Finite Group Databasebeta

← #28339673

Subgroups — #28339673

Order 13696.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×#13133817C211yesmaximal, characteristic, complemented
2C2×D4×Dic107C212yesmaximal, complemented
3C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
4#20292548C211yesmaximal, characteristic, complemented
5#20292543C212yesmaximal, complemented
6#20292541C211yesmaximal, characteristic, complemented
7#20292539C211yesmaximal, characteristic, complemented
8#20292535C212yesmaximal, complemented
9C22⋊C4×Dic107C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—28no—
13C2×#13133817—24no—
14C22×Dic214C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C22×C22⋊D107C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C23×Dic107C2212yes—
19C22×C4×D107C2211yescharacteristic
20D4×Dic107—28no—
21(C22⋊D107)⋊C4—24no—
22C2×C4⋊Dic107C2212yes—
23C2×Dic107⋊C4C2211yescharacteristic
24C2×C22⋊Dic107C2212yes—
25C2×C22⋊Dic107C2212yes—
26C2×C22⋊Dic107C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2211yescharacteristic
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C22×D4×C107C2212yes—
36C2×C22⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2211yescharacteristic
38#95×C107C2211yescharacteristic
39C23×D107C2311yescharacteristic
40#13133817—232no—
41C2×Dic214—22no—
42C2×Dic214—24no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107—22no—
45C2×C22⋊D107—22no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—24no—
48C2×C22⋊D107—22no—
49C22×Dic107—24no—
50C22×Dic107—24no—
51C22×Dic107—22no—
52C22×Dic107—24no—
53C22×Dic107D418yescomplemented
54C22×Dic107C2312yes—
55C22×Dic107C2311yescharacteristic
56C22×Dic107C2312yes—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C2×C4×D107—24no—
60C2×C4×D107D414yescomplemented
61C4⋊Dic107—24no—
62Dic107⋊C4—22no—
63C22⋊Dic107—24no—
64C22⋊Dic107—24no—
65C22⋊Dic107—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—22no—
69C4×Dic107—24no—
70C24×C107C2312yes—
71C24×C107C2311yescharacteristic
72C2×D4×C107—24no—
73C2×D4×C107—24no—
74C2×D4×C107—24no—
75C2×D4×C107—22no—
76C2×D4×C107—24no—
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2311yescharacteristic
79C22×C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C22⋊C4×C107—24no—
82C23×C107—21no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—22no—
88C23×C107—22no—
89C23×C107—22no—
90C23×C107—22no—
91C23×C107—21no—
92C23×C107C2×D414yes—
93C23×C107#4112yescomplemented
94C23×C107C2411yescharacteristic
95C22×D107—22no—
96C22×D107—22no—
97C22×D107—21no—
98C22×D107—21no—
99C22×D107C2×D412yes—
100D4×C107—216no—
101D4×C107—28no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—22no—
105C2×C4×C107—21no—
106C2×C4×C107—22no—
107C2×C4×C107C2×D412yes—
108C2×C4×C107#4114yescomplemented
109C22⋊D107—28no—
110C22⋊D107—216no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—216no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107—22no—
121C2×Dic107—44no—
122C2×Dic107C2×D414yes—
123C2×Dic107C2×D414yes—
124C2×Dic107C2×D412yes—
125C2×Dic107#4112yescomplemented
126C4×D107—28no—
127Dic214—28no—
128C22×C107—24no—
129C22×C107—22no—
130C22×C107—24no—
131C22×C107—24no—
132C22×C107—24no—
133C22×C107—41no—
134C22×C107—42no—
135C22×C107—42no—
136C22×C107—42no—
137C22×C107—42no—
138C22×C107—41no—
139C22×C107—42no—
140C22×C107C22×D411yescharacteristic
141C22×C107C22×D412yes—
142C22×C107C2×#4111yescharacteristic
143C22×C107C2×#4111yescharacteristic
144C22×C107C2×#4112yes—
145C2×D107—22no—
146C2×D107—22no—
147C2×D107—22no—
148C2×D107—42no—
149C2×D107—42no—
150C4×C107—22no—
151C4×C107—24no—
152C4×C107—41no—
153Dic107—24no—
154Dic107—22no—
155Dic107—41no—
156Dic107—42no—
157Dic107#12014yes—
158C2×C107—44no—
159C2×C107—42no—
160C2×C107C2×#12011yescharacteristic
161C2×C107#52512yes—
162C2×C107#52511yescharacteristic
163C2×C107#53212yes—
164C2×C107#53211yescharacteristic
165D107—42no—
166#2388—1071nomaximal, Sylow
167C107#238811yescharacteristic, complemented, Sylow
168C22×#41—1071no—
169C2×C4×D4—1072no—
170C2×C4×D4—1071no—
171#395—1071no—
172#394—1071no—
173#394—1072no—
174#393—1071no—
175#392—1072no—
176#392D10711yescharacteristic, complemented
177C4×C22⋊C4—1072no—
178C4×C22⋊C4—1071no—
179C2×#41—2148no—
180C2×#41—2144no—
181C22×Q8—1071no—
182C22×D4—1071no—
183C22×D4—1072no—
184C22×D4C2×D10712yes—
185C22×D4C2×D10711yescharacteristic
186C23×C4—1072no—
187C23×C4—1071no—
188C23×C4—1071no—
189C4×D4—2144no—
190C4×D4—2148no—
191C2×C4⋊C4—1071no—
192C2×C4⋊C4—1072no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1071no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4C2×D10711yescharacteristic
199C2×C22⋊C4C2×D10712yes—
200C2×C42—1071no—
201C2×C42—1072no—
202#95—1072no—
203#95—1072no—
204#95—1071no—
205#95C2×D10711yescharacteristic
206C24—1071no—
207C24C22×D10712yes—
208C24C22×D10711yescharacteristic
209#41—21432no—
210C2×Q8—2144no—
211C2×Q8—2142no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—2144no—
218C2×D4—2142no—
219C2×D4—2142no—
220C2×D4—2144no—
221C2×D4—2144no—
222C2×D4—2142no—
223C22×C4—1071no—
224C22×C4—1078no—
225C22×C4—1071no—
226C22×C4—1072no—
227C22×C4—1074no—
228C22×C4—1072no—
229C22×C4—1071no—
230C22×C4—2144no—
231C22×C4—2142no—
232C22×C4—2144no—
233C22×C4—2144no—
234C22×C4—2144no—
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10711yescharacteristic
237C22×C4C22×D10712yes—
238C4⋊C4—2144no—
239C4⋊C4—2142no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—2142no—
243C22⋊C4—2144no—
244C22⋊C4—2144no—
245C22⋊C4—2144no—
246C22⋊C4—2144no—
247C42—2144no—
248C42—2142no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—1072no—
260C23—2142no—
261C23—2141no—
262C23—2142no—
263C23—2141no—
264C23D4×D10714yes—
265C23#1313381712yescomplemented
266C23C23×D10711yescharacteristic
267Q8—2148no—
268D4—28no—
269D4—216no—
270D4—21416no—
271D4—2148no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—1074no—
278C2×C4—1072no—
279C2×C4—1074no—
280C2×C4—1072no—
281C2×C4—4284no—
282C2×C4—2148no—
283C2×C4—21416no—
284C2×C4—2142no—
285C2×C4—2142no—
286C2×C4—2141no—
287C2×C4—2142no—
288C2×C4—2142no—
289C2×C4—2141no—
290C2×C4—2142no—
291C2×C4—2142no—
292C2×C4—2141no—
293C2×C4D4×D10712yes—
294C2×C4#1313381714yescomplemented
295C22—24no—
296C22—24no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—4282no—
308C22—4282no—
309C22—2142no—
310C22—2142no—
311C22—2142no—
312C22C2×D4×D10712yes—
313C22C2×D4×D10711yescharacteristic
314C22#41×D10711yescharacteristic
315C22C2×#1313381711yescharacteristic
316C22C2×#1313381712yes—
317C4—24no—
318C4—22no—
319C4—41no—
320C4—1074no—
321C4—4282no—
322C4—4281no—
323C4—2142no—
324C4—2144no—
325C2—44no—
326C2—42no—
327C2—4282no—
328C2#120×D10711yescharacteristic
329C2#2029313312yes—
330C2#2029313711yescharacteristic
331C2#2029317212yes—
332C2#2029317511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2388
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×D4×Dic107C2semidirect
3. C2×(C22⋊D107)⋊C4C2semidirect
4. #20292548C2semidirect
5. #20292543C2semidirect
6. #20292541C2semidirect
7. #20292539C2semidirect
8. #20292535C2semidirect
9. C22⋊C4×Dic107C2semidirect
10. #20292474C2semidirect
11. #392×C107C2semidirect
53. C22×Dic107D4semidirect
60. C2×C4×D107D4semidirect
93. C23×C107#41semidirect
108. C2×C4×C107#41semidirect
125. C2×Dic107#41semidirect
167. C107#2388semidirect
176. #392D107semidirect
265. C23#13133817semidirect
294. C2×C4#13133817semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×D4×Dic1072
C2×(C22⋊D107)⋊C42
#202925482
#202925432
#202925412
#202925392
#202925352
C22⋊C4×Dic1072
#202924742
#392×C1072
#2388107