Finite Group Databasebeta

← #28344462

Subgroups — #28344462

Order 13696.

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

normal subgroups only

This group has 578 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, complemented
2C2×#16812080C211yesmaximal, characteristic, complemented
3#20292566C211yesmaximal, characteristic, complemented
4#20292564C211yesmaximal, characteristic, complemented
5#20292550C212yesmaximal, complemented
6#20292543C212yesmaximal, complemented
7#20292542C212yesmaximal, complemented
8#20292536C212yesmaximal, complemented
9#20292535C211yesmaximal, characteristic, complemented
10#20292486C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C22⋊D107C2212yes—
15C22×C4×D107C2211yescharacteristic
16#16812094—24no—
17#16812080—24no—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2212yes—
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C23×D107C2311yescharacteristic
39C2×D428—22no—
40C2×D428—42no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—42no—
43C2×C22⋊D107—42no—
44C2×C22⋊D107—42no—
45C2×C22⋊D107—42no—
46C2×C22⋊D107—42no—
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2312yes—
49C22×Dic107C2312yes—
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—42no—
52C2×C4×D107D414yescomplemented
53C4⋊Dic107—22no—
54C4⋊Dic107—42no—
55Dic107⋊C4—22no—
56Dic107⋊C4—42no—
57C22⋊Dic107—24no—
58C22⋊Dic107—42no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—42no—
63(C2×D107)⋊C4—42no—
64C24×C107C2312yes—
65C2×D4×C107—22no—
66C2×D4×C107—42no—
67C22×C4×C107C2312yes—
68C22×C4×C107C2311yescharacteristic
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—22no—
72C22⋊C4×C107—42no—
73C22⋊C4×C107—42no—
74C23×C107—22no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107C2411yescharacteristic
82C22×D107—21no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—21no—
89C22×D107—22no—
90C22×D107—21no—
91C22×D107C2×D412yes—
92D4×C107—44no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107C2×D412yes—
103C22⋊D107—48no—
104C22⋊D107—48no—
105C2×Dic107—22no—
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—
115D428—44no—
116C4×D107—44no—
117C22×C107—42no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107C22×D411yescharacteristic
125C22×C107H(5,𝔽2)12yes—
126C22×C107H(5,𝔽2)11yescharacteristic
127C22×C107H(5,𝔽2)12yes—
128C22×C1072−1+411yescharacteristic
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—22no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—41no—
135C2×D107—41no—
136C2×D107—41no—
137C2×D107—42no—
138C2×D107—42no—
139C4×C107—22no—
140C4×C107—41no—
141C4×C107—41no—
142C4×C107—42no—
143Dic107—22no—
144Dic107—41no—
145Dic107—42no—
146Dic107—42no—
147C2×C107—82no—
148C2×C107#53411yescharacteristic
149C2×C107#53511yescharacteristic
150C2×C107#53711yescharacteristic
151C2×C107#56012yes—
152C2×C107#56212yes—
153D107—42no—
154D107—81no—
155#2813—1071nomaximal, Sylow
156C107#281311yescharacteristic, complemented, Sylow
157C2×#123—1071no—
158C2×#121—1071no—
159#400—1071no—
160#399—1071no—
161#396—1072no—
162#394—1072no—
163#394—1072no—
164#394D10711yescharacteristic, complemented
165#392—1071no—
166#392—1072no—
167#380—1071no—
168C22×D4—1072no—
169C22×D4—1072no—
170C22×D4—1071no—
171C22×D4C2×D10711yescharacteristic
172C23×C4—1071no—
173#123—2144no—
174#121—2144no—
175C2×C4⋊C4—1072no—
176C2×C4⋊C4—1072no—
177C2×C4⋊C4—1071no—
178C2×C4⋊C4—1071no—
179C2×C4⋊C4C2×D10711yescharacteristic
180C2×C22⋊C4—1072no—
181C2×C22⋊C4—1072no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1072no—
184C2×C22⋊C4—1071no—
185C2×C22⋊C4—1071no—
186C2×C22⋊C4—1071no—
187C2×C22⋊C4C2×D10712yes—
188C2×C22⋊C4C2×D10712yes—
189#95—1071no—
190#95—1072no—
191#95—1072no—
192#95C2×D10711yescharacteristic
193C24—1071no—
194C24—1071no—
195C24C22×D10712yes—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—4282no—
199C2×D4—4282no—
200C2×D4—4282no—
201C2×D4—4282no—
202C2×D4—4282no—
203C2×D4—4282no—
204C2×D4—2142no—
205C2×D4—2144no—
206C22×C4—1071no—
207C22×C4—1074no—
208C22×C4—1072no—
209C22×C4—1072no—
210C22×C4—1071no—
211C22×C4—4282no—
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10711yescharacteristic
214C22×C4C22×D10712yes—
215C22×C4C22×D10711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—4282no—
218C4⋊C4—4282no—
219C4⋊C4—2142no—
220C4⋊C4—2142no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—4282no—
224C22⋊C4—4282no—
225C22⋊C4—4282no—
226C22⋊C4—2142no—
227C22⋊C4—2142no—
228C22⋊C4—2144no—
229C22⋊C4—2142no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—1072no—
238C23—2141no—
239C23—2142no—
240C23—2141no—
241C23—2142no—
242C23—2141no—
243C23—2141no—
244C23—2142no—
245C23—2142no—
246C23—2141no—
247C23C23×D10711yescharacteristic
248D4—44no—
249D4—4284no—
250D4—4288no—
251D4—4288no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—1072no—
262C2×C4—4284no—
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—2142no—
272C2×C4D4×D10712yes—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—4281no—
281C22—4281no—
282C22—4281no—
283C22—4282no—
284C22—4282no—
285C22—4282no—
286C22—4282no—
287C22—2142no—
288C22—2142no—
289C22—2142no—
290C22C2×D4×D10711yescharacteristic
291C22#1681215412yes—
292C22#1681215411yescharacteristic
293C22#1681215512yes—
294C22#1681215611yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—42no—
299C4—4282no—
300C4—4281no—
301C4—4282no—
302C4—2142no—
303C2—82no—
304C2—8561no—
305C2—4282no—
306C2#2029318711yescharacteristic
307C2#2029319511yescharacteristic
308C2#2029320611yescharacteristic
309C2#2029336512yes—
310C2#2029337412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
hypercenter#394D107
hypocenterC107#2813
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
2. C2×#16812080C2semidirect
3. #20292566C2semidirect
4. #20292564C2semidirect
5. #20292550C2semidirect
6. #20292543C2semidirect
7. #20292542C2semidirect
8. #20292536C2semidirect
9. #20292535C2semidirect
10. #20292486C2semidirect
11. #394×C107C2semidirect
52. C2×C4×D107D4semidirect
156. C107#2813semidirect
164. #394D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120802
#202925662
#202925642
#202925502
#202925432
#202925422
#202925362
#202925352
#202924862
#394×C1072
#2813107