Finite Group Databasebeta

← #28344475

Subgroups — #28344475

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×#16812091C211yesmaximal, characteristic
2C2×#16812089C211yesmaximal, characteristic, complemented
3#20292559C212yesmaximal, complemented
4#20292558C211yesmaximal, characteristic, complemented
5#20292556C212yesmaximal, complemented
6#20292550C212yesmaximal, complemented
7#20292548C211yesmaximal, characteristic, complemented
8#20292543C212yesmaximal
9#20292539C211yesmaximal, characteristic, complemented
10#20292486C211yesmaximal, characteristic, complemented
11#397×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C22⋊D107C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812091—24no—
16#16812089—24no—
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2212yes—
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2212yes—
23C2×C22⋊Dic107C2211yescharacteristic
24C2×C22⋊Dic107C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C22⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
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
47C2×C4×D107—42no—
48C2×C4×D107D414yes—
49C4⋊Dic107—24no—
50C4⋊Dic107—22no—
51C4⋊Dic107—42no—
52Dic107⋊C4—22no—
53Dic107⋊C4—42no—
54Dic107⋊C4—42no—
55C22⋊Dic107—22no—
56C22⋊Dic107—42no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—42no—
60C24×C107C2311yescharacteristic
61C22×C4×C107C2312yes—
62C22×C4×C107C2312yes—
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C4⋊C4×C107—22no—
66C22⋊C4×C107—22no—
67C22⋊C4×C107—42no—
68C23×C107—21no—
69C23×C107—21no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—21no—
73C23×C107C2411yescharacteristic
74C22×D107—21no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—21no—
78C22×D107C2×D412yes—
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—22no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—21no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107C2×D412yes—
103C4×D107—44no—
104Dic214—44no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107—42no—
109C22×C107—42no—
110C22×C107C22×D411yescharacteristic
111C22×C107H(5,𝔽2)11yescharacteristic
112C22×C107H(5,𝔽2)12yes—
113C22×C1072−1+412yes—
114C22×C1072−1+411yescharacteristic
115C2×D107—22no—
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—42no—
119C2×D107—42no—
120C4×C107—22no—
121C4×C107—41no—
122C4×C107—42no—
123C4×C107—42no—
124Dic107—22no—
125Dic107—42no—
126Dic107—41no—
127Dic107—41no—
128Dic107—42no—
129C2×C107—81no—
130C2×C107#53411yescharacteristic
131C2×C107#53611yescharacteristic
132C2×C107#53711yescharacteristic
133C2×C107#56212yes—
134C2×C107#56312yes—
135D107—42no—
136#2814—1071nomaximal, Sylow
137C107#281411yescharacteristic, complemented, Sylow
138C2×#123—1071no—
139C2×C22⋊Q8—1071no—
140#398—1071no—
141#398—1072no—
142#397—1072no—
143#397D10711yescharacteristic, complemented
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—1072no—
158C2×C4⋊C4—1071no—
159C2×C4⋊C4—1071no—
160C2×C4⋊C4C2×D10711yescharacteristic
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1071no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4—1071no—
166C2×C22⋊C4C2×D10711yescharacteristic
167C2×C22⋊C4C2×D10712yes—
168#95—1071no—
169#95—1072no—
170#95—1072no—
171#95—1072no—
172#95C2×D10711yescharacteristic
173#95C2×D10712yes—
174C24—1071no—
175C24C22×D10711yescharacteristic
176C2×Q8—4282no—
177C2×Q8—2142no—
178C2×D4—4282no—
179C2×D4—2142no—
180C22×C4—1071no—
181C22×C4—1074no—
182C22×C4—1072no—
183C22×C4—1071no—
184C22×C4—1072no—
185C22×C4—1071no—
186C22×C4—4282no—
187C22×C4C22×D10711yescharacteristic
188C22×C4C22×D10712yes—
189C22×C4C22×D10712yes—
190C22×C4C22×D10711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—4282no—
193C4⋊C4—4282no—
194C4⋊C4—4282no—
195C4⋊C4—2144no—
196C4⋊C4—2142no—
197C4⋊C4—2142no—
198C22⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—4282no—
201C22⋊C4—4282no—
202C22⋊C4—2144no—
203C22⋊C4—2142no—
204C22⋊C4—2142no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—1072no—
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—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—1072no—
228C2×C4—4284no—
229C2×C4—2141no—
230C2×C4—2142no—
231C2×C4—2141no—
232C2×C4—2142no—
233C2×C4—2141no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10712yes—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—4282no—
248C22—4282no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22C2×D4×D10711yescharacteristic
253C22#1681215411yescharacteristic
254C22#1681215512yes—
255C22#1681215611yescharacteristic
256C22#1681215612yes—
257C4—22no—
258C4—41no—
259C4—42no—
260C4—42no—
261C4—4281no—
262C4—4282no—
263C4—4281no—
264C4—4282no—
265C4—2142no—
266C2—81no—
267C2—4282no—
268C2#2029318811yescharacteristic
269C2#2029319711yescharacteristic
270C2#2029320611yescharacteristic
271C2#2029337412yes—
272C2#2029337912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#397×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#397D107
hypocenterC107#2814
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#16812089C2semidirect
3. #20292559C2semidirect
4. #20292558C2semidirect
5. #20292556C2semidirect
6. #20292550C2semidirect
7. #20292548C2semidirect
9. #20292539C2semidirect
10. #20292486C2semidirect
11. #397×C107C2semidirect
137. C107#2814semidirect
143. #397D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120912
C2×#168120892
#202925592
#202925582
#202925562
#202925502
#202925482
#202925432
#202925392
#202924862
#397×C1072
#2814107