Finite Group Databasebeta

← #28340064

Subgroups — #28340064

Order 13696.

914 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16812091C211yesmaximal, characteristic, complemented
3C2×#16812080C212yesmaximal, complemented
4#20292548C211yesmaximal, characteristic, complemented
5#20292541C211yesmaximal, characteristic, complemented
6#20292539C211yesmaximal, characteristic, complemented
7#20292507C212yesmaximal, complemented
8#20292503C212yesmaximal, complemented
9#20292502C212yesmaximal, complemented
10#95×D107C211yesmaximal, characteristic, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—28no—
13C2×#13133817—24no—
14C22×Dic214C2211yescharacteristic
15C22×C22⋊D107C2211yescharacteristic
16C22×C22⋊D107C2212yes—
17C23×Dic107C2212yes—
18C22×C4×D107C2212yes—
19C22×C4×D107C2211yescharacteristic
20#16812091—24no—
21#16812080—28no—
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2212yes—
24C2×Dic107⋊C4C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×C22⋊Dic107C2212yes—
27C2×C22⋊Dic107C2212yes—
28C2×C22⋊Dic107C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2211yescharacteristic
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C22×D4×C107C2212yes—
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2212yes—
38#95×C107C2211yescharacteristic
39C23×D107C2311yescharacteristic
40#13133817—232no—
41C2×Dic214—22no—
42C2×Dic214—24no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107D418yescomplemented
48C22×Dic107—24no—
49C22×Dic107—24no—
50C22×Dic107—24no—
51C22×Dic107C2311yescharacteristic
52C22×Dic107C2312yes—
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2311yescharacteristic
55C22×Dic107C2312yes—
56C2×C4×D107—24no—
57C2×C4×D107—24no—
58C2×C4×D107—24no—
59C2×C4×D107—42no—
60C2×C4×D107D414yescomplemented
61C4⋊Dic107—24no—
62C4⋊Dic107—22no—
63Dic107⋊C4—22no—
64C22⋊Dic107—24no—
65C22⋊Dic107—24no—
66C22⋊Dic107—22no—
67C22⋊Dic107—42no—
68(C2×D107)⋊C4—22no—
69C24×C107C2311yescharacteristic
70C24×C107C2312yes—
71C2×D4×C107—24no—
72C2×D4×C107—24no—
73C2×D4×C107—22no—
74C2×D4×C107—24no—
75C2×D4×C107—42no—
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2311yescharacteristic
78C22×C4×C107C2312yes—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—24no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—21no—
88C23×C107—22no—
89C23×C107—22no—
90C23×C107—21no—
91C23×C107—21no—
92C23×C107C2×D414yes—
93C23×C107C2411yescharacteristic
94C22×D107—22no—
95C22×D107—21no—
96C22×D107—21no—
97C22×D107C2×D412yes—
98C22×D107C2×D414yes—
99D4×C107—28no—
100D4×C107—48no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—21no—
105C2×C4×C107—22no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107#4114yescomplemented
108C22⋊D107—216no—
109C22⋊D107—44no—
110C2×Dic107—216no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107—21no—
120C2×Dic107—22no—
121C2×Dic107—21no—
122C2×Dic107C2×D412yes—
123C2×Dic107C2×D414yes—
124C4×D107—28no—
125C4×D107—48no—
126Dic214—28no—
127C22×C107—24no—
128C22×C107—24no—
129C22×C107—24no—
130C22×C107—42no—
131C22×C107—42no—
132C22×C107—42no—
133C22×C107—42no—
134C22×C107—41no—
135C22×C107—41no—
136C22×C107—41no—
137C22×C107—42no—
138C22×C107—42no—
139C22×C107C22×D411yescharacteristic
140C22×C107C22×D412yes—
141C22×C107C2×#4112yes—
142C22×C107H(5,𝔽2)11yescharacteristic
143C22×C1072−1+411yescharacteristic
144C2×D107—22no—
145C2×D107—22no—
146C2×D107—24no—
147C2×D107—24no—
148C2×D107#12014yes—
149C4×C107—22no—
150C4×C107—24no—
151C4×C107—41no—
152Dic107—24no—
153Dic107—22no—
154Dic107—41no—
155Dic107—41no—
156Dic107—42no—
157C2×C107—44no—
158C2×C107—81no—
159C2×C107C2×#12011yescharacteristic
160C2×C107#53211yescharacteristic
161C2×C107#53711yescharacteristic
162C2×C107#54612yes—
163C2×C107#54712yes—
164D107—24no—
165#2430—1071nomaximal, Sylow
166C107#243011yescharacteristic, complemented, Sylow
167C22×#41—1071no—
168C2×#123—1071no—
169C2×#121—1072no—
170#395—1071no—
171#394—1071no—
172#393—1071no—
173#392D10711yescharacteristic, complemented
174#386—1072no—
175#386—1072no—
176#385—1072no—
177C2×#95—1071no—
178C2×#41—2144no—
179C2×#41—2148no—
180C22×Q8—1071no—
181C22×D4—1072no—
182C22×D4—1071no—
183C22×D4C2×D10711yescharacteristic
184C22×D4C2×D10712yes—
185C23×C4—1071no—
186C23×C4—1072no—
187C23×C4—1072no—
188#123—2144no—
189#121—2148no—
190C2×C4⋊C4—1072no—
191C2×C4⋊C4—1071no—
192C2×C4⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1071no—
197C2×C22⋊C4—1071no—
198C2×C22⋊C4C2×D10711yescharacteristic
199C2×C22⋊C4C2×D10712yes—
200#95—1072no—
201#95—1071no—
202#95—1072no—
203#95—1072no—
204#95C2×D10711yescharacteristic
205C24—1071no—
206C24C22×D10711yescharacteristic
207C24C22×D10712yes—
208#41—21432no—
209C2×Q8—2144no—
210C2×Q8—2142no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—42no—
216C2×D4—1078no—
217C2×D4—2144no—
218C2×D4—2142no—
219C2×D4—2144no—
220C2×D4—2144no—
221C22×C4—1071no—
222C22×C4—1072no—
223C22×C4—1074no—
224C22×C4—1072no—
225C22×C4—1071no—
226C22×C4—1071no—
227C22×C4—4282no—
228C22×C4—2144no—
229C22×C4—2144no—
230C22×C4—2144no—
231C22×C4—2144no—
232C22×C4—2144no—
233C22×C4—2144no—
234C22×C4C22×D10711yescharacteristic
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10712yes—
237C4⋊C4—2144no—
238C4⋊C4—2142no—
239C4⋊C4—2142no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—4282no—
243C22⋊C4—2142no—
244C22⋊C4—2144no—
245C22⋊C4—2142no—
246C22⋊C4—2144no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—1072no—
259C23—1074no—
260C23—2142no—
261C23—2141no—
262C23—2141no—
263C23D4×D10714yes—
264C23C23×D10711yescharacteristic
265Q8—2148no—
266D4—28no—
267D4—48no—
268D4—4284no—
269D4—21416no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—1072no—
276C2×C4—1074no—
277C2×C4—4288no—
278C2×C4—2148no—
279C2×C4—2141no—
280C2×C4—2142no—
281C2×C4—2142no—
282C2×C4—2141no—
283C2×C4—2142no—
284C2×C4—2141no—
285C2×C4—2142no—
286C2×C4—2141no—
287C2×C4—21416no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4—2142no—
291C2×C4D4×D10712yes—
292C2×C4#1313381714yescomplemented
293C22—24no—
294C22—24no—
295C22—24no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—1074no—
306C22—2142no—
307C22—2142no—
308C22—2144no—
309C22—2144no—
310C22C2×D4×D10711yescharacteristic
311C22C2×D4×D10712yes—
312C22C2×#1313381712yes—
313C22#1681215411yescharacteristic
314C22#1681215611yescharacteristic
315C4—24no—
316C4—22no—
317C4—41no—
318C4—4281no—
319C4—4282no—
320C4—4281no—
321C4—2142no—
322C4—2144no—
323C2—44no—
324C2—81no—
325C2—2144no—
326C2#120×D10711yescharacteristic
327C2#2029317511yescharacteristic
328C2#2029320611yescharacteristic
329C2#2029326412yes—
330C2#2029327712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2430
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×#16812091C2semidirect
3. C2×#16812080C2semidirect
4. #20292548C2semidirect
5. #20292541C2semidirect
6. #20292539C2semidirect
7. #20292507C2semidirect
8. #20292503C2semidirect
9. #20292502C2semidirect
10. #95×D107C2semidirect
11. #392×C107C2semidirect
47. C2×C22⋊D107D4semidirect
60. C2×C4×D107D4semidirect
107. C2×C4×C107#41semidirect
166. C107#2430semidirect
173. #392D107semidirect
292. 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×#168120912
C2×#168120802
#202925482
#202925412
#202925392
#202925072
#202925032
#202925022
#95×D1072
#392×C1072
#2430107