Finite Group Databasebeta

← #28344477

Subgroups — #28344477

Order 13696.

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

normal subgroups only

This group has 478 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×#16812094C211yesmaximal, characteristic
2C2×#16812089C211yesmaximal, characteristic, complemented
3#20292561C212yesmaximal, complemented
4#20292553C212yesmaximal, complemented
5#20292553C211yesmaximal, characteristic, complemented
6#20292549C212yesmaximal, complemented
7#20292547C211yesmaximal, characteristic, complemented
8#20292545C212yesmaximal
9#20292540C211yesmaximal, characteristic, complemented
10#20292486C211yesmaximal, characteristic, complemented
11#398×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812094—24no—
16#16812089—24no—
17C2×C4⋊Dic107C2212yes—
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2211yescharacteristic
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C23×D107C2311yescharacteristic
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×D428—22no—
42C2×D428—42no—
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—42no—
48C2×C4×D107D414yes—
49C4⋊Dic107—24no—
50Dic107⋊C4—22no—
51Dic107⋊C4—42no—
52Dic107⋊C4—42no—
53(C2×D107)⋊C4—24no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—42no—
58(C2×D107)⋊C4—42no—
59(C2×D107)⋊C4—42no—
60C22×C4×C107C2312yes—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C4⋊C4×C107—22no—
66C4⋊C4×C107—22no—
67C4⋊C4×C107—42no—
68C23×C107C2411yescharacteristic
69C22×D107—21no—
70C22×D107—22no—
71C22×D107—21no—
72C22×D107—22no—
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—22no—
76C22×D107—21no—
77C22×D107—22no—
78C22×D107C2×D412yes—
79C2×C4×C107—22no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D412yes—
92C2×Dic107—22no—
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—
102D428—44no—
103C4×D107—44no—
104Dic214—44no—
105C22×C107C22×D411yescharacteristic
106C22×C107H(5,𝔽2)11yescharacteristic
107C22×C107H(5,𝔽2)12yes—
108C22×C1072−1+411yescharacteristic
109C22×C1072−1+412yes—
110C2×D107—22no—
111C2×D107—22no—
112C2×D107—22no—
113C2×D107—42no—
114C2×D107—42no—
115C2×D107—41no—
116C2×D107—41no—
117C2×D107—41no—
118C2×D107—42no—
119C2×D107—42no—
120C4×C107—22no—
121C4×C107—41no—
122C4×C107—41no—
123C4×C107—42no—
124C4×C107—42no—
125Dic107—22no—
126Dic107—42no—
127Dic107—41no—
128Dic107—42no—
129C2×C107#53411yescharacteristic
130C2×C107#53611yescharacteristic
131C2×C107#53711yescharacteristic
132C2×C107#56212yes—
133C2×C107#56312yes—
134D107—42no—
135D107—81no—
136#2814—1071nomaximal, Sylow
137C107#281411yescharacteristic, complemented, Sylow
138C2×#123—1071no—
139C2×C22⋊Q8—1071no—
140#398—1072no—
141#398D10711yescharacteristic, complemented
142#397—1071no—
143#397—1072no—
144#396—1072no—
145#395—1071no—
146#394—1072no—
147#393—1071no—
148#380—1071no—
149C22×Q8—1071no—
150C22×D4—1071no—
151C23×C4—1071no—
152#123—2144no—
153C22⋊Q8—2144no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4—1072no—
156C2×C4⋊C4—1072no—
157C2×C4⋊C4—1071no—
158C2×C4⋊C4C2×D10711yescharacteristic
159C2×C4⋊C4C2×D10711yescharacteristic
160C2×C4⋊C4C2×D10712yes—
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4—1071no—
166C2×C22⋊C4—1071no—
167C2×C22⋊C4—1071no—
168#95—1071no—
169#95—1072no—
170#95—1072no—
171#95—1072no—
172#95C2×D10711yescharacteristic
173#95C2×D10712yes—
174C24—1071no—
175C24—1071no—
176C2×Q8—4282no—
177C2×Q8—2142no—
178C2×D4—4282no—
179C2×D4—2142no—
180C22×C4—1071no—
181C22×C4—1074no—
182C22×C4—1071no—
183C22×C4—1072no—
184C22×C4—1072no—
185C22×C4—4282no—
186C22×C4C22×D10711yescharacteristic
187C22×C4C22×D10712yes—
188C22×C4C22×D10712yes—
189C22×C4C22×D10711yescharacteristic
190C22×C4C22×D10711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—4282no—
195C4⋊C4—4282no—
196C4⋊C4—2144no—
197C4⋊C4—2142no—
198C22⋊C4—4282no—
199C22⋊C4—4282no—
200C22⋊C4—4282no—
201C22⋊C4—2142no—
202C22⋊C4—2144no—
203C22⋊C4—2142no—
204C22⋊C4—2142no—
205C23—1072no—
206C23—2141no—
207C23—2141no—
208C23—2142no—
209C23—2141no—
210C23—2142no—
211C23—2141no—
212C23—2142no—
213C23—2141no—
214C23—2142no—
215C23C23×D10711yescharacteristic
216Q8—4284no—
217D4—4284no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—1072no—
231C2×C4—4284no—
232C2×C4—2141no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10712yes—
242C22—4282no—
243C22—4281no—
244C22—4281no—
245C22—4281no—
246C22—4282no—
247C22—4282no—
248C22—4282no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22C2×D4×D10711yescharacteristic
253C22#1681215411yescharacteristic
254C22#1681215512yes—
255C22#1681215611yescharacteristic
256C22#1681215712yes—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—42no—
262C4—4282no—
263C4—4281no—
264C4—4282no—
265C4—2142no—
266C2—8561no—
267C2—4282no—
268C2#2029318811yescharacteristic
269C2#2029320211yescharacteristic
270C2#2029320611yescharacteristic
271C2#2029337312yes—
272C2#2029338112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2814
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#16812089C2semidirect
3. #20292561C2semidirect
4. #20292553C2semidirect
5. #20292553C2semidirect
6. #20292549C2semidirect
7. #20292547C2semidirect
9. #20292540C2semidirect
10. #20292486C2semidirect
11. #398×C107C2semidirect
137. C107#2814semidirect
141. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120892
#202925612
#202925532
#202925492
#202925472
#202925452
#202925402
#202924862
#398×C1072
#2814107