Finite Group Databasebeta

← #21455224

Subgroups — #21455224

Order 7808.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#16840598C211yesmaximal, characteristic
2#16840336C212yesmaximal, complemented
3#16839951C218yesmaximal, complemented
4#16839736C211yesmaximal, characteristic, complemented
5#16839735C212yesmaximal, complemented
6C61×#536C211yesmaximal, characteristic, complemented
7#14519825—21no—
8#14519824—22no—
9#14519823—22no—
10#14519822—21no—
11#14519822C2214yes—
12C2×#11758222—22no—
13C2×#11758222C2212yes—
14#41×D61—24no—
15#14519775—22no—
16#14519766—22no—
17#14519760C2218yescomplemented
18#14519753—24no—
19#14519736C2212yescomplemented
20#14519692C2214yescomplemented
21#14519691C2218yescomplemented
22C2×#41×C61C2211yescharacteristic
23C22×Q8×C61C2211yescharacteristic, complemented
24C4⋊Q8×C61—21no—
25#124×C61—21no—
26#123×C61C2214yescomplemented
27C22⋊Q8×C61—22no—
28#117×C61C2211yescharacteristic, complemented
29#41×C61—22no—
30#41×C61—22no—
31C2×Q8×C61—21no—
32C2×Q8×C61—21no—
33C2×Q8×C61—22no—
34C2×Q8×C61—22no—
35C2×Q8×C61D411yescharacteristic
36C2×D4×C61D411yescharacteristic
37C2×D4×C61C2312yes—
38C22×C4×C61D412yes—
39C22×C4×C61C2312yes—
40C22×C4×C61C2311yescharacteristic
41C2×C22⋊D61—24no—
42C2×C22⋊D61C2314yes—
43#11758226—24no—
44Q8×D61—24no—
45#11758224—24no—
46#11758224—28no—
47D4×D61—24no—
48D4×D61—28no—
49#11758222—24no—
50#11758222—24no—
51#11758222—22no—
52#11758222—22no—
53#11758222—24no—
54#11758222—22no—
55#11758222—22no—
56C2×D244—22no—
57C2×D244D412yes—
58C2×C4×D61—24no—
59C2×C4×D61D414yes—
60C2×Dic122—22no—
61C2×Dic122D412yes—
62C4⋊C4×C61—21no—
63C4⋊C4×C61—24no—
64C22⋊C4×C61—24no—
65C22⋊C4×C61C2312yes—
66C42×C61—21no—
67C22⋊Dic61C2314yes—
68#11758198—28no—
69C4⋊Dic61—22no—
70#11758196—28no—
71C4×Dic61—22no—
72C23×C61C2×D412yes—
73C23×C61C2411yescharacteristic
74C22×D61—24no—
75C22×D61C2×D414yes—
76Q8×C61—22no—
77Q8×C61—41no—
78Q8×C61—42no—
79Q8×C61—41no—
80D4×C61—24no—
81D4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—24no—
85C2×C4×C61—22no—
86C2×C4×C61—22no—
87C2×C4×C61C2×D412yes—
88C2×C4×C61C2×D412yes—
89C2×C4×C61C2×D412yes—
90C2×C4×C61C2×D411yescharacteristic
91C2×C4×C61C2×D411yescharacteristic
92C22⋊D61—28no—
93C22⋊D61—24no—
94C22⋊D61—24no—
95C2×Dic61—24no—
96C2×Dic61—24no—
97C2×Dic61C2×D414yes—
98D244—24no—
99D244—22no—
100D244—22no—
101C4×D61—28no—
102C4×D61—24no—
103C4×D61—24no—
104Dic122—24no—
105Dic122—22no—
106Dic122—22no—
107C22×C61—22no—
108C22×C61—41no—
109C22×C61—42no—
110C22×C61C22×D412yes—
111C22×C61C22×D411yescharacteristic
112C2×D61—24no—
113C2×D61—44no—
114C2×D61—84no—
115C4×C61—22no—
116C4×C61—42no—
117C4×C61—42no—
118C4×C61#12011yescharacteristic
119C4×C61#12012yes—
120C4×C61#12011yescharacteristic
121Dic61—24no—
122Dic61—44no—
123#3028—611nomaximal, Sylow
124C2×C61—22no—
125C2×C61—21no—
126C2×C61—42no—
127C2×C61C2×#12011yescharacteristic
128D61—44no—
129#585—611no—
130#536—612no—
131#536D6111yescharacteristic, complemented
132#458—618no—
133#410—611no—
134#410—612no—
135C61#302811yescharacteristic, complemented, Sylow
1362−1+4—1222no—
1372−1+4—1221no—
138H(5,𝔽2)—614no—
139H(5,𝔽2)—1222no—
140H(5,𝔽2)—1221no—
141C2×#41—612no—
142C2×#41—1224no—
143C2×#41—1222no—
144C2×#41C2×D6111yescharacteristic
145C22×Q8C2×D6111yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—1222no—
148#124—21no—
149#124—1222no—
150#123—618no—
151#123C2×D6114yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—1224no—
154#117—612no—
155#117C2×D6111yescharacteristic, complemented
156#99—614no—
157#99—618no—
158#41—22no—
159#41—22no—
160#41—1224no—
161#41—1224no—
162#41—1228no—
163#41—1224no—
164#41—1224no—
165#41—1222no—
166#41—1222no—
167#41—1222no—
168#41—1222no—
169#41—1224no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—612no—
175C2×Q8—1222no—
176C2×Q8—1224no—
177C2×Q8C22⋊D6111yescharacteristic
178C2×D4—614no—
179C2×D4—612no—
180C2×D4—1222no—
181C2×D4—1228no—
182C2×D4—1224no—
183C2×D4—1224no—
184C2×D4C22⋊D6111yescharacteristic
185C2×D4C22×D6112yes—
186C22×C4—614no—
187C22×C4—1224no—
188C22×C4C22⋊D6112yes—
189C22×C4C22×D6112yes—
190C22×C4C22×D6111yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—1228no—
194C4⋊C4—1222no—
195C22⋊C4—24no—
196C22⋊C4—614no—
197C22⋊C4—1228no—
198C22⋊C4C22×D6112yes—
199C42—21no—
200C42—1222no—
201C23—614no—
202C23—1224no—
203C23C2×C22⋊D6112yes—
204C23C23×D6111yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—1224no—
210Q8—1222no—
211Q8—1222no—
212D4—24no—
213D4—22no—
214D4—1224no—
215D4—1228no—
216D4—1224no—
217D4—1224no—
218D4—1222no—
219D4—1222no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—614no—
226C2×C4—1228no—
227C2×C4—1224no—
228C2×C4—1224no—
229C2×C4—1224no—
230C2×C4—1224no—
231C2×C4D4×D6112yes—
232C2×C4D4×D6112yes—
233C2×C4C2×C22⋊D6112yes—
234C2×C4C2×C22⋊D6111yescharacteristic
235C2×C4C2×C22⋊D6111yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—1224no—
240C22—4884no—
241C22—2444no—
242C22C2×D4×D6112yes—
243C22C22×C22⋊D6111yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—1224no—
248C4—2444no—
249C4#1451974511yescharacteristic
250C4#1451974511yescharacteristic
251C4#1451974512yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—2444no—
256C2C2×#1451974511yescharacteristic
subgroupG/H
centerC2C2×#14519745
commutatorC23×C61C24
FrattiniC23C23×D61
FittingC61×#536C2
socleC2×C61C2×#120
hypercenter#536D61
hypocenterC61#3028
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #16840336C2semidirect
3. #16839951C2semidirect
4. #16839736C2semidirect
5. #16839735C2semidirect
6. C61×#536C2semidirect
17. #14519760C22semidirect
19. #14519736C22semidirect
20. #14519692C22semidirect
21. #14519691C22semidirect
23. C22×Q8×C61C22semidirect
26. #123×C61C22semidirect
28. #117×C61C22semidirect
131. #536D61semidirect
135. C61#3028semidirect
145. C22×Q8C2×D61semidirect
151. #123C2×D61semidirect
155. #117C2×D61semidirect