Finite Group Databasebeta

← #28344494

Subgroups — #28344494

Order 13696.

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

normal subgroups only

This group has 528 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×#16812087C211yesmaximal, characteristic, complemented
2C2×#16812084C211yesmaximal, characteristic
3#20292562C212yesmaximal, complemented
4#20292558C211yesmaximal, characteristic, complemented
5#20292545C212yesmaximal, complemented
6#20292545C211yesmaximal, characteristic, complemented
7#20292544C212yesmaximal, complemented
8#20292544C211yesmaximal, characteristic, complemented
9#20292540C212yesmaximal, complemented
10#20292489C211yesmaximal, characteristic, complemented
11#398×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2212yes—
14C22×D428C2211yescharacteristic
15C22×C4×D107C2211yescharacteristic
16#16812087—24no—
17#16812084—24no—
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2211yescharacteristic
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2312yes—
38C23×D107C2311yescharacteristic
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×D428—24no—
42C2×D428—22no—
43C2×D428—42no—
44C2×D428—42no—
45C2×D428—42no—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C2×C4×D107—42no—
51C2×C4×D107D414yes—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—42no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—42no—
59(C2×D107)⋊C4—42no—
60(C2×D107)⋊C4—42no—
61(C2×D107)⋊C4—42no—
62(C2×D107)⋊C4—42no—
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2312yes—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107—22no—
70C4⋊C4×C107—42no—
71C23×C107C2411yescharacteristic
72C22×D107—22no—
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—22no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107C2×D412yes—
84C2×C4×C107—22no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107C2×D412yes—
97C2×Dic107—22no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—21no—
105C2×Dic107—22no—
106C2×Dic107C2×D412yes—
107D428—44no—
108D428—48no—
109C4×D107—44no—
110Dic214—44no—
111C22×C107C22×D411yescharacteristic
112C22×C107H(5,𝔽2)11yescharacteristic
113C22×C107H(5,𝔽2)12yes—
114C22×C107H(5,𝔽2)11yescharacteristic
115C22×C1072−1+412yes—
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—42no—
120C2×D107—42no—
121C2×D107—42no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C4×C107—22no—
129C4×C107—41no—
130C4×C107—42no—
131C4×C107—41no—
132C4×C107—42no—
133Dic107—22no—
134Dic107—41no—
135Dic107—41no—
136Dic107—42no—
137C2×C107#53411yescharacteristic
138C2×C107#53511yescharacteristic
139C2×C107#53611yescharacteristic
140C2×C107#56212yes—
141C2×C107#56212yes—
142D107—42no—
143D107—82no—
144#2816—1071nomaximal, Sylow
145C107#281611yescharacteristic, complemented, Sylow
146C2×C22⋊Q8—1071no—
147C2×#121—1071no—
148#399—1072no—
149#398—1071no—
150#398D10711yescharacteristic, complemented
151#394—1071no—
152#394—1072no—
153#394—1071no—
154#394—1072no—
155#393—1072no—
156#381—1071no—
157C22×Q8—1071no—
158C22×D4—1072no—
159C22×D4—1071no—
160C23×C4—1071no—
161C22⋊Q8—2144no—
162#121—2144no—
163C2×C4⋊C4—1072no—
164C2×C4⋊C4—1071no—
165C2×C4⋊C4—1071no—
166C2×C4⋊C4C2×D10711yescharacteristic
167C2×C4⋊C4C2×D10711yescharacteristic
168C2×C4⋊C4C2×D10712yes—
169C2×C22⋊C4—1072no—
170C2×C22⋊C4—1072no—
171C2×C22⋊C4—1072no—
172C2×C22⋊C4—1072no—
173C2×C22⋊C4—1072no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1071no—
176C2×C22⋊C4—1071no—
177#95—1072no—
178#95—1071no—
179#95—1072no—
180#95C2×D10711yescharacteristic
181#95C2×D10712yes—
182C24—1071no—
183C24—1072no—
184C2×Q8—4282no—
185C2×Q8—2142no—
186C2×D4—4282no—
187C2×D4—4282no—
188C2×D4—4282no—
189C2×D4—2144no—
190C2×D4—2142no—
191C22×C4—1071no—
192C22×C4—1074no—
193C22×C4—1072no—
194C22×C4—1071no—
195C22×C4—1071no—
196C22×C4—4282no—
197C22×C4C22×D10712yes—
198C22×C4C22×D10712yes—
199C22×C4C22×D10711yescharacteristic
200C22×C4C22×D10711yescharacteristic
201C22×C4C22×D10711yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—4282no—
206C4⋊C4—2142no—
207C4⋊C4—2142no—
208C22⋊C4—4282no—
209C22⋊C4—4282no—
210C22⋊C4—4282no—
211C22⋊C4—4282no—
212C22⋊C4—4282no—
213C22⋊C4—2144no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C23—1072no—
217C23—2142no—
218C23—2142no—
219C23—2142no—
220C23—2142no—
221C23—2142no—
222C23—2142no—
223C23—2142no—
224C23—2141no—
225C23—2142no—
226C23—2141no—
227C23—2142no—
228C23C23×D10711yescharacteristic
229Q8—4284no—
230D4—4284no—
231D4—4288no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—1072no—
245C2×C4—4284no—
246C2×C4—2141no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4—2142no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2141no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10712yes—
256C22—4282no—
257C22—4282no—
258C22—4282no—
259C22—4282no—
260C22—4282no—
261C22—4282no—
262C22—4282no—
263C22—4282no—
264C22—4282no—
265C22—2142no—
266C22—2142no—
267C22—2142no—
268C22C2×D4×D10711yescharacteristic
269C22#1681215411yescharacteristic
270C22#1681215512yes—
271C22#1681215511yescharacteristic
272C22#1681215712yes—
273C4—22no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—4281no—
279C4—4281no—
280C4—4282no—
281C4—2142no—
282C2—8562no—
283C2—4282no—
284C2#2029318811yescharacteristic
285C2#2029319311yescharacteristic
286C2#2029320211yescharacteristic
287C2#2029337312yes—
288C2#2029337512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC2×#16812084C2
hypercenter#398D107
hypocenterC107#2816
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16812087C2semidirect
3. #20292562C2semidirect
4. #20292558C2semidirect
5. #20292545C2semidirect
6. #20292545C2semidirect
7. #20292544C2semidirect
8. #20292544C2semidirect
9. #20292540C2semidirect
10. #20292489C2semidirect
11. #398×C107C2semidirect
145. C107#2816semidirect
150. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120872
C2×#168120842
#202925622
#202925582
#202925452
#202925442
#202925402
#202924892
#398×C1072
#2816107