Finite Group Databasebeta

← #28339622

Subgroups — #28339622

Order 13696.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812085C211yesmaximal, characteristic
2#20292534C211yesmaximal, characteristic, complemented
3#20292524C211yesmaximal, characteristic, complemented
4#20292522C211yesmaximal, characteristic, complemented
5#20292519C211yesmaximal, characteristic
6#20292518C211yesmaximal, characteristic
7#20292517C211yesmaximal, characteristic
8#20292516C211yesmaximal, characteristic
9#20292512C211yesmaximal, characteristic, complemented
10#20292511C211yesmaximal, characteristic, complemented
11#20292510C211yesmaximal, characteristic, complemented
12#20292509C211yesmaximal, characteristic, complemented
13#20292489C211yesmaximal, characteristic
14#20292486C211yesmaximal, characteristic
15#391×C107C211yesmaximal, characteristic, complemented
16C22×C4×D107C2211yescharacteristic
17#16812085C418yescomplemented
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2211yescharacteristic
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×C4×Dic107C2211yescharacteristic
33C2×C4×Dic107C2211yescharacteristic
34C2×C4×Dic107C2211yescharacteristic
35C107⋊#95C2211yescharacteristic
36C107⋊#95C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C22×Q8×C107C2211yescharacteristic
46C2×C4⋊C4×C107C2211yescharacteristic
47C2×C42×C107C2211yescharacteristic
48#95×C107C2211yescharacteristic
49#95×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52C23×D107C2311yescharacteristic
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2311yescharacteristic
55C22×Dic107C2311yescharacteristic
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C2×C4×D107—41no—
61C2×C4×D107—41no—
62C2×C4×D107C2×C414yes—
63C4⋊Dic107—41no—
64C4⋊Dic107C2×C414yes—
65Dic107⋊C4—41no—
66Dic107⋊C4—41no—
67Dic107⋊C4—41no—
68Dic107⋊C4—41no—
69Dic107⋊C4C2×C414yes—
70Dic107⋊C4C2×C414yes—
71(C2×D107)⋊C4—41no—
72(C2×D107)⋊C4—41no—
73(C2×D107)⋊C4—41no—
74(C2×D107)⋊C4—41no—
75(C2×D107)⋊C4C2×C414yes—
76(C2×D107)⋊C4C2×C414yes—
77C4×Dic107—41no—
78C4×Dic107—41no—
79C4×Dic107—41no—
80C2×Q8×C107—41no—
81C2×Q8×C107—41no—
82C2×Q8×C107C2×C414yes—
83C22×C4×C107C2311yescharacteristic
84C22×C4×C107C2311yescharacteristic
85C22×C4×C107C2311yescharacteristic
86C22×C4×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C4⋊C4×C107—41no—
91C42×C107—41no—
92C23×C107C2411yescharacteristic
93C22×D107—21no—
94C22×D107—21no—
95C22×D107—21no—
96C22×D107—21no—
97C22×D107—21no—
98C22×D107—21no—
99C22×D107C22×C412yes—
100Q8×C107—44no—
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×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107C22×C412yes—
120C2×C4×C107C22×C412yes—
121C2×C4×C107C22×C412yes—
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—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107C22×C412yes—
141C2×Dic107C22×C412yes—
142C2×Dic107C22×C412yes—
143C4×D107—44no—
144C22×C107C23×C411yescharacteristic
145C22×C107H(5,𝔽2)11yescharacteristic
146C22×C107H(5,𝔽2)11yescharacteristic
147C22×C107H(5,𝔽2)11yescharacteristic
148C22×C1072−1+411yescharacteristic
149C22×C1072−1+411yescharacteristic
150C22×C1072−1+411yescharacteristic
151C2×D107—22no—
152C2×D107—22no—
153C2×D107—22no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C4×C107—22no—
159C4×C107—22no—
160C4×C107—22no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107—41no—
164C4×C107—41no—
165Dic107—22no—
166Dic107—22no—
167Dic107—22no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171Dic107—41no—
172C2×C107#51811yescharacteristic
173C2×C107#51911yescharacteristic
174C2×C107#52011yescharacteristic
175C2×C107#56211yescharacteristic
176C2×C107#56211yescharacteristic
177C2×C107#56311yescharacteristic
178C2×C107#56311yescharacteristic
179D107—42no—
180#2378—1071nomaximal, Sylow
181C107#237811yescharacteristic, complemented, Sylow
182C2×C22⋊Q8—1071no—
183#391—1071no—
184#391D10711yescharacteristic, complemented
185#389—1071no—
186#389—1071no—
187#388—1071no—
188#388—1071no—
189#388—1071no—
190#388—1071no—
191#387—1071no—
192#387—1071no—
193#387—1071no—
194#387—1071no—
195#381—1071no—
196#380—1071no—
197C22×Q8C2×D10711yescharacteristic
198C23×C4—1071no—
199C22⋊Q8—1078no—
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⋊C4—1071no—
207C2×C4⋊C4—1071no—
208C2×C4⋊C4C2×D10711yescharacteristic
209C2×C22⋊C4—1071no—
210C2×C22⋊C4—1071no—
211C2×C22⋊C4—1071no—
212C2×C22⋊C4—1071no—
213C2×C22⋊C4—1071no—
214C2×C22⋊C4—1071no—
215C2×C42—1071no—
216C2×C42—1071no—
217C2×C42—1071no—
218C2×C42C2×D10711yescharacteristic
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#95C2×D10711yescharacteristic
230#95C2×D10711yescharacteristic
231#95C2×D10711yescharacteristic
232#95C2×D10711yescharacteristic
233C24—1071no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8C4×D10714yes—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1074no—
241C22×C4—1071no—
242C22×C4—1071no—
243C22×C4—1071no—
244C22×C4—1071no—
245C22×C4—4281no—
246C22×C4—4281no—
247C22×C4C22×D10711yescharacteristic
248C22×C4C22×D10711yescharacteristic
249C22×C4C22×D10711yescharacteristic
250C22×C4C22×D10711yescharacteristic
251C22×C4C22×D10711yescharacteristic
252C22×C4C22×D10711yescharacteristic
253C22×C4C22×D10711yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—1074no—
256C4⋊C4—1074no—
257C4⋊C4—1074no—
258C4⋊C4—4281no—
259C4⋊C4—4281no—
260C4⋊C4—4281no—
261C4⋊C4—4281no—
262C4⋊C4—4281no—
263C22⋊C4—1074no—
264C22⋊C4—1074no—
265C22⋊C4—4281no—
266C22⋊C4—4281no—
267C22⋊C4—4281no—
268C22⋊C4—4281no—
269C42—41no—
270C42—4281no—
271C42—4281no—
272C42—4281no—
273C23—1072no—
274C23—2141no—
275C23—2141no—
276C23—2141no—
277C23—2141no—
278C23—2141no—
279C23—2141no—
280C23C23×D10711yescharacteristic
281Q8—44no—
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—21no—
299C2×C4—21no—
300C2×C4—1072no—
301C2×C4—1072no—
302C2×C4—1072no—
303C2×C4—4284no—
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×C4C2×C4×D10712yes—
323C2×C4C2×C4×D10712yes—
324C2×C4C2×C4×D10712yes—
325C22—4281no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22—2142no—
330C22—2142no—
331C22—2142no—
332C22C22×C4×D10711yescharacteristic
333C22#1681215411yescharacteristic
334C22#1681215411yescharacteristic
335C22#1681215511yescharacteristic
336C22#1681215611yescharacteristic
337C22#1681215711yescharacteristic
338C22#1681215711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4281no—
347C4—4281no—
348C4—4281no—
349C4—4281no—
350C4—2142no—
351C4—2142no—
352C4—2142no—
353C2—4282no—
354C2#2029310011yescharacteristic
355C2#2029310311yescharacteristic
356C2#2029310911yescharacteristic
357C2#2029337111yescharacteristic
358C2#2029337311yescharacteristic
359C2#2029337611yescharacteristic
360C2#2029337811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#391×C107C2
socle, cosocleC23×C107C24
prime radicalC23×D107C23
hypercenter#391D107
hypocenterC107#2378
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20292534C2semidirect
3. #20292524C2semidirect
4. #20292522C2semidirect
9. #20292512C2semidirect
10. #20292511C2semidirect
11. #20292510C2semidirect
12. #20292509C2semidirect
15. #391×C107C2semidirect
17. #16812085C4semidirect
181. C107#2378semidirect
184. #391D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
#202925342
#202925242
#202925222
#202925192
#202925182
#202925172
#202925162
#202925122
#202925112
#202925102
#202925092
#202924892
#202924862
#391×C1072
#2378107