Finite Group Databasebeta

← #28339675

Subgroups — #28339675

Order 13696.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16812071C211yesmaximal, characteristic, complemented
3C2×(C22⋊D107)⋊C4C212yesmaximal, complemented
4#20292547C211yesmaximal, characteristic, complemented
5#20292542C212yesmaximal, complemented
6#20292540C211yesmaximal, characteristic, complemented
7#20292536C212yesmaximal, complemented
8#20292535C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic107C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—28no—
13C2×#13133817—24no—
14C22×Dic214C2211yescharacteristic
15C22×D428C2211yescharacteristic
16C22×C22⋊D107C2212yes—
17C22×C22⋊D107C2212yes—
18C23×Dic107C2212yes—
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21#16812071—24no—
22(C22⋊D107)⋊C4—28no—
23C2×Dic107⋊C4C2212yes—
24C2×C22⋊Dic107C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2211yescharacteristic
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2212yes—
38#95×C107C2211yescharacteristic
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41#13133817—232no—
42C2×Dic214—22no—
43C2×Dic214—24no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428—22no—
47C2×C22⋊D107—24no—
48C2×C22⋊D107—24no—
49C2×C22⋊D107—24no—
50C2×C22⋊D107—24no—
51C2×C22⋊D107—24no—
52C2×C22⋊D107—24no—
53C22×Dic107—24no—
54C22×Dic107—24no—
55C22×Dic107D418yescomplemented
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2312yes—
58C22×Dic107C2312yes—
59C22×Dic107C2311yescharacteristic
60C2×C4×D107—24no—
61C2×C4×D107—24no—
62C2×C4×D107—22no—
63C2×C4×D107D414yescomplemented
64Dic107⋊C4—24no—
65C22⋊Dic107—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—24no—
69(C2×D107)⋊C4—24no—
70C4×Dic107—24no—
71C4×Dic107—22no—
72C24×C107C2312yes—
73C2×D4×C107—24no—
74C2×D4×C107—22no—
75C22×C4×C107C2312yes—
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2311yescharacteristic
79C4⋊C4×C107—22no—
80C22⋊C4×C107—24no—
81C22⋊C4×C107—24no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—22no—
88C23×C107C2×D414yes—
89C23×C107C2411yescharacteristic
90C22×D107—22no—
91C22×D107—21no—
92C22×D107—22no—
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—21no—
96C22×D107—21no—
97C22×D107—21no—
98C22×D107C2×D412yes—
99C22×D107#4112yescomplemented
100D4×C107—28no—
101C2×C4×C107—21no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—22no—
105C2×C4×C107—22no—
106C2×C4×C107—21no—
107C2×C4×C107—22no—
108C2×C4×C107—22no—
109C2×C4×C107C2×D412yes—
110C2×C4×C107#4112yescomplemented
111C22⋊D107—216no—
112C22⋊D107—216no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—216no—
118C2×Dic107—21no—
119C2×Dic107—22no—
120C2×Dic107C2×D414yes—
121C2×Dic107C2×D414yes—
122C2×Dic107C2×D412yes—
123C2×Dic107#4114yescomplemented
124D428—28no—
125C4×D107—28no—
126C4×D107—44no—
127Dic214—28no—
128C22×C107—24no—
129C22×C107—24no—
130C22×C107—24no—
131C22×C107—42no—
132C22×C107—42no—
133C22×C107—42no—
134C22×C107—42no—
135C22×C107C22×D411yescharacteristic
136C22×C107C22×D412yes—
137C22×C107C2×#4111yescharacteristic
138C22×C107C2×#4112yes—
139C22×C107C2×#4111yescharacteristic
140C2×D107—22no—
141C2×D107—22no—
142C2×D107—22no—
143C2×D107—24no—
144C2×D107—22no—
145C2×D107—41no—
146C2×D107—42no—
147C2×D107—42no—
148C2×D107—42no—
149C2×D107—41no—
150C4×C107—22no—
151C4×C107—22no—
152C4×C107—41no—
153C4×C107—42no—
154Dic107—24no—
155Dic107—24no—
156Dic107—41no—
157Dic107#12014yes—
158C2×C107—44no—
159C2×C107C2×#12011yescharacteristic
160C2×C107#52512yes—
161C2×C107#52511yescharacteristic
162C2×C107#53212yes—
163C2×C107#53211yescharacteristic
164D107—42no—
165D107—42no—
166#2388—1071nomaximal, Sylow
167C107#238811yescharacteristic, complemented, Sylow
168C22×#41—1071no—
169C2×C4×D4—1071no—
170C2×C4×D4—1072no—
171#395—1071no—
172#394—1072no—
173#394D10711yescharacteristic, complemented
174#393—1071no—
175#392—1071no—
176#392—1072no—
177C4×C22⋊C4—1072no—
178C4×C22⋊C4—1071no—
179C2×#41—2148no—
180C2×#41—2144no—
181C22×Q8—1071no—
182C22×D4—1072no—
183C22×D4—1072no—
184C22×D4—1071no—
185C22×D4C2×D10711yescharacteristic
186C23×C4—1072no—
187C23×C4—1071no—
188C23×C4—1071no—
189C4×D4—2148no—
190C4×D4—2144no—
191C2×C4⋊C4—1072no—
192C2×C4⋊C4C2×D10711yescharacteristic
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1071no—
196C2×C22⋊C4—1071no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4C2×D10712yes—
199C2×C22⋊C4C2×D10712yes—
200C2×C42—1072no—
201C2×C42—1071no—
202#95—1071no—
203#95—1072no—
204#95—1072no—
205#95C2×D10711yescharacteristic
206C24—1071no—
207C24—1071no—
208C24C22×D10712yes—
209#41—21432no—
210C2×Q8—2144no—
211C2×Q8—2142no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—2142no—
215C2×D4—2142no—
216C2×D4—2144no—
217C2×D4—2144no—
218C2×D4—2144no—
219C2×D4—2144no—
220C2×D4—2144no—
221C2×D4—2144no—
222C2×D4—2142no—
223C22×C4—1071no—
224C22×C4—1078no—
225C22×C4—1072no—
226C22×C4—1072no—
227C22×C4—1074no—
228C22×C4—1071no—
229C22×C4—2144no—
230C22×C4—2142no—
231C22×C4—2144no—
232C22×C4—2144no—
233C22×C4—2144no—
234C22×C4C22×D10712yes—
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10711yescharacteristic
237C22×C4C22×D10711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2144no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—2144no—
243C22⋊C4—2144no—
244C22⋊C4—2142no—
245C22⋊C4—2144no—
246C22⋊C4—2142no—
247C42—2142no—
248C42—2144no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—1072no—
256C23—1072no—
257C23—2142no—
258C23—2141no—
259C23—2142no—
260C23—2142no—
261C23—2141no—
262C23—2142no—
263C23—2141no—
264C23—2141no—
265C23D4×D10714yes—
266C23C23×D10711yescharacteristic
267Q8—2148no—
268D4—28no—
269D4—2148no—
270D4—21416no—
271D4—21416no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—1074no—
281C2×C4—1072no—
282C2×C4—1074no—
283C2×C4—1074no—
284C2×C4—4284no—
285C2×C4—2148no—
286C2×C4—21416no—
287C2×C4—2141no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4—2142no—
291C2×C4—2142no—
292C2×C4—2142no—
293C2×C4D4×D10712yes—
294C2×C4#1313381712yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—4282no—
303C22—4282no—
304C22—4282no—
305C22—4281no—
306C22—4281no—
307C22—2144no—
308C22—2142no—
309C22—2142no—
310C22—2142no—
311C22—2142no—
312C22C2×D4×D10712yes—
313C22C2×D4×D10711yescharacteristic
314C22#41×D10711yescharacteristic
315C22#41×D10712yes—
316C22C2×#1313381711yescharacteristic
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—1074no—
322C4—4281no—
323C4—2144no—
324C4—2144no—
325C2—44no—
326C2—4282no—
327C2—4282no—
328C2#120×D10711yescharacteristic
329C2#2029313512yes—
330C2#2029313711yescharacteristic
331C2#2029317212yes—
332C2#2029317611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
hypercenter#394D107
hypocenterC107#2388
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812071C2semidirect
3. C2×(C22⋊D107)⋊C4C2semidirect
4. #20292547C2semidirect
5. #20292542C2semidirect
6. #20292540C2semidirect
7. #20292536C2semidirect
8. #20292535C2semidirect
9. C22⋊C4×Dic107C2semidirect
10. #20292474C2semidirect
11. #394×C107C2semidirect
55. C22×Dic107D4semidirect
63. C2×C4×D107D4semidirect
99. C22×D107#41semidirect
110. C2×C4×C107#41semidirect
123. C2×Dic107#41semidirect
167. C107#2388semidirect
173. #394D107semidirect
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×#168120712
C2×(C22⋊D107)⋊C42
#202925472
#202925422
#202925402
#202925362
#202925352
C22⋊C4×Dic1072
#202924742
#394×C1072
#2388107