Finite Group Databasebeta

← #28339797

Subgroups — #28339797

Order 13696.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#13133819C211yesmaximal, characteristic
2C2×#16812094C212yesmaximal, complemented
3C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292546C211yesmaximal, characteristic, complemented
5#20292545C211yesmaximal, characteristic, complemented
6#20292544C211yesmaximal, characteristic, complemented
7#20292505C212yesmaximal, complemented
8#20292501C212yesmaximal, complemented
9#20292498C212yesmaximal, complemented
10#20292489C211yesmaximal, characteristic, complemented
11#395×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—28no—
13C2×#13133819—24no—
14C22×D428C2211yescharacteristic
15C22×D428C2212yes—
16C22×C4×D107C2211yescharacteristic
17C22×C4×D107C2212yes—
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812094—28no—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C4⋊C4×D107—24no—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2211yescharacteristic
33C107⋊#95C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yes—
36C2×C4⋊C4×C107C2211yescharacteristic
37#95×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2312yes—
40C23×D107C2311yescharacteristic
41#13133819—232no—
42C2×D428—24no—
43C2×D428—24no—
44C2×D428—24no—
45C2×D428—22no—
46C2×D428D418yes—
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107—24no—
53C2×C4×D107—24no—
54C2×C4×D107—24no—
55C2×C4×D107—22no—
56C2×C4×D107—24no—
57C2×C4×D107—22no—
58C2×C4×D107—22no—
59C2×C4×D107—24no—
60C2×C4×D107D414yes—
61C4⋊Dic107—22no—
62Dic107⋊C4—22no—
63Dic107⋊C4—24no—
64Dic107⋊C4—22no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—24no—
67(C2×D107)⋊C4—24no—
68(C2×D107)⋊C4—24no—
69(C2×D107)⋊C4—24no—
70C2×Q8×C107—24no—
71C2×Q8×C107—22no—
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C4⋊C4×C107—22no—
78C4⋊C4×C107—24no—
79C23×C107C2411yescharacteristic
80C22×D107—21no—
81C22×D107—21no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107C2×D414yes—
90C22×D107C2×D414yes—
91C22×D107C2×D412yes—
92Q8×C107—28no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107C2×D414yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yescomplemented
106C2×Dic107—21no—
107C2×Dic107—22no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107C2×D412yes—
115C2×Dic107#4112yescomplemented
116D428—28no—
117D428—216no—
118C4×D107—28no—
119C4×D107—216no—
120C4×D107—28no—
121C4×D107—44no—
122C22×C107C22×D411yescharacteristic
123C22×C107C22×D412yes—
124C22×C107C2×#4111yescharacteristic
125C22×C107C2×#4111yescharacteristic
126C22×C1072−1+412yes—
127C2×D107—22no—
128C2×D107—22no—
129C2×D107—24no—
130C2×D107—24no—
131C2×D107—24no—
132C2×D107—24no—
133C2×D107—24no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107#12014yes—
139C4×C107—24no—
140C4×C107—22no—
141C4×C107—22no—
142C4×C107—41no—
143C4×C107—42no—
144Dic107—22no—
145Dic107—22no—
146Dic107—41no—
147Dic107—42no—
148C2×C107C2×#12011yescharacteristic
149C2×C107#52511yescharacteristic
150C2×C107#53611yescharacteristic
151C2×C107#54712yes—
152C2×C107#54712yes—
153D107—24no—
154D107—44no—
155#2406—1071nomaximal, Sylow
156C107#240611yescharacteristic, complemented, Sylow
157C22×#41—1071no—
158C2×#123—1072no—
159C22×C4⋊C4—1071no—
160#395—1071no—
161#395D10711yescharacteristic, complemented
162#394—1071no—
163#394—1071no—
164#386—1072no—
165#385—1072no—
166#385—1072no—
167#381—1071no—
168C2×#41—2144no—
169C2×#41—2148no—
170C22×Q8C2×D10711yescharacteristic
171C22×D4—1072no—
172C22×D4—1071no—
173C23×C4—1071no—
174C23×C4—1071no—
175C23×C4—1071no—
176C23×C4—1072no—
177#123—2148no—
178C2×C4⋊C4—1071no—
179C2×C4⋊C4—1071no—
180C2×C4⋊C4—1071no—
181C2×C4⋊C4—1072no—
182C2×C4⋊C4—2144no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C2×C4⋊C4C2×D10712yes—
185C2×C22⋊C4—1072no—
186C2×C22⋊C4—1072no—
187C2×C22⋊C4—1072no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190#95—1072no—
191#95—1071no—
192#95—1072no—
193#95C2×D10711yescharacteristic
194#95C2×D10712yes—
195C24—1071no—
196C24—1072no—
197#41—21432no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—1078no—
201C2×D4—2144no—
202C2×D4—2144no—
203C2×D4—2144no—
204C2×D4—2142no—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—1074no—
208C22×C4—1072no—
209C22×C4—1071no—
210C22×C4—2144no—
211C22×C4—2142no—
212C22×C4—2142no—
213C22×C4—2144no—
214C22×C4—2142no—
215C22×C4—2144no—
216C22×C4—2144no—
217C22×C4—2144no—
218C22×C4—2142no—
219C22×C4C22×D10712yes—
220C22×C4C22×D10711yescharacteristic
221C22×C4C22×D10711yescharacteristic
222C22×C4C22×D10712yes—
223C22×C4C22×D10711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—2142no—
227C4⋊C4—2142no—
228C4⋊C4—2142no—
229C4⋊C4—2144no—
230C22⋊C4—2144no—
231C22⋊C4—2144no—
232C22⋊C4—2144no—
233C22⋊C4—2144no—
234C22⋊C4—2144no—
235C23—1074no—
236C23—1072no—
237C23—1074no—
238C23—2142no—
239C23—2142no—
240C23—2142no—
241C23—2142no—
242C23—2142no—
243C23—2142no—
244C23—2141no—
245C23—2141no—
246C23—2142no—
247C23C23×D10711yescharacteristic
248Q8—28no—
249D4—21416no—
250D4—2148no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—1072no—
262C2×C4—1072no—
263C2×C4—4284no—
264C2×C4—2148no—
265C2×C4—21416no—
266C2×C4—2148no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4—2142no—
272C2×C4—2141no—
273C2×C4—2142no—
274C2×C4—2141no—
275C2×C4D4×D10712yes—
276C2×C4D4×D10714yes—
277C2×C4#1313381912yescomplemented
278C22—1074no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22—4282no—
283C22—2144no—
284C22—2144no—
285C22—2144no—
286C22—2142no—
287C22—2142no—
288C22—2144no—
289C22—2144no—
290C22C2×D4×D10711yescharacteristic
291C22C2×D4×D10712yes—
292C22#41×D10711yescharacteristic
293C22C2×#1313381911yescharacteristic
294C22#1681215712yes—
295C4—22no—
296C4—22no—
297C4—24no—
298C4—42no—
299C4—41no—
300C4—4282no—
301C4—4281no—
302C4—2142no—
303C4—2142no—
304C2—4284no—
305C2—2144no—
306C2#120×D10711yescharacteristic
307C2#2029314011yescharacteristic
308C2#2029320211yescharacteristic
309C2#2029327912yes—
310C2#2029328012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#395×C107C2
prime radicalC22×#13133819C2
hypercenter#395D107
hypocenterC107#2406
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#16812094C2semidirect
3. C2×C4⋊C4×D107C2semidirect
4. #20292546C2semidirect
5. #20292545C2semidirect
6. #20292544C2semidirect
7. #20292505C2semidirect
8. #20292501C2semidirect
9. #20292498C2semidirect
10. #20292489C2semidirect
11. #395×C107C2semidirect
105. C2×C4×C107#41semidirect
115. C2×Dic107#41semidirect
156. C107#2406semidirect
161. #395D107semidirect
277. C2×C4#13133819semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168120942
C2×C4⋊C4×D1072
#202925462
#202925452
#202925442
#202925052
#202925012
#202924982
#202924892
#395×C1072
#2406107