Finite Group Databasebeta

← #28344524

Subgroups — #28344524

Order 13696.

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

normal subgroups only

This group has 430 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×#16812108C211yesmaximal, characteristic, complemented
2C2×#16812102C211yesmaximal, characteristic
3#20292567C211yesmaximal, characteristic, complemented
4#20292565C211yesmaximal, characteristic, complemented
5#20292563C211yesmaximal, characteristic
6#20292560C212yesmaximal, complemented
7#20292555C212yesmaximal
8#20292547C212yesmaximal, complemented
9#20292539C212yesmaximal
10#20292494C211yesmaximal, characteristic, complemented
11#399×C107C211yesmaximal, characteristic
12C22×Dic214C2212yes—
13#16812108—24no—
14#16812102—24no—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C22⋊Dic107C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C107⋊#95C2211yescharacteristic
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C2×Dic214—24no—
38C2×Dic214—42no—
39C2×Dic214—42no—
40C22×Dic107C2312yes—
41C22×Dic107C2311yescharacteristic
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2312yes—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2311yescharacteristic
46C4⋊Dic107—22no—
47C4⋊Dic107—22no—
48Dic107⋊C4—22no—
49Dic107⋊C4—22no—
50Dic107⋊C4—24no—
51Dic107⋊C4—42no—
52C22⋊Dic107—22no—
53C22⋊Dic107—22no—
54C22⋊Dic107—42no—
55C4×Dic107D414yes—
56C24×C107C2311yescharacteristic
57C22×C4×C107C2311yescharacteristic
58C22×C4×C107C2312yes—
59C22×C4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107—42no—
63C22⋊C4×C107—22no—
64C22⋊C4×C107—42no—
65C23×C107—21no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107—21no—
69C23×C107—21no—
70C23×C107C2411yescharacteristic
71C2×C4×C107—22no—
72C2×C4×C107—21no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107C2×D412yes—
81C2×Dic107—22no—
82C2×Dic107—21no—
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107C2×D412yes—
96C2×Dic107C2×D412yes—
97Dic214—48no—
98C22×C107—41no—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107C22×D411yescharacteristic
104C22×C107H(5,𝔽2)12yes—
105C22×C1072−1+412yes—
106C22×C1072−1+411yescharacteristic
107C22×C1072−1+411yescharacteristic
108C4×C107—22no—
109C4×C107—42no—
110C4×C107—41no—
111C4×C107—42no—
112Dic107—22no—
113Dic107—22no—
114Dic107—42no—
115Dic107—41no—
116Dic107—41no—
117Dic107—42no—
118C2×C107—81no—
119C2×C107#53511yescharacteristic
120C2×C107#53611yescharacteristic
121C2×C107#53611yescharacteristic
122C2×C107#56312yes—
123C2×C107#56312yes—
124#2821—1071nomaximal, Sylow
125C107#282111yescharacteristic, complemented, Sylow
126C2×C4⋊Q8—1071no—
127C2×#126—1071no—
128#400—1071no—
129#400—1071no—
130#399—1071no—
131#399D10711yescharacteristic
132#398—1072no—
133#397—1072no—
134#395—1072no—
135#393—1072no—
136#383—1071no—
137C22×Q8—1072no—
138C4⋊Q8—2144no—
139#126—2144no—
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—1071no—
142C2×C4⋊C4—1071no—
143C2×C4⋊C4—1071no—
144C2×C4⋊C4—1072no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4C2×D10711yescharacteristic
147C2×C4⋊C4C2×D10712yes—
148C2×C22⋊C4—1072no—
149C2×C22⋊C4—1071no—
150C2×C22⋊C4—1071no—
151C2×C22⋊C4C2×D10711yescharacteristic
152C2×C22⋊C4C2×D10712yes—
153C2×C42—1071no—
154#95—1072no—
155#95—1072no—
156#95—1071no—
157#95—1072no—
158#95—1072no—
159#95—1072no—
160#95—1072no—
161#95C2×D10711yescharacteristic
162C24C22×D10711yescharacteristic
163C2×Q8—4282no—
164C2×Q8—4282no—
165C2×Q8—2144no—
166C22×C4—1071no—
167C22×C4—1071no—
168C22×C4—1072no—
169C22×C4—1071no—
170C22×C4—1071no—
171C22×C4—1072no—
172C22×C4C22×D10712yes—
173C22×C4C22×D10712yes—
174C22×C4C22×D10711yescharacteristic
175C22×C4C22×D10711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—4282no—
179C4⋊C4—2142no—
180C4⋊C4—2142no—
181C4⋊C4—2144no—
182C4⋊C4—2142no—
183C4⋊C4—2142no—
184C22⋊C4—22no—
185C22⋊C4—42no—
186C22⋊C4—4282no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C42—1074no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C23×D10711yescharacteristic
196Q8—4288no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—1072no—
207C2×C4—1072no—
208C2×C4—2141no—
209C2×C4—2141no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2141no—
217C2×C4—2142no—
218C2×C4—2141no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4—2141no—
222C2×C4D4×D10712yes—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22C2×D4×D10711yescharacteristic
229C22#1681215412yes—
230C22#1681215612yes—
231C22#1681215611yescharacteristic
232C22#1681215711yescharacteristic
233C4—22no—
234C4—42no—
235C4—42no—
236C4—41no—
237C4—4282no—
238C4—4282no—
239C4—4281no—
240C4—4281no—
241C4—2142no—
242C4—2142no—
243C2—81no—
244C2#2029319011yescharacteristic
245C2#2029319911yescharacteristic
246C2#2029320011yescharacteristic
247C2#2029337812yes—
248C2#2029338012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radicalC24×C107C23
hypercenter#399D107
hypocenterC107#2821
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812108C2semidirect
3. #20292567C2semidirect
4. #20292565C2semidirect
6. #20292560C2semidirect
8. #20292547C2semidirect
10. #20292494C2semidirect
125. C107#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#168121082
C2×#168121022
#202925672
#202925652
#202925632
#202925602
#202925552
#202925472
#202925392
#202924942
#399×C1072
#2821107