Finite Group Databasebeta

← #21456639

Subgroups — #21456639

Order 7808.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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⋊Q8×D61C212yesmaximal
2#16840094C212yesmaximal, complemented
3#16840091C212yesmaximal, complemented
4#16840020C211yesmaximal, characteristic, complemented
5#16840018C212yesmaximal, complemented
6#16839919C212yesmaximal, complemented
7#16839916C212yesmaximal, complemented
8#16839725C211yesmaximal, characteristic
9C61×#470C211yesmaximal, characteristic, complemented
10C2×Q8×D61C2212yes—
11C22×C4×D61C2211yescharacteristic
12C2×Dic244C2211yescharacteristic, complemented
13C2×#11758202C2212yescomplemented
14C2×#11758194C2211yescharacteristic
15#14519757C2212yes—
16#14519756C2212yes—
17#14519755—22no—
18#14519754—22no—
19#14519753C2212yes—
20C4⋊C4×D61—22no—
21C4⋊C4×D61C2212yes—
22C22⋊C4×D61—22no—
23#14519709C2211yescharacteristic, complemented
24#14519708C2212yescomplemented
25#14519702C2212yescomplemented
26#14519701C2212yescomplemented
27#14519700C2212yescomplemented
28#14519689C2212yes—
29#14519689C2212yes—
30C2×Q16×C61C2211yescharacteristic, complemented
31C2×M16×C61C2211yescharacteristic
32C22⋊Q8×C61C2212yes—
33#106×C61C2211yescharacteristic, complemented
34#103×C61C2212yescomplemented
35C23×D61D411yescharacteristic
36C2×Q8×C61C2312yes—
37C22×C4×C61C2311yescharacteristic
38C22×Dic61D411yescharacteristic
39Q8×D61—42no—
40Q8×D61—42no—
41C2×C4×D61—22no—
42C2×C4×D61—22no—
43C2×C4×D61—41no—
44C2×C4×D61—41no—
45C2×C4×D61D412yes—
46C2×C4×D61C2312yes—
47C2×Dic122—22no—
48C2×Dic122C2312yes—
49Q16×C61—41no—
50M16×C61—41no—
51C2×C8×C61C2311yescharacteristic
52C2×C8×C61C2311yescharacteristic
53C4⋊C4×C61—22no—
54C4⋊C4×C61C2312yes—
55C22⋊C4×C61—22no—
56C22⋊Dic61—22no—
57#11758202—42no—
58#11758198—22no—
59#11758198—22no—
60C4⋊Dic61—22no—
61C4⋊Dic61C2312yes—
62#11758196—22no—
63#11758196—22no—
64#11758196—22no—
65#11758194—41no—
66C2×#92418C2312yes—
67Dic244—41no—
68C23×C61C2×D411yescharacteristic
69C22×D61—24no—
70C22×D61—41no—
71C22×D61C2×D412yes—
72Q8×C61—22no—
73Q8×C61—22no—
74C2×C4×C61—21no—
75C2×C4×C61—21no—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61C2×D411yescharacteristic
79C2×C4×C61C2411yescharacteristic
80C2×Dic61—22no—
81C2×Dic61—22no—
82C2×Dic61—41no—
83C2×Dic61C2×D412yes—
84C4×D61—22no—
85C4×D61—22no—
86C4×D61—42no—
87C4×D61—44no—
88C4×D61—44no—
89Dic122—22no—
90Dic122—22no—
91Dic122—44no—
92C8×C61—21no—
93C8×C61—21no—
94#92418—22no—
95C22×C61—21no—
96C22×C61—22no—
97C22×C61C22×D411yescharacteristic
98C2×D61—22no—
99C2×D61—22no—
100C2×D61—44no—
101C2×D61#13714yes—
102C4×C61—21no—
103C4×C61—42no—
104C4×C61—42no—
105C4×C61H(5,𝔽2)11yescharacteristic
106C4×C61H(5,𝔽2)11yescharacteristic
107Dic61—22no—
108Dic61—42no—
109Dic61—42no—
110#3195—611nomaximal, Sylow
111C2×C61—41no—
112C2×C61#53411yescharacteristic
113C2×C61C2×#13712yes—
114D61—24no—
115C2×C22⋊Q8—612no—
116#483—612no—
117#483—612no—
118#470—611no—
119#470—612no—
120#470D6111yescharacteristic, complemented
121#451—612no—
122#451—612no—
123#407—611no—
124C61#319511yescharacteristic, complemented, Sylow
125C22×Q8—612no—
126C23×C4—611no—
127C2×Q16—612no—
128C2×Q16—611no—
129C2×Q16C2×D6111yescharacteristic, complemented
130C2×M16—611no—
131C2×M16C2×D6111yescharacteristic
132C22⋊Q8—612no—
133C22⋊Q8—612no—
134C22⋊Q8—612no—
135C22⋊Q8—1222no—
136C22⋊Q8—1222no—
137C22⋊Q8C2×D6112yes—
138C2×C4⋊C4—612no—
139C2×C4⋊C4—1222no—
140C2×C22⋊C4—1222no—
141#106—611no—
142#106—612no—
143#106C2×D6111yescharacteristic, complemented
144#103—612no—
145#103—612no—
146#103—612no—
147#103C2×D6112yescomplemented
148C22⋊C8—612no—
149C22⋊C8—612no—
150C24—611no—
151C2×Q8—612no—
152C2×Q8—1222no—
153C2×Q8—2442no—
154C2×Q8—2442no—
155C2×Q8C22×D6112yes—
156C22×C4—611no—
157C22×C4—612no—
158C22×C4—612no—
159C22×C4—1222no—
160C22×C4—1222no—
161C22×C4—2441no—
162C22×C4—2441no—
163C22×C4C22×D6111yescharacteristic
164Q16—41no—
165Q16—2441no—
166Q16—2442no—
167M16—41no—
168M16—2441no—
169C2×C8—612no—
170C2×C8C22×D6111yescharacteristic
171C2×C8C22×D6111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—612no—
174C4⋊C4—1222no—
175C4⋊C4—1222no—
176C4⋊C4—1222no—
177C4⋊C4—1222no—
178C4⋊C4C22×D6112yes—
179C22⋊C4—22no—
180C22⋊C4—1222no—
181C22⋊C4—1222no—
182C22⋊C4—1222no—
183C23—612no—
184C23—1224no—
185C23—2441no—
186C23D4×D6111yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—1222no—
190Q8—1222no—
191Q8—2444no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—612no—
197C2×C4—1222no—
198C2×C4—1222no—
199C2×C4—1222no—
200C2×C4—1222no—
201C2×C4—2442no—
202C2×C4—2444no—
203C2×C4—2444no—
204C2×C4—2441no—
205C2×C4D4×D6111yescharacteristic
206C2×C4C23×D6111yescharacteristic
207C8—21no—
208C8—21no—
209C8—1222no—
210C22—22no—
211C22—21no—
212C22—614no—
213C22—1222no—
214C22—1222no—
215C22—2444no—
216C22C2×D4×D6111yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—1222no—
221C4—2442no—
222C4—2442no—
223C4#1451982211yescharacteristic
224C4#1451982311yescharacteristic
225C2—41no—
226C2—1224no—
227C2#1684032011yescharacteristic
228C2#137×D6112yes—
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#470C2
socleC22×C61C22×D4
hypercenter#470D61
hypocenterC61#3195
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #16840094C2semidirect
3. #16840091C2semidirect
4. #16840020C2semidirect
5. #16840018C2semidirect
6. #16839919C2semidirect
7. #16839916C2semidirect
9. C61×#470C2semidirect
12. C2×Dic244C22semidirect
13. C2×#11758202C22semidirect
23. #14519709C22semidirect
24. #14519708C22semidirect
25. #14519702C22semidirect
26. #14519701C22semidirect
27. #14519700C22semidirect
30. C2×Q16×C61C22semidirect
33. #106×C61C22semidirect
34. #103×C61C22semidirect
120. #470D61semidirect
124. C61#3195semidirect
129. C2×Q16C2×D61semidirect
143. #106C2×D61semidirect
147. #103C2×D61semidirect