Finite Group Databasebeta

← #28344200

Subgroups — #28344200

Order 13696.

416 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20292565C211yesmaximal, characteristic, complemented
2#20292565C211yesmaximal, characteristic, complemented
3#20292558C211yesmaximal, characteristic, complemented
4#20292547C211yesmaximal, characteristic, complemented
5#20292539C211yesmaximal, characteristic
6#20292531C211yesmaximal, characteristic, complemented
7#20292522C211yesmaximal, characteristic, complemented
8#20292522C211yesmaximal, characteristic, complemented
9#20292515C211yesmaximal, characteristic
10#20292515C211yesmaximal, characteristic
11#20292509C211yesmaximal, characteristic, complemented
12#20292499C211yesmaximal, characteristic
13#20292499C211yesmaximal, characteristic
14#20292485C211yesmaximal, characteristic
15#396×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2211yescharacteristic
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C4×Dic107C2211yescharacteristic
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2211yescharacteristic
32C107⋊#95C2211yescharacteristic
33C107⋊#95C2211yescharacteristic
34C107⋊#95C2211yescharacteristic
35C107⋊#95C2211yescharacteristic
36C107⋊#95C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C2×C4⋊C4×C107C2211yescharacteristic
45C2×C4⋊C4×C107C2211yescharacteristic
46C2×C4⋊C4×C107C2211yescharacteristic
47C2×C22⋊C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51C2×Dic214—22no—
52C2×Dic214—41no—
53C2×Dic214—41no—
54C22×Dic107—22no—
55C22×Dic107—41no—
56C22×Dic107—41no—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65Dic107⋊C4—22no—
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68Dic107⋊C4—41no—
69Dic107⋊C4—41no—
70Dic107⋊C4—41no—
71Dic107⋊C4—41no—
72Dic107⋊C4—41no—
73C22⋊Dic107—22no—
74C22⋊Dic107—41no—
75C22⋊Dic107Q814yescomplemented
76C4×Dic107—41no—
77C4×Dic107—41no—
78C24×C107C2311yescharacteristic
79C22×C4×C107C2311yescharacteristic
80C22×C4×C107C2311yescharacteristic
81C22×C4×C107C2311yescharacteristic
82C22×C4×C107C2311yescharacteristic
83C22×C4×C107C2311yescharacteristic
84C22×C4×C107C2311yescharacteristic
85C4⋊C4×C107—41no—
86C4⋊C4×C107—41no—
87C4⋊C4×C107—41no—
88C22⋊C4×C107—41no—
89C22⋊C4×C107—41no—
90C22⋊C4×C107—41no—
91C23×C107—21no—
92C23×C107—21no—
93C23×C107—21no—
94C23×C107—21no—
95C23×C107—21no—
96C23×C107—21no—
97C23×C107C2×Q812yes—
98C23×C107C2411yescharacteristic
99C2×C4×C107—21no—
100C2×C4×C107—21no—
101C2×C4×C107—21no—
102C2×C4×C107—21no—
103C2×C4×C107—21no—
104C2×C4×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
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×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—44no—
138C2×Dic107C2×Q812yes—
139C2×Dic107C2×Q812yes—
140C2×Dic107#4112yes—
141C2×Dic107#4112yes—
142Dic214—44no—
143C22×C107—22no—
144C22×C107—22no—
145C22×C107—22no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C22×Q811yescharacteristic
151C22×C107C2×#4111yescharacteristic
152C22×C107C2×#4111yescharacteristic
153C22×C107H(5,𝔽2)11yescharacteristic
154C22×C107H(5,𝔽2)11yescharacteristic
155C22×C107H(5,𝔽2)11yescharacteristic
156C22×C1072−1+411yescharacteristic
157C4×C107—41no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—41no—
163Dic107—22no—
164Dic107—22no—
165Dic107—22no—
166Dic107—22no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171C2×C107—42no—
172C2×C107#53811yescharacteristic
173C2×C107#53811yescharacteristic
174C2×C107#54411yescharacteristic
175C2×C107#55411yescharacteristic
176C2×C107#55411yescharacteristic
177C2×C107#55611yescharacteristic
178C2×C107#56211yescharacteristic
179#2788—1071nomaximal, Sylow
180C107#278811yescharacteristic, complemented, Sylow
181#400—1071no—
182#400—1071no—
183#398—1071no—
184#396D10711yescharacteristic
185#395—1071no—
186#393—1071no—
187#391—1071no—
188#389—1071no—
189#389—1071no—
190#388—1071no—
191#388—1071no—
192#387—1071no—
193#385—1071no—
194#385—1071no—
195#380—1071no—
196C22×Q8—1071no—
197C23×C4—1071no—
198C2×C4⋊C4—1071no—
199C2×C4⋊C4—1071no—
200C2×C4⋊C4—1071no—
201C2×C4⋊C4—1071no—
202C2×C4⋊C4—1071no—
203C2×C4⋊C4—1071no—
204C2×C4⋊C4—1071no—
205C2×C4⋊C4—1071no—
206C2×C4⋊C4C2×D10711yescharacteristic
207C2×C4⋊C4C2×D10711yescharacteristic
208C2×C4⋊C4C2×D10711yescharacteristic
209C2×C22⋊C4—1071no—
210C2×C22⋊C4—1071no—
211C2×C22⋊C4—1071no—
212C2×C22⋊C4C2×D10711yescharacteristic
213C2×C22⋊C4C2×D10711yescharacteristic
214C2×C22⋊C4C2×D10711yescharacteristic
215C2×C42—1071no—
216C2×C42—1071no—
217#95—1071no—
218#95—1071no—
219#95—1071no—
220#95—1071no—
221#95—1071no—
222#95—1071no—
223#95—1071no—
224#95—1071no—
225#95—1071no—
226#95—1071no—
227#95—1071no—
228#95—1071no—
229#95—1071no—
230#95C2×D10711yescharacteristic
231C24C22×D10711yescharacteristic
232C2×Q8—4281no—
233C2×Q8—4281no—
234C2×Q8—2142no—
235C22×C4—1071no—
236C22×C4—1071no—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—1071no—
243C22×C4—4281no—
244C22×C4—4281no—
245C22×C4—2142no—
246C22×C4C22×D10711yescharacteristic
247C22×C4C22×D10711yescharacteristic
248C22×C4C22×D10711yescharacteristic
249C22×C4C22×D10711yescharacteristic
250C22×C4C22×D10711yescharacteristic
251C22×C4C22×D10711yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—4281no—
256C4⋊C4—4281no—
257C4⋊C4—4281no—
258C4⋊C4—4281no—
259C4⋊C4—4281no—
260C4⋊C4—2142no—
261C4⋊C4—2142no—
262C4⋊C4—2142no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—1074no—
267C22⋊C4—4281no—
268C22⋊C4—2142no—
269C42—4281no—
270C42—4281no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23Q8×D10712yes—
278C23C23×D10711yescharacteristic
279Q8—4284no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—1072no—
299C2×C4—1072no—
300C2×C4—1072no—
301C2×C4—1072no—
302C2×C4—4284no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C2×Q8×D10711yescharacteristic
331C22#41×D10711yescharacteristic
332C22#41×D10711yescharacteristic
333C22#1681215411yescharacteristic
334C22#1681215411yescharacteristic
335C22#1681215411yescharacteristic
336C22#1681215611yescharacteristic
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4—2142no—
348C4—2142no—
349C4—2142no—
350C4—2142no—
351C2—42no—
352C2#2029320911yescharacteristic
353C2#2029320911yescharacteristic
354C2#2029325611yescharacteristic
355C2#2029333111yescharacteristic
356C2#2029333111yescharacteristic
357C2#2029334411yescharacteristic
358C2#2029337211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#396×C107C2
prime radicalC24×C107C23
hypercenter#396D107
hypocenterC107#2788
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #20292565C2semidirect
2. #20292565C2semidirect
3. #20292558C2semidirect
4. #20292547C2semidirect
6. #20292531C2semidirect
7. #20292522C2semidirect
8. #20292522C2semidirect
11. #20292509C2semidirect
75. C22⋊Dic107Q8semidirect
180. C107#2788semidirect

Maximal subgroups

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

subgroupindex
#202925652
#202925582
#202925472
#202925392
#202925312
#202925222
#202925152
#202925092
#202924992
#202924852
#396×C1072
#2788107