Finite Group Databasebeta

← #28339570

Subgroups — #28339570

Order 13696.

560 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#16812096C211yesmaximal, characteristic
2#20292531C212yesmaximal, complemented
3#20292521C212yesmaximal, complemented
4#20292515C212yesmaximal, complemented
5#20292507C211yesmaximal, characteristic
6C22⋊C4×Dic107C212yesmaximal, complemented
7C4×C22⋊Dic107C212yesmaximal, complemented
8#20292472C212yesmaximal, complemented
9#386×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C23×Dic107C2211yescharacteristic
12#16812096C418yescomplemented
13C2×Dic107⋊C4C2212yes—
14C2×C22⋊Dic107C2212yes—
15C2×C22⋊Dic107C2211yescharacteristic
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C4×Dic107C2212yes—
19C2×C4×Dic107C2212yes—
20C2×C4×Dic107C2212yes—
21C2×C4×Dic107C2211yescharacteristic
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C22×D4×C107C2211yescharacteristic
28C23×C4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2212yes—
30C2×C22⋊C4×C107C2211yescharacteristic
31#95×C107C2212yes—
32C2×Dic214—41no—
33C2×Dic214—41no—
34C2×Dic214C2×C414yescomplemented
35C22×Dic107—24no—
36C22×Dic107—41no—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41Dic107⋊C4—42no—
42C22⋊Dic107—24no—
43C22⋊Dic107—41no—
44C22⋊Dic107C2×C418yescomplemented
45C22⋊Dic107C2×C418yescomplemented
46C4×Dic107—24no—
47C4×Dic107—24no—
48C4×Dic107—42no—
49C4×Dic107C2×C414yescomplemented
50C24×C107C2311yescharacteristic
51C24×C107C2311yescharacteristic
52C2×D4×C107—41no—
53C2×D4×C107—41no—
54C2×D4×C107C2×C414yes—
55C22×C4×C107—24no—
56C22×C4×C107—41no—
57C22×C4×C107C2312yes—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2312yes—
60C22⋊C4×C107—24no—
61C22⋊C4×C107—41no—
62C23×C107—22no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—21no—
67C23×C107—21no—
68C23×C107—21no—
69C23×C107—21no—
70C23×C107C22×C412yes—
71C23×C107C22×C412yes—
72C23×C107C2411yescharacteristic
73D4×C107—44no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—21no—
77C2×C4×C107—21no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—44no—
82C2×C4×C107C22×C412yes—
83C2×C4×C107#4114yes—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—44no—
94C2×Dic107C22×C414yes—
95C2×Dic107C22×C414yes—
96C2×Dic107#4114yes—
97Dic214—44no—
98C22×C107—24no—
99C22×C107—24no—
100C22×C107—22no—
101C22×C107—22no—
102C22×C107—41no—
103C22×C107—41no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107C23×C411yescharacteristic
109C22×C107C2×#4112yes—
110C22×C107C2×#4112yes—
111C22×C107H(5,𝔽2)11yescharacteristic
112C22×C107H(5,𝔽2)11yescharacteristic
113C4×C107—22no—
114C4×C107—24no—
115C4×C107—42no—
116Dic107—24no—
117Dic107—24no—
118Dic107—24no—
119Dic107—42no—
120C2×C107—42no—
121C2×C107—42no—
122C2×C107C4×#4112yes—
123C2×C107#51811yescharacteristic
124C2×C107#55111yescharacteristic
125C2×C107#55111yescharacteristic
126C2×C107#55512yes—
127#2372—1071nomaximal, Sylow
128C107#237211yescharacteristic, complemented, Sylow
129C2×#124—1071no—
130#391—1072no—
131#388—1072no—
132#388—1072no—
133#386—1071no—
134#386D10711yescharacteristic
135C4×C22⋊C4—1072no—
136C4×C22⋊C4—1072no—
137#376—1072no—
138C22×Q8—1071no—
139C22×D4C2×D10711yescharacteristic
140C23×C4—1071no—
141C23×C4C2×D10711yescharacteristic
142#124—1078no—
143C2×C4⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1071no—
147C2×C22⋊C4—1072no—
148C2×C22⋊C4C2×D10712yes—
149C2×C22⋊C4C2×D10711yescharacteristic
150C2×C42—1071no—
151C2×C42—1072no—
152C2×C42—1072no—
153C2×C42—1072no—
154#95—1072no—
155#95—1072no—
156#95—1072no—
157#95—1072no—
158#95—1072no—
159#95C2×D10712yes—
160C24C22×D10711yescharacteristic
161C24C22×D10711yescharacteristic
162C2×Q8—1074no—
163C2×Q8—4281no—
164C2×Q8—4281no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4C4×D10714yes—
168C22×C4—24no—
169C22×C4—41no—
170C22×C4—1072no—
171C22×C4—1072no—
172C22×C4—1072no—
173C22×C4—1072no—
174C22×C4—4281no—
175C22×C4—2144no—
176C22×C4C22×D10712yes—
177C22×C4C22×D10711yescharacteristic
178C22×C4C22×D10712yes—
179C4⋊C4—4282no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—1078no—
183C22⋊C4—1078no—
184C22⋊C4—4281no—
185C22⋊C4—2144no—
186C42—1074no—
187C42—4282no—
188C42—2144no—
189C42—2144no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23C2×C4×D10712yes—
199C23C2×C4×D10712yes—
200C23C23×D10711yescharacteristic
201Q8—4284no—
202D4—44no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—1074no—
212C2×C4—1074no—
213C2×C4—1074no—
214C2×C4—4284no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4C2×C4×D10712yes—
225C2×C4#1313381714yes—
226C22—24no—
227C22—24no—
228C22—22no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—42no—
236C22C22×C4×D10711yescharacteristic
237C22#41×D10712yes—
238C22C2×#1313381712yes—
239C22#1681215411yescharacteristic
240C22#1681215411yescharacteristic
241C4—24no—
242C4—22no—
243C4—42no—
244C4—4282no—
245C4—2144no—
246C4—2144no—
247C4—2144no—
248C2—42no—
249C2—42no—
250C2C4×#1313381712yes—
251C2#2029310011yescharacteristic
252C2#2029330511yescharacteristic
253C2#2029330811yescharacteristic
254C2#2029334112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#386×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D4×C107C22
hypercenter#386D107
hypocenterC107#2372
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20292531C2semidirect
3. #20292521C2semidirect
4. #20292515C2semidirect
6. C22⋊C4×Dic107C2semidirect
7. C4×C22⋊Dic107C2semidirect
8. #20292472C2semidirect
12. #16812096C4semidirect
34. C2×Dic214C2×C4semidirect
44. C22⋊Dic107C2×C4semidirect
45. C22⋊Dic107C2×C4semidirect
49. C4×Dic107C2×C4semidirect
128. C107#2372semidirect

Maximal subgroups

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

subgroupindex
C2×#168120962
#202925312
#202925212
#202925152
#202925072
C22⋊C4×Dic1072
C4×C22⋊Dic1072
#202924722
#386×C1072
#2372107