Finite Group Databasebeta

← #28344509

Subgroups — #28344509

Order 13696.

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

normal subgroups only

This group has 580 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×#16812098C211yesmaximal, characteristic, complemented
2C2×#16812095C211yesmaximal, characteristic, complemented
3#20292545C212yesmaximal, complemented
4#20292542C212yesmaximal, complemented
5#20292540C212yesmaximal, complemented
6#20292538C211yesmaximal, characteristic, complemented
7#20292537C211yesmaximal, characteristic, complemented
8#20292536C212yesmaximal, complemented
9#20292536C211yesmaximal, characteristic, complemented
10#20292491C211yesmaximal, characteristic, complemented
11#393×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×D428C2212yes—
15C22×C22⋊D107C2211yescharacteristic
16C22×C22⋊D107C2212yes—
17#16812098—24no—
18#16812095—24no—
19C2×Dic107⋊C4C2212yes—
20C2×C22⋊Dic107C2211yescharacteristic
21C2×(C2×D107)⋊C4C2211yescharacteristic
22C2×(C2×D107)⋊C4C2211yescharacteristic
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×Q8×C107C2211yescharacteristic
33C2×C22⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C23×D107C2312yes—
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×D428—22no—
42C2×D428—42no—
43C2×D428—42no—
44C2×D428—42no—
45C2×D428—42no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—42no—
48C2×C22⋊D107—42no—
49C2×C22⋊D107—42no—
50C2×C22⋊D107—42no—
51C22×Dic107C2311yescharacteristic
52C22×Dic107C2312yes—
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2311yescharacteristic
55Dic107⋊C4—42no—
56C22⋊Dic107—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—42no—
62(C2×D107)⋊C4—42no—
63(C2×D107)⋊C4—42no—
64C4×Dic107D414yescomplemented
65C24×C107C2311yescharacteristic
66C2×Q8×C107—22no—
67C2×Q8×C107—42no—
68C22×C4×C107C2311yescharacteristic
69C22×C4×C107C2312yes—
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22⋊C4×C107—22no—
73C22⋊C4×C107—42no—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—21no—
78C23×C107—21no—
79C23×C107C2411yescharacteristic
80C22×D107—21no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107—22no—
92Q8×C107—44no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107C2×D412yes—
103C22⋊D107—44no—
104C22⋊D107—48no—
105C2×Dic107—21no—
106C2×Dic107—22no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107C2×D412yes—
114C2×Dic107C2×D412yes—
115D428—44no—
116D428—48no—
117Dic214—44no—
118C22×C107—41no—
119C22×C107—41no—
120C22×C107—41no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107C22×D411yescharacteristic
124C22×C107H(5,𝔽2)12yes—
125C22×C107H(5,𝔽2)12yes—
126C22×C107H(5,𝔽2)11yescharacteristic
127C22×C1072−1+411yescharacteristic
128C2×D107—42no—
129C2×D107—41no—
130C2×D107—41no—
131C2×D107—42no—
132C2×D107—41no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—42no—
139C2×D107—42no—
140C4×C107—22no—
141C4×C107—41no—
142C4×C107—42no—
143C4×C107—42no—
144Dic107—22no—
145Dic107—22no—
146Dic107—41no—
147Dic107—42no—
148C2×C107—81no—
149C2×C107#53511yescharacteristic
150C2×C107#53511yescharacteristic
151C2×C107#53711yescharacteristic
152C2×C107H(3,𝔽4)12yes—
153C2×C107#56212yes—
154D107—82no—
155D107—81no—
156#2819—1071nomaximal, Sylow
157C107#281911yescharacteristic, complemented, Sylow
158C2×#124—1071no—
159C2×#124—1071no—
160#394—1072no—
161#394—1072no—
162#393—1071no—
163#393—1072no—
164#393D10711yescharacteristic, complemented
165#392—1071no—
166#392—1072no—
167#392—1071no—
168#382—1071no—
169C22×Q8—1071no—
170C22×Q8C2×D10711yescharacteristic
171C22×D4—1072no—
172C22×D4—1071no—
173C22×D4—1072no—
174C22×D4—1071no—
175#124—2144no—
176#124—2144no—
177C2×C4⋊C4—1072no—
178C2×C22⋊C4—1072no—
179C2×C22⋊C4—1072no—
180C2×C22⋊C4—1072no—
181C2×C22⋊C4—1071no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1071no—
184C2×C22⋊C4—1072no—
185C2×C22⋊C4—1071no—
186C2×C22⋊C4C2×D10711yescharacteristic
187C2×C22⋊C4C2×D10712yes—
188C2×C42—1071no—
189#95—1072no—
190#95—1071no—
191#95—1072no—
192#95C2×D10711yescharacteristic
193#95C2×D10712yes—
194C24—1071no—
195C24—1072no—
196C24C22×D10711yescharacteristic
197C2×Q8—22no—
198C2×Q8—42no—
199C2×Q8—4282no—
200C2×Q8—2142no—
201C2×D4—4282no—
202C2×D4—4282no—
203C2×D4—4282no—
204C2×D4—4282no—
205C2×D4—4282no—
206C2×D4—4282no—
207C2×D4—4282no—
208C2×D4—4282no—
209C2×D4—2142no—
210C2×D4—2142no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1072no—
214C22×C4—1071no—
215C22×C4C22×D10712yes—
216C22×C4C22×D10712yes—
217C22×C4C22×D10711yescharacteristic
218C22×C4C22×D10711yescharacteristic
219C4⋊C4—4282no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—4282no—
223C22⋊C4—4282no—
224C22⋊C4—4282no—
225C22⋊C4—2142no—
226C22⋊C4—2144no—
227C22⋊C4—2144no—
228C22⋊C4—2142no—
229C22⋊C4—2142no—
230C42—1074no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—2142no—
237C23—2141no—
238C23—2142no—
239C23—2142no—
240C23—2142no—
241C23—2142no—
242C23—2142no—
243C23—2142no—
244C23—2141no—
245C23—2142no—
246C23—2141no—
247C23—2142no—
248C23C23×D10711yescharacteristic
249Q8—44no—
250Q8—4284no—
251D4—4284no—
252D4—4288no—
253D4—4288no—
254D4—4284no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—1072no—
265C2×C4—1072no—
266C2×C4—2141no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4—2141no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4D4×D10712yes—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—4282no—
281C22—4281no—
282C22—4281no—
283C22—4282no—
284C22—4282no—
285C22—4281no—
286C22—4282no—
287C22—4282no—
288C22—4282no—
289C22—4282no—
290C22—4282no—
291C22—4282no—
292C22C2×D4×D10711yescharacteristic
293C22#1681215412yes—
294C22#1681215512yes—
295C22#1681215511yescharacteristic
296C22#1681215711yescharacteristic
297C4—22no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—4281no—
302C4—4282no—
303C4—2142no—
304C4—2142no—
305C2—81no—
306C2—8562no—
307C2—8561no—
308C2#2029319211yescharacteristic
309C2#2029319511yescharacteristic
310C2#2029320711yescharacteristic
311C2#2029336912yes—
312C2#2029337312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#393×C107C2
hypercenter#393D107
hypocenterC107#2819
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#16812098C2semidirect
2. C2×#16812095C2semidirect
3. #20292545C2semidirect
4. #20292542C2semidirect
5. #20292540C2semidirect
6. #20292538C2semidirect
7. #20292537C2semidirect
8. #20292536C2semidirect
9. #20292536C2semidirect
10. #20292491C2semidirect
11. #393×C107C2semidirect
64. C4×Dic107D4semidirect
157. C107#2819semidirect
164. #393D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
C2×#168120952
#202925452
#202925422
#202925402
#202925382
#202925372
#202925362
#202924912
#393×C1072
#2819107