Finite Group Databasebeta

← #28345866

Subgroups — #28345866

Order 13696.

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#20293463C211yesmaximal, characteristic
2#20293201C212yesmaximal, complemented
3#20292816C218yesmaximal, complemented
4#20292601C211yesmaximal, characteristic, complemented
5#20292600C212yesmaximal, complemented
6#536×C107C211yesmaximal, characteristic, complemented
7#16812157—21no—
8#16812156—22no—
9#16812155—22no—
10#16812154—21no—
11#16812154C2214yes—
12C2×#13133818—22no—
13C2×#13133818C2212yes—
14#41×D107—24no—
15#16812107—22no—
16#16812098—22no—
17#16812092C2218yescomplemented
18#16812085—24no—
19#16812068C2212yescomplemented
20#16812024C2214yescomplemented
21#16812023C2218yescomplemented
22C2×#41×C107C2211yescharacteristic
23C22×Q8×C107C2211yescharacteristic, complemented
24C4⋊Q8×C107—21no—
25#124×C107—21no—
26#123×C107C2214yescomplemented
27C22⋊Q8×C107—22no—
28#117×C107C2211yescharacteristic, complemented
29#13133819—24no—
30#13133818—24no—
31#13133818—24no—
32#13133818—22no—
33#13133818—22no—
34#13133818—24no—
35#13133818—22no—
36#13133818—22no—
37#13133817—24no—
38#13133817—28no—
39C2×Dic214—22no—
40C2×Dic214D412yes—
41Q8×D107—24no—
42C2×D428—22no—
43C2×D428D412yes—
44C2×C22⋊D107—24no—
45C2×C22⋊D107C2314yes—
46D4×D107—24no—
47D4×D107—28no—
48C2×C4×D107—24no—
49C2×C4×D107D414yes—
50C4⋊Dic107—22no—
51Dic107⋊C4—28no—
52C22⋊Dic107C2314yes—
53(C2×D107)⋊C4—28no—
54C4×Dic107—22no—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107D411yescharacteristic
62C2×D4×C107D411yescharacteristic
63C2×D4×C107C2312yes—
64C22×C4×C107D412yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—21no—
68C4⋊C4×C107—24no—
69C22⋊C4×C107—24no—
70C22⋊C4×C107C2312yes—
71C42×C107—21no—
72C23×C107C2×D412yes—
73C23×C107C2411yescharacteristic
74C22×D107—24no—
75C22×D107C2×D414yes—
76Q8×C107—22no—
77Q8×C107—41no—
78Q8×C107—42no—
79Q8×C107—41no—
80D4×C107—24no—
81D4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—24no—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C22⋊D107—28no—
93C22⋊D107—24no—
94C22⋊D107—24no—
95C2×Dic107—24no—
96C2×Dic107—24no—
97C2×Dic107C2×D414yes—
98D428—24no—
99D428—22no—
100D428—22no—
101C4×D107—28no—
102C4×D107—24no—
103C4×D107—24no—
104Dic214—24no—
105Dic214—22no—
106Dic214—22no—
107C22×C107—22no—
108C22×C107—41no—
109C22×C107—42no—
110C22×C107C22×D412yes—
111C22×C107C22×D411yescharacteristic
112C2×D107—24no—
113C2×D107—44no—
114C2×D107—84no—
115C4×C107—22no—
116C4×C107—42no—
117C4×C107—42no—
118C4×C107#12011yescharacteristic
119C4×C107#12012yes—
120C4×C107#12011yescharacteristic
121Dic107—24no—
122Dic107—44no—
123C2×C107—22no—
124C2×C107—21no—
125C2×C107—42no—
126C2×C107C2×#12011yescharacteristic
127D107—44no—
128#3028—1071nomaximal, Sylow
129C107#302811yescharacteristic, complemented, Sylow
130#585—1071no—
131#536—1072no—
132#536D10711yescharacteristic, complemented
133#458—1078no—
134#410—1071no—
135#410—1072no—
1362−1+4—2142no—
1372−1+4—2141no—
138H(5,𝔽2)—1074no—
139H(5,𝔽2)—2142no—
140H(5,𝔽2)—2141no—
141C2×#41—1072no—
142C2×#41—2144no—
143C2×#41—2142no—
144C2×#41C2×D10711yescharacteristic
145C22×Q8C2×D10711yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—2142no—
148#124—21no—
149#124—2142no—
150#123—1078no—
151#123C2×D10714yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—2144no—
154#117—1072no—
155#117C2×D10711yescharacteristic, complemented
156#99—1074no—
157#99—1078no—
158#41—22no—
159#41—22no—
160#41—2144no—
161#41—2144no—
162#41—2148no—
163#41—2144no—
164#41—2144no—
165#41—2142no—
166#41—2142no—
167#41—2142no—
168#41—2142no—
169#41—2144no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—1072no—
175C2×Q8—2142no—
176C2×Q8—2144no—
177C2×Q8C22⋊D10711yescharacteristic
178C2×D4—1074no—
179C2×D4—1072no—
180C2×D4—2142no—
181C2×D4—2148no—
182C2×D4—2144no—
183C2×D4—2144no—
184C2×D4C22⋊D10711yescharacteristic
185C2×D4C22×D10712yes—
186C22×C4—1074no—
187C22×C4—2144no—
188C22×C4C22⋊D10712yes—
189C22×C4C22×D10712yes—
190C22×C4C22×D10711yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—2148no—
194C4⋊C4—2142no—
195C22⋊C4—24no—
196C22⋊C4—1074no—
197C22⋊C4—2148no—
198C22⋊C4C22×D10712yes—
199C42—21no—
200C42—2142no—
201C23—1074no—
202C23—2144no—
203C23C2×C22⋊D10712yes—
204C23C23×D10711yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—2144no—
210Q8—2142no—
211Q8—2142no—
212D4—24no—
213D4—22no—
214D4—2144no—
215D4—2148no—
216D4—2144no—
217D4—2144no—
218D4—2142no—
219D4—2142no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—1074no—
226C2×C4—2148no—
227C2×C4—2144no—
228C2×C4—2144no—
229C2×C4—2144no—
230C2×C4—2144no—
231C2×C4D4×D10712yes—
232C2×C4D4×D10712yes—
233C2×C4C2×C22⋊D10712yes—
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×C22⋊D10711yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—8564no—
240C22—4284no—
241C22—2144no—
242C22C2×D4×D10712yes—
243C22C22×C22⋊D10711yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—4284no—
248C4—2144no—
249C4#1681207711yescharacteristic
250C4#1681207711yescharacteristic
251C4#1681207712yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—4284no—
256C2C2×#1681207711yescharacteristic
subgroupG/H
centerC2C2×#16812077
commutator, cosocleC23×C107C24
FrattiniC23C23×D107
Fitting#536×C107C2
socleC2×C107C2×#120
prime radical#20293463C2
hypercenter#536D107
hypocenterC107#3028
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20293201C2semidirect
3. #20292816C2semidirect
4. #20292601C2semidirect
5. #20292600C2semidirect
6. #536×C107C2semidirect
17. #16812092C22semidirect
19. #16812068C22semidirect
20. #16812024C22semidirect
21. #16812023C22semidirect
23. C22×Q8×C107C22semidirect
26. #123×C107C22semidirect
28. #117×C107C22semidirect
129. C107#3028semidirect
132. #536D107semidirect
145. C22×Q8C2×D107semidirect
151. #123C2×D107semidirect
155. #117C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202934632
#202932012
#202928162
#202926012
#202926002
#536×C1072
#3028107