Finite Group Databasebeta

← #28339859

Subgroups — #28339859

Order 13696.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16812092C211yesmaximal, characteristic, complemented
2C2×#16812079C211yesmaximal, characteristic
3C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292556C212yesmaximal, complemented
6#20292545C212yesmaximal, complemented
7#20292505C212yesmaximal, complemented
8#20292504C211yesmaximal, characteristic, complemented
9#20292499C211yesmaximal, characteristic, complemented
10#20292486C212yesmaximal, complemented
11#381×C107C211yesmaximal, characteristic, complemented
12C24×D107C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×C22⋊D107C2211yescharacteristic
16C23×Dic107C2211yescharacteristic
17C22×C4×D107C2212yes—
18#16812092—24no—
19#16812079—24no—
20C2×Dic107⋊C4C2212yescomplemented
21C2×Dic107⋊C4C2212yescomplemented
22C2×C22⋊Dic107C2211yescharacteristic
23C2×(C2×D107)⋊C4—24no—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2212yescomplemented
30C2×(C2×D107)⋊C4C2212yescomplemented
31C22⋊C4×D107—24no—
32C107⋊#95C2212yescomplemented
33C107⋊#95C2212yescomplemented
34C23×C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37#95×C107C2212yescomplemented
38#95×C107C2212yescomplemented
39C23×D107—22no—
40C23×D107—22no—
41C23×D107—24no—
42C23×D107—42no—
43C23×D107D414yes—
44C23×D107C2311yescharacteristic
45C23×D107C2311yescharacteristic
46C23×D107C2311yescharacteristic
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—24no—
50C2×D428—42no—
51C2×C22⋊D107—22no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—22no—
54C22×Dic107—22no—
55C22×Dic107—42no—
56C22×Dic107C2312yes—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2312yes—
59C2×C4×D107—24no—
60C2×C4×D107—24no—
61C2×C4×D107—42no—
62Dic107⋊C4—24no—
63Dic107⋊C4—42no—
64C22⋊Dic107D414yescomplemented
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—24no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—24no—
70(C2×D107)⋊C4—24no—
71(C2×D107)⋊C4—42no—
72C24×C107C2311yescharacteristic
73C22×C4×C107—22no—
74C22×C4×C107—22no—
75C22×C4×C107—22no—
76C22×C4×C107C2312yes—
77C22×C4×C107C2312yes—
78C22×C4×C107C2311yescharacteristic
79C22×C4×C107C2311yescharacteristic
80C22⋊C4×C107—22no—
81C22⋊C4×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107C2×D412yes—
85C23×C107C2×D412yes—
86C23×C107#4112yes—
87C23×C107C2411yescharacteristic
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—22no—
91C22×D107—28no—
92C22×D107—21no—
93C22×D107—22no—
94C22×D107—21no—
95C22×D107—22no—
96C22×D107—28no—
97C22×D107—21no—
98C22×D107—21no—
99C22×D107—21no—
100C22×D107—22no—
101C22×D107—22no—
102C22×D107—44no—
103C22×D107—48no—
104C22×D107—48no—
105C22×D107C2×D412yes—
106C22×D107C2×D412yes—
107C22×D107#4114yes—
108C2×C4×C107—22no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—22no—
112C2×C4×C107—22no—
113C2×C4×C107—22no—
114C2×C4×C107—22no—
115C2×C4×C107—22no—
116C2×C4×C107—22no—
117C2×C4×C107—22no—
118C2×C4×C107—44no—
119C22⋊D107—44no—
120C2×Dic107—22no—
121C2×Dic107—22no—
122C2×Dic107—22no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—22no—
127C2×Dic107—22no—
128C2×Dic107—44no—
129C2×Dic107C2×D414yes—
130D428—44no—
131D428—44no—
132C4×D107—48no—
133C22×C107—22no—
134C22×C107—22no—
135C22×C107—24no—
136C22×C107—24no—
137C22×C107#12114yes—
138C22×C107C22×D411yescharacteristic
139C22×C107C22×D411yescharacteristic
140C22×C107C2×#4112yes—
141C22×C107C2×#4111yescharacteristic
142C22×C107H(5,𝔽2)12yes—
143C2×D107—22no—
144C2×D107—22no—
145C2×D107—24no—
146C2×D107—24no—
147C2×D107—22no—
148C2×D107—22no—
149C2×D107—22no—
150C2×D107—416no—
151C2×D107—42no—
152C2×D107—42no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—42no—
157C2×D107—42no—
158C2×D107#12314yescomplemented
159C4×C107—42no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—42no—
163Dic107—24no—
164Dic107—41no—
165Dic107—42no—
166C2×C107—24no—
167C2×C107C2×#12111yescharacteristic
168C2×C107C2×#12311yescharacteristic
169C2×C107#53411yescharacteristic
170C2×C107#54612yes—
171C2×C107#55112yes—
172D107—24no—
173D107—42no—
174D107—81no—
175#2412—1071nomaximal, Sylow
176C107#241211yescharacteristic, complemented, Sylow
177C2×#123—1071no—
178C2×#120—1071no—
179C22×C22⋊C4—1071no—
180C22×C22⋊C4—1071no—
181#397—1072no—
182#394—1072no—
183#386—1071no—
184#386—1072no—
185#385—1071no—
186#381D10711yescharacteristic, complemented
187#380—1072no—
188C25—1071no—
189C22×D4—1071no—
190C22×D4—1071no—
191C22×D4—1071no—
192C23×C4—1071no—
193C23×C4—1072no—
194C23×C4C2×D10711yescharacteristic
195#123—2144no—
196#120—2144no—
197C2×C4⋊C4—1072no—
198C2×C4⋊C4—1072no—
199C2×C22⋊C4—1071no—
200C2×C22⋊C4—1072no—
201C2×C22⋊C4—1072no—
202C2×C22⋊C4—1072no—
203C2×C22⋊C4—1072no—
204C2×C22⋊C4—1071no—
205C2×C22⋊C4—1071no—
206C2×C22⋊C4—1072no—
207C2×C22⋊C4—2144no—
208C2×C22⋊C4—2144no—
209C2×C22⋊C4C2×D10711yescharacteristic
210C2×C22⋊C4C2×D10711yescharacteristic
211#95—1072no—
212#95—1072no—
213#95C2×D10712yescomplemented
214#95C2×D10712yescomplemented
215C24—1071no—
216C24—1074no—
217C24—1071no—
218C24—1071no—
219C24—4282no—
220C24—2144no—
221C24—2142no—
222C24—2142no—
223C24C22×D10711yescharacteristic
224C2×D4—4282no—
225C2×D4—2142no—
226C2×D4—2142no—
227C2×D4—2144no—
228C2×D4—2142no—
229C2×D4—2142no—
230C2×D4—2142no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—1072no—
235C22×C4—1072no—
236C22×C4—1071no—
237C22×C4—4282no—
238C22×C4—4282no—
239C22×C4—2144no—
240C22×C4—2144no—
241C22×C4—2142no—
242C22×C4C22×D10711yescharacteristic
243C22×C4C22×D10712yes—
244C22×C4C22×D10712yes—
245C22×C4C22×D10711yescharacteristic
246C4⋊C4—4282no—
247C4⋊C4—2144no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—1074no—
251C22⋊C4—4282no—
252C22⋊C4—2144no—
253C22⋊C4—2144no—
254C22⋊C4—2144no—
255C22⋊C4—2144no—
256C22⋊C4—2142no—
257C22⋊C4—2142no—
258C23—22no—
259C23—22no—
260C23—1072no—
261C23—1074no—
262C23—1072no—
263C23—4288no—
264C23—4288no—
265C23—4284no—
266C23—2148no—
267C23—2141no—
268C23—2141no—
269C23—2142no—
270C23—2141no—
271C23—2142no—
272C23—2148no—
273C23—2141no—
274C23—2142no—
275C23—2141no—
276C23—2142no—
277C23—2141no—
278C23—2141no—
279C23—2142no—
280C23D4×D10712yes—
281C23D4×D10712yes—
282C23#1313381812yes—
283C23C23×D10711yescharacteristic
284D4—4284no—
285D4—4284no—
286D4—4284no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—1074no—
299C2×C4—4288no—
300C2×C4—4284no—
301C2×C4—2142no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2142no—
306C2×C4—2142no—
307C2×C4—2142no—
308C2×C4—2142no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—1074no—
314C22—42816no—
315C22—4281no—
316C22—4281no—
317C22—4282no—
318C22—4282no—
319C22—4281no—
320C22—4282no—
321C22—4282no—
322C22—2142no—
323C22—2142no—
324C22—2142no—
325C22—2142no—
326C22—2144no—
327C22—2144no—
328C22—2142no—
329C22#1681208214yes—
330C22C2×D4×D10711yescharacteristic
331C22C2×D4×D10711yescharacteristic
332C22#41×D10712yes—
333C22C2×#1313381811yescharacteristic
334C22#1681215512yes—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—4282no—
340C4—4281no—
341C4—2144no—
342C2—24no—
343C2—8561no—
344C2—4282no—
345C2—2144no—
346C2C2×#1681208211yescharacteristic
347C2#123×D10711yescharacteristic
348C2#2029318811yescharacteristic
349C2#2029326912yes—
350C2#2029330112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#381×C107C2
prime radicalC2×#16812079C2
hypercenter#381D107
hypocenterC107#2412
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#16812092C2semidirect
3. C22×(C2×D107)⋊C4C2semidirect
4. C2×C22⋊C4×D107C2semidirect
5. #20292556C2semidirect
6. #20292545C2semidirect
7. #20292505C2semidirect
8. #20292504C2semidirect
9. #20292499C2semidirect
10. #20292486C2semidirect
11. #381×C107C2semidirect
20. C2×Dic107⋊C4C22semidirect
21. C2×Dic107⋊C4C22semidirect
29. C2×(C2×D107)⋊C4C22semidirect
30. C2×(C2×D107)⋊C4C22semidirect
32. C107⋊#95C22semidirect
33. C107⋊#95C22semidirect
37. #95×C107C22semidirect
38. #95×C107C22semidirect
64. C22⋊Dic107D4semidirect
158. C2×D107#123semidirect
176. C107#2412semidirect
186. #381D107semidirect
213. #95C2×D107semidirect
214. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120922
C2×#168120792
C22×(C2×D107)⋊C42
C2×C22⋊C4×D1072
#202925562
#202925452
#202925052
#202925042
#202924992
#202924862
#381×C1072
#2412107