Finite Group Databasebeta

← #28344515

Subgroups — #28344515

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×#16812100C211yesmaximal, characteristic, complemented
2C2×#16812095C211yesmaximal, characteristic
3#20292562C211yesmaximal, characteristic, complemented
4#20292554C211yesmaximal, characteristic, complemented
5#20292552C211yesmaximal, characteristic, complemented
6#20292550C212yesmaximal, complemented
7#20292549C212yesmaximal, complemented
8#20292547C212yesmaximal, complemented
9#20292542C212yesmaximal
10#20292491C211yesmaximal, characteristic, complemented
11#399×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C22⋊D107C2211yescharacteristic
14#16812100—24no—
15#16812095—24no—
16C2×C4⋊Dic107C2212yes—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2212yes—
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C2×Dic214—22no—
39C2×Dic214—42no—
40C2×C22⋊D107—22no—
41C2×C22⋊D107—42no—
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C4⋊Dic107—22no—
48C4⋊Dic107—24no—
49Dic107⋊C4—24no—
50Dic107⋊C4—42no—
51Dic107⋊C4—42no—
52C22⋊Dic107—22no—
53C22⋊Dic107—42no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—42no—
57(C2×D107)⋊C4—42no—
58C4×Dic107D414yes—
59C24×C107C2311yescharacteristic
60C22×C4×C107C2312yes—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2311yescharacteristic
64C4⋊C4×C107—22no—
65C4⋊C4×C107—42no—
66C22⋊C4×C107—22no—
67C22⋊C4×C107—42no—
68C23×C107—21no—
69C23×C107—22no—
70C23×C107—21no—
71C23×C107—22no—
72C23×C107—21no—
73C23×C107C2411yescharacteristic
74C22×D107—21no—
75C22×D107—21no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—21no—
79C2×C4×C107—22no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D412yes—
89C22⋊D107—44no—
90C2×Dic107—21no—
91C2×Dic107—22no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107C2×D412yes—
102C2×Dic107C2×D412yes—
103Dic214—44no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—42no—
108C22×C107—42no—
109C22×C107C22×D411yescharacteristic
110C22×C107H(5,𝔽2)12yes—
111C22×C107H(5,𝔽2)11yescharacteristic
112C22×C1072−1+412yes—
113C22×C1072−1+411yescharacteristic
114C2×D107—41no—
115C2×D107—41no—
116C2×D107—41no—
117C2×D107—42no—
118C2×D107—42no—
119C4×C107—22no—
120C4×C107—41no—
121C4×C107—42no—
122C4×C107—42no—
123Dic107—22no—
124Dic107—22no—
125Dic107—41no—
126Dic107—42no—
127Dic107—42no—
128C2×C107—81no—
129C2×C107#53511yescharacteristic
130C2×C107#53611yescharacteristic
131C2×C107#53711yescharacteristic
132C2×C107#56212yes—
133C2×C107#56312yes—
134D107—81no—
135#2820—1071nomaximal, Sylow
136C107#282011yescharacteristic, complemented, Sylow
137C2×#125—1071no—
138C2×#124—1071no—
139#399—1071no—
140#399D10711yescharacteristic, complemented
141#397—1071no—
142#397—1071no—
143#396—1072no—
144#396—1072no—
145#395—1072no—
146#394—1072no—
147#382—1071no—
148C22×Q8—1071no—
149C22×D4—1071no—
150#125—2144no—
151#124—2144no—
152C2×C4⋊C4—1072no—
153C2×C4⋊C4—1072no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4—1072no—
156C2×C4⋊C4—1071no—
157C2×C4⋊C4C2×D10711yescharacteristic
158C2×C4⋊C4C2×D10712yes—
159C2×C22⋊C4—1072no—
160C2×C22⋊C4—1072no—
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1071no—
163C2×C22⋊C4—1071no—
164C2×C22⋊C4—1071no—
165C2×C22⋊C4C2×D10711yescharacteristic
166C2×C22⋊C4C2×D10712yes—
167C2×C42—1071no—
168#95—1072no—
169#95—1071no—
170#95—1072no—
171#95—1072no—
172#95C2×D10711yescharacteristic
173C24—1071no—
174C24C22×D10711yescharacteristic
175C2×Q8—4282no—
176C2×Q8—2142no—
177C2×D4—4282no—
178C2×D4—2142no—
179C22×C4—1071no—
180C22×C4—1071no—
181C22×C4—1072no—
182C22×C4—1072no—
183C22×C4—1071no—
184C22×C4C22×D10712yes—
185C22×C4C22×D10711yescharacteristic
186C22×C4C22×D10712yes—
187C22×C4C22×D10711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—4282no—
191C4⋊C4—4282no—
192C4⋊C4—2144no—
193C4⋊C4—2144no—
194C4⋊C4—2142no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—4282no—
198C22⋊C4—4282no—
199C22⋊C4—4282no—
200C22⋊C4—2142no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C42—1074no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—2141no—
210C23—2142no—
211C23—2141no—
212C23—2142no—
213C23—2141no—
214C23C23×D10711yescharacteristic
215Q8—4284no—
216D4—4284no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—1072no—
227C2×C4—1072no—
228C2×C4—2141no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4D4×D10712yes—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—4282no—
246C22—4281no—
247C22—4281no—
248C22—4281no—
249C22—4282no—
250C22C2×D4×D10711yescharacteristic
251C22#1681215412yes—
252C22#1681215511yescharacteristic
253C22#1681215612yes—
254C22#1681215711yescharacteristic
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—4281no—
260C4—4282no—
261C4—4282no—
262C4—2142no—
263C4—2142no—
264C2—81no—
265C2—8561no—
266C2#2029319011yescharacteristic
267C2#2029319911yescharacteristic
268C2#2029320711yescharacteristic
269C2#2029337412yes—
270C2#2029337812yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#399D107
hypocenterC107#2820
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812100C2semidirect
3. #20292562C2semidirect
4. #20292554C2semidirect
5. #20292552C2semidirect
6. #20292550C2semidirect
7. #20292549C2semidirect
8. #20292547C2semidirect
10. #20292491C2semidirect
11. #399×C107C2semidirect
136. C107#2820semidirect
140. #399D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121002
C2×#168120952
#202925622
#202925542
#202925522
#202925502
#202925492
#202925472
#202925422
#202924912
#399×C1072
#2820107