Finite Group Databasebeta

← #28340028

Subgroups — #28340028

Order 13696.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812096C211yesmaximal, characteristic, complemented
2C2×#16812089C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic
4C2×C4×Dic214C211yesmaximal, characteristic, complemented
5#20292550C212yesmaximal, complemented
6#20292542C212yesmaximal, complemented
7#20292527C212yesmaximal, complemented
8#20292523C211yesmaximal, characteristic, complemented
9#20292520C212yesmaximal, complemented
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
11#390×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C4×D107C2211yescharacteristic
15#16812096—24no—
16#16812089—24no—
17#16812080—24no—
18C4×Dic214—24no—
19C2×C4⋊Dic107C2212yes—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×Dic107⋊C4C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×C22⋊Dic107C2212yes—
25C2×C22⋊Dic107C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2211yescharacteristic
30C2×C4×Dic107C2212yes—
31C107⋊#95C2211yescharacteristic
32C107⋊#95C2211yescharacteristic
33C107⋊#95C2212yes—
34C22×D4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37C2×C22⋊C4×C107C2212yes—
38C2×C42×C107C2211yescharacteristic
39C23×D107C2311yescharacteristic
40C2×Dic214—22no—
41C2×Dic214—22no—
42C2×Dic214D414yescomplemented
43C2×C22⋊D107—24no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—42no—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2312yes—
50C22×Dic107C2312yes—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C4⋊Dic107—24no—
55C4⋊Dic107D414yes—
56Dic107⋊C4—24no—
57Dic107⋊C4—42no—
58C22⋊Dic107—24no—
59C22⋊Dic107—24no—
60C22⋊Dic107—24no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—24no—
64C4×Dic107—22no—
65C4×Dic107—24no—
66C24×C107C2312yes—
67C2×D4×C107—22no—
68C2×D4×C107—22no—
69C2×D4×C107—22no—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—22no—
75C22⋊C4×C107—24no—
76C22⋊C4×C107—42no—
77C42×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107C2411yescharacteristic
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107—21no—
92C22×D107#4112yes—
93D4×C107—44no—
94C2×C4×C107—21no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2×D412yes—
104C2×C4×C107#4112yes—
105C2×C4×C107#4112yes—
106C22⋊D107—48no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—21no—
113C2×Dic107—21no—
114C2×Dic107—21no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107C2×D414yes—
121C2×Dic107C2×D414yes—
122C4×D107—44no—
123Dic214—28no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107—42no—
131C22×C107C22×D411yescharacteristic
132C22×C107C22×D411yescharacteristic
133C22×C107C2×#4111yescharacteristic
134C22×C107C2×#4111yescharacteristic
135C22×C107C2×#4111yescharacteristic
136C22×C107H(5,𝔽2)11yescharacteristic
137C22×C1072−1+411yescharacteristic
138C2×D107—22no—
139C2×D107—22no—
140C2×D107—22no—
141C2×D107—41no—
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C4×C107—22no—
146C4×C107—41no—
147C4×C107—42no—
148C4×C107#12114yes—
149Dic107—24no—
150Dic107—24no—
151Dic107—41no—
152Dic107—41no—
153Dic107—41no—
154C2×C107—82no—
155C2×C107C2×#12111yescharacteristic
156C2×C107#52911yescharacteristic
157C2×C107#54211yescharacteristic
158C2×C107#54611yescharacteristic
159C2×C107#54811yescharacteristic
160C2×C107#54811yescharacteristic
161C2×C107#55011yescharacteristic
162D107—42no—
163#2426—1071nomaximal, Sylow
164C107#242611yescharacteristic, complemented, Sylow
165C2×#124—1071no—
166C2×C22⋊Q8—1071no—
167C2×#121—1071no—
168C2×C4×Q8—1071no—
169#396—1072no—
170#394—1072no—
171#390—1072no—
172#390D10711yescharacteristic, complemented
173#389—1071no—
174#388—1072no—
175C4×C22⋊C4—1071no—
176C22×Q8—1071no—
177C22×D4—1072no—
178C22×D4C2×D10711yescharacteristic
179C23×C4—1071no—
180#124—2144no—
181C22⋊Q8—2144no—
182#121—2144no—
183C4×Q8—2144no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1072no—
186C2×C4⋊C4—1072no—
187C2×C4⋊C4—1072no—
188C2×C4⋊C4C2×D10711yescharacteristic
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1072no—
192C2×C22⋊C4—1072no—
193C2×C22⋊C4—1071no—
194C2×C22⋊C4—1071no—
195C2×C22⋊C4C2×D10712yes—
196C2×C22⋊C4C2×D10712yes—
197C2×C42—1072no—
198C2×C42—1071no—
199C2×C42C2×D10711yescharacteristic
200#95—1071no—
201#95—1071no—
202#95—1072no—
203C24—1071no—
204C24C22×D10712yes—
205C2×Q8—1074no—
206C2×Q8—2142no—
207C2×Q8—2142no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—4282no—
212C2×D4—2144no—
213C2×D4—2144no—
214C22×C4—1072no—
215C22×C4—1072no—
216C22×C4—1071no—
217C22×C4—1071no—
218C22×C4—1071no—
219C22×C4—2142no—
220C22×C4—2142no—
221C22×C4—2142no—
222C22×C4C22×D10711yescharacteristic
223C22×C4C22×D10712yes—
224C22×C4C22×D10711yescharacteristic
225C22×C4C22×D10711yescharacteristic
226C4⋊C4—22no—
227C4⋊C4—1074no—
228C4⋊C4—4282no—
229C4⋊C4—2144no—
230C4⋊C4—2144no—
231C22⋊C4—24no—
232C22⋊C4—42no—
233C22⋊C4—2144no—
234C22⋊C4—2142no—
235C22⋊C4—2144no—
236C22⋊C4—2144no—
237C22⋊C4—2144no—
238C22⋊C4—2142no—
239C42—22no—
240C42—2144no—
241C42—2142no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—1072no—
250C23—2141no—
251C23—2141no—
252C23—2141no—
253C23—2141no—
254C23—2141no—
255C23—2141no—
256C23C23×D10711yescharacteristic
257Q8—2148no—
258D4—44no—
259D4—4288no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—1074no—
269C2×C4—1074no—
270C2×C4—4284no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2141no—
274C2×C4—2141no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2141no—
279C2×C4—2141no—
280C2×C4—2142no—
281C2×C4—2142no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4D4×D10712yes—
285C2×C4D4×D10712yes—
286C2×C4#1313381712yes—
287C2×C4#1313381812yes—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—4281no—
296C22—4281no—
297C22—4281no—
298C22—4281no—
299C22—2142no—
300C22—2142no—
301C22—2142no—
302C22C2×D4×D10711yescharacteristic
303C22C2×D4×D10711yescharacteristic
304C22#41×D10711yescharacteristic
305C22C2×#1313381711yescharacteristic
306C22C2×#1313381811yescharacteristic
307C22#1681215411yescharacteristic
308C22#1681215611yescharacteristic
309C4—22no—
310C4—41no—
311C4—42no—
312C4—4281no—
313C4—4281no—
314C4—4281no—
315C4—2144no—
316C4—2144no—
317C4#1681208214yes—
318C2—82no—
319C2—4282no—
320C2C2×#1681208211yescharacteristic
321C2#2029315911yescharacteristic
322C2#2029324211yescharacteristic
323C2#2029327411yescharacteristic
324C2#2029328311yescharacteristic
325C2#2029328511yescharacteristic
326C2#2029329711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
prime radicalC2×#16812080C2
hypercenter#390D107
hypocenterC107#2426
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16812096C2semidirect
2. C2×#16812089C2semidirect
4. C2×C4×Dic214C2semidirect
5. #20292550C2semidirect
6. #20292542C2semidirect
7. #20292527C2semidirect
8. #20292523C2semidirect
9. #20292520C2semidirect
10. C4×(C2×D107)⋊C4C2semidirect
11. #390×C107C2semidirect
42. C2×Dic214D4semidirect
164. C107#2426semidirect
172. #390D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120962
C2×#168120892
C2×#168120802
C2×C4×Dic2142
#202925502
#202925422
#202925272
#202925232
#202925202
C4×(C2×D107)⋊C42
#390×C1072
#2426107