—
201C22—2442no—
202C22—2442no—
203C22—2444no—
204C22#1451976614yes—
205C22C22×C22⋊D6111yescharacteristic
206C22#41×D6112yes—
207C22C2×#1175822612yes—
208C22#1451982212yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—42no—
213C4—2442no—
214C4—2444no—
215C2—24no—
216C2—2444no—
217C2#1684026911yescharacteristic
218C2C2×#1451976611yescharacteristic
219C2#1684032111yescharacteristic
220C2#1684043914yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×C22⋊Q8×C61C2
hypercenterC2×C22⋊Q8D61
hypocenterC61#2561
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16839688C2semidirect
3. #16839673C2semidirect
4. #16839652C2semidirect
5. #16839623C2semidirect
7. C4×C22⋊Dic61C2semidirect
8. C2×C22⋊Q8×C61C2semidirect
17. C2×C4×Dic61C22semidirect
18. #14519684C22semidirect
19. #14519684C22semidirect
20. #14519684C22semidirect
21. C22×Q8×C61C22semidirect
24. C2×C4⋊C4×C61C22semidirect
25. C2×C4⋊C4×C61C22semidirect
111. C2×C22⋊Q8D61semidirect
119. C61#2561semidirect
121. C22×Q8C2×D61semidirect
125. C2×C4⋊C4C2×D61semidirect
126. C2×C4⋊C4C2×D61semidirect