Finite Group Databasebeta

← #28344508

Subgroups — #28344508

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×#16812096C211yesmaximal, characteristic, complemented
2C2×#16812095C211yesmaximal, characteristic, complemented
3#20292542C212yesmaximal, complemented
4#20292541C212yesmaximal, complemented
5#20292540C211yesmaximal, characteristic, complemented
6#20292539C212yesmaximal, complemented
7#20292539C211yesmaximal, characteristic, complemented
8#20292535C211yesmaximal, characteristic, complemented
9#20292535C212yesmaximal, complemented
10#20292491C211yesmaximal, characteristic, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C22×C22⋊D107C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16#16812096—24no—
17#16812095—24no—
18C2×Dic107⋊C4C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×C22⋊Dic107C2212yes—
21C2×C22⋊Dic107C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2212yes—
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C22×D4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C2×Dic214—22no—
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×Dic214—42no—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—42no—
44C2×C22⋊D107—42no—
45C2×C22⋊D107—42no—
46C2×C22⋊D107—42no—
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2312yes—
49C22×Dic107C2312yes—
50C22×Dic107C2311yescharacteristic
51C22×Dic107C2311yescharacteristic
52Dic107⋊C4—42no—
53C22⋊Dic107—22no—
54C22⋊Dic107—24no—
55C22⋊Dic107—24no—
56C22⋊Dic107—42no—
57C22⋊Dic107—42no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—42no—
61C4×Dic107D414yescomplemented
62C24×C107C2311yescharacteristic
63C24×C107C2312yes—
64C2×D4×C107—22no—
65C2×D4×C107—42no—
66C2×D4×C107—42no—
67C2×D4×C107—42no—
68C2×D4×C107—42no—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C22⋊C4×C107—22no—
73C22⋊C4×C107—42no—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—21no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—21no—
85C23×C107—22no—
86C23×C107C2411yescharacteristic
87C22×D107—21no—
88C22×D107—22no—
89C22×D107—22no—
90C22×D107—21no—
91C22×D107—21no—
92D4×C107—44no—
93D4×C107—48no—
94C2×C4×C107—22no—
95C2×C4×C107—21no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107C2×D412yes—
101C22⋊D107—44no—
102C22⋊D107—48no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—21no—
106C2×Dic107—21no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107C2×D412yes—
115C2×Dic107C2×D412yes—
116Dic214—44no—
117Dic214—44no—
118C22×C107—41no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107C22×D411yescharacteristic
131C22×C107H(5,𝔽2)12yes—
132C22×C107H(5,𝔽2)12yes—
133C22×C107H(5,𝔽2)11yescharacteristic
134C22×C1072−1+411yescharacteristic
135C2×D107—41no—
136C2×D107—41no—
137C2×D107—41no—
138C2×D107—42no—
139C2×D107—42no—
140C4×C107—22no—
141C4×C107—41no—
142C4×C107—42no—
143Dic107—22no—
144Dic107—22no—
145Dic107—41no—
146Dic107—42no—
147Dic107—42no—
148C2×C107—81no—
149C2×C107—82no—
150C2×C107#53511yescharacteristic
151C2×C107#53511yescharacteristic
152C2×C107#53711yescharacteristic
153C2×C107H(3,𝔽4)12yes—
154C2×C107#56212yes—
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#393—1071no—
165#392—1071no—
166#392—1072no—
167#392D10711yescharacteristic, complemented
168#382—1071no—
169C22×Q8—1071no—
170C22×Q8—1071no—
171C22×D4—1072no—
172C22×D4—1071no—
173C22×D4C2×D10711yescharacteristic
174C22×D4C2×D10712yes—
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—1072no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1071no—
184C2×C22⋊C4—1071no—
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#95—1072no—
193#95C2×D10711yescharacteristic
194C24—1071no—
195C24C22×D10711yescharacteristic
196C24C22×D10712yes—
197C2×Q8—4282no—
198C2×Q8—4282no—
199C2×Q8—2142no—
200C2×Q8—2142no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—4282no—
207C2×D4—4282no—
208C2×D4—4282no—
209C2×D4—4282no—
210C2×D4—2142no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1072no—
214C22×C4—1071no—
215C22×C4—1072no—
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—2142no—
227C22⋊C4—2144no—
228C22⋊C4—2144no—
229C22⋊C4—2142no—
230C42—1074no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—22no—
243C23—2141no—
244C23—2142no—
245C23—2142no—
246C23—2141no—
247C23—2141no—
248C23C23×D10711yescharacteristic
249Q8—4284no—
250Q8—4284no—
251D4—44no—
252D4—48no—
253D4—4288no—
254D4—4284no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—1072no—
262C2×C4—1072no—
263C2×C4—2141no—
264C2×C4—2142no—
265C2×C4—2142no—
266C2×C4—2142no—
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—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—4281no—
288C22—4281no—
289C22—4281no—
290C22—4282no—
291C22—4282no—
292C22C2×D4×D10711yescharacteristic
293C22#1681215412yes—
294C22#1681215412yes—
295C22#1681215411yescharacteristic
296C22#1681215611yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—4281no—
301C4—4282no—
302C4—4282no—
303C4—2142no—
304C4—2142no—
305C2—81no—
306C2—82no—
307C2—8561no—
308C2#2029319011yescharacteristic
309C2#2029319211yescharacteristic
310C2#2029320611yescharacteristic
311C2#2029337012yes—
312C2#2029337212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2819
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16812096C2semidirect
2. C2×#16812095C2semidirect
3. #20292542C2semidirect
4. #20292541C2semidirect
5. #20292540C2semidirect
6. #20292539C2semidirect
7. #20292539C2semidirect
8. #20292535C2semidirect
9. #20292535C2semidirect
10. #20292491C2semidirect
11. #392×C107C2semidirect
61. C4×Dic107D4semidirect
157. C107#2819semidirect
167. #392D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120962
C2×#168120952
#202925422
#202925412
#202925402
#202925392
#202925352
#202924912
#392×C1072
#2819107