Finite Group Databasebeta

← #28344514

Subgroups — #28344514

Order 13696.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812099C211yesmaximal, characteristic, complemented
2C2×#16812096C211yesmaximal, characteristic
3#20292563C212yesmaximal, complemented
4#20292555C212yesmaximal, complemented
5#20292550C212yesmaximal, complemented
6#20292550C212yesmaximal, complemented
7#20292548C211yesmaximal, characteristic, complemented
8#20292547C211yesmaximal, characteristic, complemented
9#20292541C211yesmaximal, characteristic
10#20292491C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic
12C22×Dic214C2211yescharacteristic
13#16812099—24no—
14#16812096—24no—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×C4⋊Dic107C2212yes—
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×C22⋊Dic107C2212yes—
25C2×C22⋊Dic107C2212yes—
26C2×C22⋊Dic107C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C2×Dic214—22no—
38C2×Dic214—42no—
39C22×Dic107C2311yescharacteristic
40C22×Dic107C2311yescharacteristic
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2311yescharacteristic
45C4⋊Dic107—22no—
46C4⋊Dic107—42no—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49Dic107⋊C4—22no—
50Dic107⋊C4—22no—
51Dic107⋊C4—42no—
52Dic107⋊C4—42no—
53C22⋊Dic107—24no—
54C22⋊Dic107—24no—
55C22⋊Dic107—42no—
56C22⋊Dic107—42no—
57C4×Dic107D414yes—
58C24×C107C2312yes—
59C2×D4×C107—22no—
60C2×D4×C107—42no—
61C22×C4×C107C2311yescharacteristic
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2312yes—
64C22×C4×C107C2311yescharacteristic
65C4⋊C4×C107—22no—
66C22⋊C4×C107—42no—
67C22⋊C4×C107—42no—
68C23×C107—22no—
69C23×C107—22no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107C2411yescharacteristic
76D4×C107—44no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D412yes—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—21no—
100C2×Dic107—22no—
101C2×Dic107C2×D412yes—
102C2×Dic107C2×D412yes—
103Dic214—44no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107—42no—
109C22×C107—42no—
110C22×C107—42no—
111C22×C107C22×D411yescharacteristic
112C22×C107H(5,𝔽2)12yes—
113C22×C107H(5,𝔽2)11yescharacteristic
114C22×C1072−1+412yes—
115C22×C1072−1+411yescharacteristic
116C4×C107—22no—
117C4×C107—41no—
118C4×C107—41no—
119C4×C107—42no—
120Dic107—22no—
121Dic107—22no—
122Dic107—42no—
123Dic107—41no—
124Dic107—41no—
125Dic107—42no—
126C2×C107—82no—
127C2×C107#53511yescharacteristic
128C2×C107#53611yescharacteristic
129C2×C107#53711yescharacteristic
130C2×C107#56212yes—
131C2×C107#56312yes—
132#2820—1071nomaximal, Sylow
133C107#282011yescharacteristic, complemented, Sylow
134C2×#125—1071no—
135C2×#124—1071no—
136#399—1072no—
137#397—1072no—
138#396—1072no—
139#396—1072no—
140#395—1071no—
141#395—1071no—
142#394—1071no—
143#394D10711yescharacteristic
144#382—1071no—
145C22×Q8—1071no—
146C22×D4C2×D10711yescharacteristic
147#125—2144no—
148#124—2144no—
149C2×C4⋊C4—1072no—
150C2×C4⋊C4—1072no—
151C2×C4⋊C4—1072no—
152C2×C4⋊C4—1071no—
153C2×C4⋊C4—1071no—
154C2×C4⋊C4—1071no—
155C2×C4⋊C4—1071no—
156C2×C4⋊C4—1071no—
157C2×C4⋊C4C2×D10711yescharacteristic
158C2×C22⋊C4—1072no—
159C2×C22⋊C4—1072no—
160C2×C22⋊C4—1072no—
161C2×C22⋊C4—1072no—
162C2×C22⋊C4C2×D10712yes—
163C2×C22⋊C4C2×D10712yes—
164C2×C42—1071no—
165#95—1071no—
166#95—1072no—
167#95—1072no—
168#95—1072no—
169#95C2×D10711yescharacteristic
170C24C22×D10712yes—
171C2×Q8—4282no—
172C2×Q8—2142no—
173C2×D4—22no—
174C2×D4—42no—
175C22×C4—1071no—
176C22×C4—1071no—
177C22×C4—1072no—
178C22×C4—1071no—
179C22×C4—1072no—
180C22×C4—1071no—
181C22×C4C22×D10712yes—
182C22×C4C22×D10711yescharacteristic
183C22×C4C22×D10711yescharacteristic
184C22×C4C22×D10711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—4282no—
187C4⋊C4—4282no—
188C4⋊C4—4282no—
189C4⋊C4—2142no—
190C4⋊C4—2142no—
191C4⋊C4—2142no—
192C4⋊C4—2142no—
193C4⋊C4—2142no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—4282no—
197C22⋊C4—4282no—
198C22⋊C4—2144no—
199C22⋊C4—2144no—
200C42—1074no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23C23×D10711yescharacteristic
209Q8—4284no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—1072no—
221C2×C4—1072no—
222C2×C4—2142no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2141no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2141no—
232C2×C4—2142no—
233C2×C4—2141no—
234C2×C4—2142no—
235C2×C4—2141no—
236C2×C4D4×D10712yes—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D10711yescharacteristic
245C22#1681215412yes—
246C22#1681215411yescharacteristic
247C22#1681215611yescharacteristic
248C22#1681215612yes—
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—4282no—
254C4—4281no—
255C4—4281no—
256C4—4282no—
257C4—2142no—
258C4—2142no—
259C2—82no—
260C2#2029319211yescharacteristic
261C2#2029319711yescharacteristic
262C2#2029320611yescharacteristic
263C2#2029337212yes—
264C2#2029338012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
prime radicalC22×D4×C107C22
hypercenter#394D107
hypocenterC107#2820
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16812099C2semidirect
3. #20292563C2semidirect
4. #20292555C2semidirect
5. #20292550C2semidirect
6. #20292550C2semidirect
7. #20292548C2semidirect
8. #20292547C2semidirect
10. #20292491C2semidirect
133. C107#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#168120992
C2×#168120962
#202925632
#202925552
#202925502
#202925482
#202925472
#202925412
#202924912
#394×C1072
#2820107