Finite Group Databasebeta

← #28340055

Subgroups — #28340055

Order 13696.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812108C211yesmaximal, characteristic, complemented
2C2×#16812087C211yesmaximal, characteristic
3C2×#16812087C211yesmaximal, characteristic
4C2×#16812074C211yesmaximal, characteristic, complemented
5#20292561C211yesmaximal, characteristic, complemented
6#20292551C211yesmaximal, characteristic
7#20292549C211yesmaximal, characteristic
8#20292531C211yesmaximal, characteristic, complemented
9#20292524C211yesmaximal, characteristic, complemented
10#20292523C211yesmaximal, characteristic, complemented
11#20292522C211yesmaximal, characteristic, complemented
12#20292519C211yesmaximal, characteristic
13#20292517C211yesmaximal, characteristic
14#20292474C211yesmaximal, characteristic
15#398×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19#16812108—24no—
20#16812087—24no—
21#16812087—24no—
22#16812074—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C2×C4×Dic107C2211yescharacteristic
42C2×C4×Dic107C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C4⋊C4×C107C2211yescharacteristic
50C2×C4⋊C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54#95×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C2×Dic214—22no—
57C2×Dic214—22no—
58C2×Dic214—22no—
59C2×Dic214—22no—
60C2×Dic214—22no—
61C2×Dic214D414yes—
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C2×C4×D107—22no—
72C4⋊Dic107—22no—
73C4⋊Dic107—22no—
74Dic107⋊C4—22no—
75Dic107⋊C4—22no—
76Dic107⋊C4—22no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4D414yes—
82(C2×D107)⋊C4—22no—
83(C2×D107)⋊C4—22no—
84(C2×D107)⋊C4—22no—
85(C2×D107)⋊C4—22no—
86(C2×D107)⋊C4—22no—
87(C2×D107)⋊C4Q814yescomplemented
88C4×Dic107—22no—
89C4×Dic107—22no—
90C4×Dic107—22no—
91C4×Dic107—22no—
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96C22×C4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C4⋊C4×C107—22no—
100C4⋊C4×C107—22no—
101C4⋊C4×C107—22no—
102C4⋊C4×C107—22no—
103C23×C107C2411yescharacteristic
104C22×D107—21no—
105C22×D107—21no—
106C22×D107—21no—
107C22×D107—21no—
108C22×D107—21no—
109C22×D107—21no—
110C22×D107C2×Q812yes—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107C2×D412yes—
130C2×C4×C107C2×D412yes—
131C2×C4×C107C2×Q812yes—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107C2×D412yes—
147C2×Dic107C2×D412yes—
148C2×Dic107C2×D412yes—
149C2×Dic107C2×D412yes—
150C2×Dic107C2×Q812yes—
151C2×Dic107#4112yes—
152C2×Dic107#4112yes—
153C4×D107—44no—
154Dic214—28no—
155Dic214—44no—
156C22×C107C22×D411yescharacteristic
157C22×C107C22×D411yescharacteristic
158C22×C107C22×Q811yescharacteristic
159C22×C107C2×#4111yescharacteristic
160C22×C107C2×#4111yescharacteristic
161C22×C1072−1+411yescharacteristic
162C22×C1072−1+411yescharacteristic
163C2×D107—22no—
164C2×D107—22no—
165C2×D107—22no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C4×C107—22no—
171C4×C107—22no—
172C4×C107—22no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107—41no—
176C4×C107—41no—
177Dic107—22no—
178Dic107—22no—
179Dic107—22no—
180Dic107—22no—
181Dic107—41no—
182Dic107—41no—
183Dic107#12114yes—
184C2×C107C2×#12111yescharacteristic
185C2×C107#53311yescharacteristic
186C2×C107#54111yescharacteristic
187C2×C107#54711yescharacteristic
188C2×C107#54811yescharacteristic
189C2×C107D4×Q811yescharacteristic
190C2×C107D4×Q811yescharacteristic
191D107—42no—
192#2429—1071nomaximal, Sylow
193C107#242911yescharacteristic, complemented, Sylow
194C2×C4⋊Q8—1071no—
195C2×C22⋊Q8—1071no—
196C2×C22⋊Q8—1071no—
197C2×C4×Q8—1071no—
198#398—1071no—
199#398D10711yescharacteristic, complemented
200#396—1071no—
201#396—1071no—
202#391—1071no—
203#389—1071no—
204#389—1071no—
205#389—1071no—
206#388—1071no—
207#388—1071no—
208C4×C22⋊C4—1071no—
209C22×Q8—1071no—
210C22×Q8—1071no—
211C23×C4—1071no—
212C4⋊Q8—2144no—
213C22⋊Q8—2144no—
214C22⋊Q8—2144no—
215C4×Q8—2144no—
216C2×C4⋊C4—1071no—
217C2×C4⋊C4—1071no—
218C2×C4⋊C4—1071no—
219C2×C4⋊C4—1071no—
220C2×C4⋊C4—1071no—
221C2×C4⋊C4—1071no—
222C2×C4⋊C4—1071no—
223C2×C4⋊C4—1071no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1071no—
226C2×C4⋊C4C2×D10711yescharacteristic
227C2×C4⋊C4C2×D10711yescharacteristic
228C2×C4⋊C4C2×D10711yescharacteristic
229C2×C4⋊C4C2×D10711yescharacteristic
230C2×C22⋊C4—1071no—
231C2×C22⋊C4—1071no—
232C2×C22⋊C4—1071no—
233C2×C22⋊C4—1071no—
234C2×C22⋊C4—1071no—
235C2×C22⋊C4—1071no—
236C2×C42—1071no—
237C2×C42—1071no—
238C2×C42—1071no—
239C2×C42—1071no—
240#95—1071no—
241#95—1071no—
242#95—1071no—
243#95—1071no—
244#95—1071no—
245#95C2×D10711yescharacteristic
246#95C2×D10711yescharacteristic
247#95C2×D10711yescharacteristic
248C24—1071no—
249C2×Q8—1074no—
250C2×Q8—2142no—
251C2×Q8—2142no—
252C2×Q8—2142no—
253C2×Q8—2142no—
254C2×Q8—2142no—
255C22×C4—1071no—
256C22×C4—1071no—
257C22×C4—1071no—
258C22×C4—1071no—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—1071no—
262C22×C4—2142no—
263C22×C4—2142no—
264C22×C4—2142no—
265C22×C4C22×D10711yescharacteristic
266C22×C4C22×D10711yescharacteristic
267C22×C4C22×D10711yescharacteristic
268C22×C4C22×D10711yescharacteristic
269C22×C4C22×D10711yescharacteristic
270C22×C4C22×D10711yescharacteristic
271C22×C4C22×D10711yescharacteristic
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—1074no—
277C4⋊C4—2142no—
278C4⋊C4—2142no—
279C4⋊C4—2142no—
280C4⋊C4—2142no—
281C4⋊C4—2142no—
282C4⋊C4—2142no—
283C4⋊C4—2142no—
284C4⋊C4—2142no—
285C4⋊C4—2142no—
286C22⋊C4—1074no—
287C22⋊C4—2142no—
288C22⋊C4—2142no—
289C22⋊C4—2142no—
290C22⋊C4—2142no—
291C22⋊C4—2142no—
292C42—2142no—
293C42—2142no—
294C42—2142no—
295C42—2142no—
296C23—1072no—
297C23—2141no—
298C23—2141no—
299C23—2141no—
300C23—2141no—
301C23—2141no—
302C23—2141no—
303C23C23×D10711yescharacteristic
304Q8—4284no—
305Q8—2148no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—1072no—
325C2×C4—1072no—
326C2×C4—1072no—
327C2×C4—1072no—
328C2×C4—1072no—
329C2×C4—1072no—
330C2×C4—1072no—
331C2×C4—4284no—
332C2×C4—2141no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4D4×D10712yes—
347C2×C4D4×D10712yes—
348C2×C4Q8×D10712yes—
349C22—4281no—
350C22—4281no—
351C22—4281no—
352C22—4281no—
353C22—2142no—
354C22—2142no—
355C22—2142no—
356C22C2×D4×D10711yescharacteristic
357C22C2×D4×D10711yescharacteristic
358C22C2×Q8×D10711yescharacteristic
359C22#41×D10711yescharacteristic
360C22#41×D10711yescharacteristic
361C22#1681215611yescharacteristic
362C22#1681215711yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—1074no—
371C4—4281no—
372C4—4281no—
373C4—2142no—
374C4—2142no—
375C4—2142no—
376C4—2142no—
377C2—4282no—
378C2#121×D10711yescharacteristic
379C2#2029318311yescharacteristic
380C2#2029323611yescharacteristic
381C2#2029327511yescharacteristic
382C2#2029328711yescharacteristic
383C2#2029329011yescharacteristic
384C2#2029329211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC23×D107C23
hypercenter#398D107
hypocenterC107#2429
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16812108C2semidirect
4. C2×#16812074C2semidirect
5. #20292561C2semidirect
8. #20292531C2semidirect
9. #20292524C2semidirect
10. #20292523C2semidirect
11. #20292522C2semidirect
15. #398×C107C2semidirect
87. (C2×D107)⋊C4Q8semidirect
193. C107#2429semidirect
199. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121082
C2×#168120872
C2×#168120742
#202925612
#202925512
#202925492
#202925312
#202925242
#202925232
#202925222
#202925192
#202925172
#202924742
#398×C1072
#2429107