Finite Group Databasebeta

← #28344405

Subgroups — #28344405

Order 13696.

480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 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#20292569C212yesmaximal, complemented
2#20292540C212yesmaximal, complemented
3#20292538C212yesmaximal, complemented
4#20292533C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic
6#20292505C212yesmaximal
7#20292501C212yesmaximal, complemented
8#20292498C212yesmaximal, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C22×C4×D107C2212yes—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×Q8×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C42×C107C2211yescharacteristic
29#95×C107C2212yes—
30#95×C107C2212yes—
31C23×D107C2312yes—
32C2×Dic214—22no—
33C2×Dic214—41no—
34C2×Dic214—41no—
35C2×D428—41no—
36C2×D428—41no—
37C2×D428Q814yescomplemented
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C2×C4×D107—42no—
42C2×C4×D107—42no—
43C2×C4×D107—42no—
44C4⋊Dic107—41no—
45C4⋊Dic107—41no—
46Dic107⋊C4—42no—
47(C2×D107)⋊C4—24no—
48(C2×D107)⋊C4—24no—
49(C2×D107)⋊C4—42no—
50(C2×D107)⋊C4—42no—
51(C2×D107)⋊C4—42no—
52C2×Q8×C107—22no—
53C2×Q8×C107—41no—
54C2×Q8×C107—41no—
55C22×C4×C107C2312yes—
56C22×C4×C107C2312yes—
57C22×C4×C107C2311yescharacteristic
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C4⋊C4×C107—41no—
61C42×C107—41no—
62C23×C107C2411yescharacteristic
63C22×D107—22no—
64C22×D107—22no—
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—22no—
68C22×D107—22no—
69C22×D107C2×Q814yes—
70Q8×C107—44no—
71C2×C4×C107—21no—
72C2×C4×C107—21no—
73C2×C4×C107—21no—
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107—21no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107C2×Q812yes—
84C2×C4×C107#4112yes—
85C2×C4×C107#4112yes—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95D428—44no—
96C4×D107—48no—
97Dic214—44no—
98C22×C107C22×Q811yescharacteristic
99C22×C107C2×#4111yescharacteristic
100C22×C107C2×#4111yescharacteristic
101C22×C107H(5,𝔽2)11yescharacteristic
102C22×C107H(5,𝔽2)11yescharacteristic
103C22×C107H(5,𝔽2)11yescharacteristic
104C22×C107H(5,𝔽2)11yescharacteristic
105C2×D107—24no—
106C2×D107—24no—
107C2×D107—24no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—42no—
111C2×D107—42no—
112C4×C107—22no—
113C4×C107—22no—
114C4×C107—22no—
115C4×C107—42no—
116C4×C107—42no—
117Dic107—42no—
118Dic107—42no—
119Dic107—42no—
120C2×C107#53811yescharacteristic
121C2×C107#53811yescharacteristic
122C2×C107#54311yescharacteristic
123C2×C107#55411yescharacteristic
124C2×C107#55411yescharacteristic
125C2×C107#55911yescharacteristic
126C2×C107H(3,𝔽4)11yescharacteristic
127D107—44no—
128#2804—1071nomaximal, Sylow
129C107#280411yescharacteristic, complemented, Sylow
130#400—1072no—
131#393—1072no—
132#393—1072no—
133#391—1071no—
134#391D10711yescharacteristic, complemented
135#390—1071no—
136#386—1072no—
137#385—1072no—
138#385—1072no—
139C22×Q8—1071no—
140C22×Q8C2×D10711yescharacteristic
141C22×D4—1071no—
142C23×C4—1072no—
143C2×C4⋊C4—1072no—
144C2×C4⋊C4—1071no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4C2×D10711yescharacteristic
147C2×C22⋊C4—1072no—
148C2×C22⋊C4—1072no—
149C2×C22⋊C4—1072no—
150C2×C22⋊C4—1072no—
151C2×C22⋊C4—1072no—
152C2×C42C2×D10711yescharacteristic
153#95—1072no—
154#95—1072no—
155#95—1072no—
156#95—1072no—
157#95—1072no—
158#95C2×D10712yes—
159#95C2×D10712yes—
160C24—1072no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—4281no—
165C2×Q8—4281no—
166C2×Q8—2142no—
167C2×D4—1074no—
168C2×D4—4281no—
169C2×D4—4281no—
170C22×C4—1072no—
171C22×C4—1072no—
172C22×C4—1072no—
173C22×C4—4282no—
174C22×C4—4282no—
175C22×C4—4282no—
176C22×C4C22×D10712yes—
177C22×C4C22×D10712yes—
178C22×C4C22×D10711yescharacteristic
179C22×C4C22×D10711yescharacteristic
180C22×C4C22×D10711yescharacteristic
181C4⋊C4—41no—
182C4⋊C4—4282no—
183C4⋊C4—4281no—
184C4⋊C4—4281no—
185C22⋊C4—4282no—
186C22⋊C4—4282no—
187C22⋊C4—4282no—
188C22⋊C4—2144no—
189C22⋊C4—2144no—
190C42—41no—
191C23—1074no—
192C23—2142no—
193C23—2142no—
194C23—2142no—
195C23—2142no—
196C23—2142no—
197C23—2142no—
198C23C23×D10711yescharacteristic
199Q8—44no—
200Q8—4284no—
201D4—4284no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—4288no—
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×C4Q8×D10712yes—
225C2×C4#1313381812yes—
226C2×C4#1313381912yes—
227C22—4282no—
228C22—4282no—
229C22—4282no—
230C22—4282no—
231C22—2144no—
232C22—2144no—
233C22—2144no—
234C22C2×Q8×D10711yescharacteristic
235C22C2×#1313381811yescharacteristic
236C22C2×#1313381911yescharacteristic
237C22#1681215411yescharacteristic
238C22#1681215411yescharacteristic
239C22#1681215511yescharacteristic
240C22#1681215511yescharacteristic
241C4—22no—
242C4—22no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—4282no—
247C4—4282no—
248C4—4282no—
249C2—4284no—
250C2#2029321511yescharacteristic
251C2#2029321711yescharacteristic
252C2#2029325211yescharacteristic
253C2#2029333711yescharacteristic
254C2#2029333811yescharacteristic
255C2#2029336211yescharacteristic
256C2#2029336911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
prime radicalC22×D428C22
hypercenter#391D107
hypocenterC107#2804
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20292569C2semidirect
2. #20292540C2semidirect
3. #20292538C2semidirect
4. #20292533C2semidirect
7. #20292501C2semidirect
8. #20292498C2semidirect
9. #391×C107C2semidirect
37. C2×D428Q8semidirect
129. C107#2804semidirect
134. #391D107semidirect

Maximal subgroups

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

subgroupindex
#202925692
#202925402
#202925382
#202925332
#202925292
#202925052
#202925012
#202924982
#391×C1072
#2804107