Finite Group Databasebeta

← #28339142

Subgroups — #28339142

Order 13696.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812087C211yesmaximal, characteristic
2C2×C4×Dic214C211yesmaximal, characteristic, complemented
3C2×#16812067C211yesmaximal, characteristic
4#20292531C211yesmaximal, characteristic, complemented
5#20292526C211yesmaximal, characteristic, complemented
6#20292524C211yesmaximal, characteristic, complemented
7#20292524C211yesmaximal, characteristic, complemented
8#20292519C211yesmaximal, characteristic
9#20292519C211yesmaximal, characteristic
10#20292518C211yesmaximal, characteristic
11#20292516C211yesmaximal, characteristic
12#20292508C211yesmaximal, characteristic, complemented
13#20292486C211yesmaximal, characteristic
14#20292478C211yesmaximal, characteristic, complemented
15#387×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18#16812087C418yescomplemented
19C4×Dic214—24no—
20#16812067—24no—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×C4×Dic107C2211yescharacteristic
37C2×C4×Dic107C2211yescharacteristic
38C2×C4×Dic107C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C2×C4⋊C4×C107C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C42×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54C23×D107C2311yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—22no—
57C2×Dic214C2×C414yes—
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C2×C4×D107—22no—
66C2×C4×D107—22no—
67C2×C4×D107C2×C414yes—
68C4⋊Dic107—22no—
69C4⋊Dic107—22no—
70C4⋊Dic107—41no—
71Dic107⋊C4—22no—
72Dic107⋊C4—22no—
73Dic107⋊C4—22no—
74Dic107⋊C4—41no—
75Dic107⋊C4C2×C414yes—
76Dic107⋊C4C2×C414yes—
77(C2×D107)⋊C4—22no—
78(C2×D107)⋊C4—22no—
79(C2×D107)⋊C4—41no—
80(C2×D107)⋊C4—41no—
81(C2×D107)⋊C4C2×C414yes—
82(C2×D107)⋊C4C2×C414yes—
83C4×Dic107—22no—
84C4×Dic107—22no—
85C4×Dic107—41no—
86C4×Dic107—41no—
87C4×Dic107D414yes—
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C4⋊C4×C107—22no—
96C4⋊C4×C107C2×C414yes—
97C42×C107—22no—
98C23×C107C2411yescharacteristic
99C22×D107—21no—
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—21no—
103C22×D107—21no—
104C22×D107—21no—
105C22×D107C22×C412yes—
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×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C22×C412yes—
124C2×C4×C107C22×C412yes—
125C2×C4×C107C22×C412yes—
126C2×C4×C107C2×D412yes—
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×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107C22×C412yes—
143C2×Dic107C22×C412yes—
144C2×Dic107C22×C412yes—
145C2×Dic107C2×D412yes—
146C2×Dic107C2×D412yes—
147C2×Dic107#4112yes—
148C4×D107—28no—
149Dic214—28no—
150C22×C107C23×C411yescharacteristic
151C22×C107C22×D411yescharacteristic
152C22×C107C2×#4111yescharacteristic
153C22×C107H(5,𝔽2)11yescharacteristic
154C22×C1072−1+411yescharacteristic
155C22×C1072−1+411yescharacteristic
156C22×C1072−1+411yescharacteristic
157C2×D107—22no—
158C2×D107—22no—
159C2×D107—22no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C4×C107—22no—
165C4×C107—22no—
166C4×C107—22no—
167C4×C107—22no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107—41no—
171Dic107—22no—
172Dic107—22no—
173Dic107—22no—
174Dic107—41no—
175Dic107—41no—
176Dic107—41no—
177Dic107C4×D414yes—
178C2×C107C2×C4×D411yescharacteristic
179C2×C107#51911yescharacteristic
180C2×C107#52011yescharacteristic
181C2×C107#53611yescharacteristic
182C2×C107#53711yescharacteristic
183C2×C107#54111yescharacteristic
184C2×C107#54211yescharacteristic
185D107—42no—
186#2323—1071nomaximal, Sylow
187C107#232311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1071no—
189C2×C4×Q8—1071no—
190C2×#117—1071no—
191#391—1071no—
192#389—1071no—
193#389—1071no—
194#389—1071no—
195#388—1071no—
196#388—1071no—
197#388—1071no—
198#388—1071no—
199#387—1071no—
200#387D10711yescharacteristic, complemented
201#380—1071no—
202C4×C4⋊C4—1071no—
203C22×Q8—1071no—
204C23×C4—1071no—
205C22⋊Q8—1078no—
206C4×Q8—2144no—
207#117—2144no—
208C2×C4⋊C4—1071no—
209C2×C4⋊C4—1071no—
210C2×C4⋊C4—1071no—
211C2×C4⋊C4—1071no—
212C2×C4⋊C4—1071no—
213C2×C4⋊C4—1071no—
214C2×C4⋊C4—1071no—
215C2×C4⋊C4—1071no—
216C2×C4⋊C4—1071no—
217C2×C4⋊C4C2×D10711yescharacteristic
218C2×C4⋊C4C2×D10711yescharacteristic
219C2×C22⋊C4—1071no—
220C2×C22⋊C4—1071no—
221C2×C22⋊C4—1071no—
222C2×C22⋊C4—1071no—
223C2×C22⋊C4—1071no—
224C2×C22⋊C4—1071no—
225C2×C42—1071no—
226C2×C42—1071no—
227C2×C42—1071no—
228C2×C42—1071no—
229C2×C42—1071no—
230C2×C42C2×D10711yescharacteristic
231#95—1071no—
232#95—1071no—
233#95—1071no—
234#95—1071no—
235#95—1071no—
236#95—1071no—
237#95C2×D10711yescharacteristic
238#95C2×D10711yescharacteristic
239#95C2×D10711yescharacteristic
240#95C2×D10711yescharacteristic
241C24—1071no—
242C2×Q8—1074no—
243C2×Q8—2142no—
244C2×Q8—2142no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1071no—
249C22×C4—1074no—
250C22×C4—1071no—
251C22×C4—1071no—
252C22×C4—1071no—
253C22×C4—2142no—
254C22×C4—2142no—
255C22×C4C22×D10711yescharacteristic
256C22×C4C22×D10711yescharacteristic
257C22×C4C22×D10711yescharacteristic
258C22×C4C22×D10711yescharacteristic
259C22×C4C22×D10711yescharacteristic
260C22×C4C22×D10711yescharacteristic
261C22×C4C22×D10711yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—1074no—
264C4⋊C4—1074no—
265C4⋊C4—4281no—
266C4⋊C4—4281no—
267C4⋊C4—2142no—
268C4⋊C4—2142no—
269C4⋊C4—2142no—
270C4⋊C4—2142no—
271C4⋊C4—2142no—
272C4⋊C4C4×D10714yes—
273C22⋊C4—1074no—
274C22⋊C4—1074no—
275C22⋊C4—4281no—
276C22⋊C4—4281no—
277C22⋊C4—2142no—
278C22⋊C4—2142no—
279C42—22no—
280C42—1074no—
281C42—4281no—
282C42—4281no—
283C42—2142no—
284C42—2142no—
285C23—1072no—
286C23—2141no—
287C23—2141no—
288C23—2141no—
289C23—2141no—
290C23—2141no—
291C23—2141no—
292C23C23×D10711yescharacteristic
293Q8—2148no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1072no—
312C2×C4—1072no—
313C2×C4—1072no—
314C2×C4—1072no—
315C2×C4—1072no—
316C2×C4—1072no—
317C2×C4—2148no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4—2141no—
333C2×C4C2×C4×D10712yes—
334C2×C4C2×C4×D10712yes—
335C2×C4C2×C4×D10712yes—
336C2×C4D4×D10712yes—
337C22—4281no—
338C22—4281no—
339C22—4281no—
340C22—4281no—
341C22—2142no—
342C22—2142no—
343C22—2142no—
344C22C22×C4×D10711yescharacteristic
345C22C2×D4×D10711yescharacteristic
346C22#41×D10711yescharacteristic
347C22#1681215511yescharacteristic
348C22#1681215611yescharacteristic
349C22#1681215611yescharacteristic
350C22#1681215711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—1074no—
359C4—4281no—
360C4—4281no—
361C4—4281no—
362C4—2142no—
363C4—2142no—
364C4—2142no—
365C2—4282no—
366C2C4×D4×D10711yescharacteristic
367C2#2029310311yescharacteristic
368C2#2029310711yescharacteristic
369C2#2029319911yescharacteristic
370C2#2029320711yescharacteristic
371C2#2029323611yescharacteristic
372C2#2029324311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#387×C107C2
socle, cosocleC23×C107C24
prime radicalC23×D107C23
hypercenter#387D107
hypocenterC107#2323
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic214C2semidirect
4. #20292531C2semidirect
5. #20292526C2semidirect
6. #20292524C2semidirect
7. #20292524C2semidirect
12. #20292508C2semidirect
14. #20292478C2semidirect
15. #387×C107C2semidirect
18. #16812087C4semidirect
187. C107#2323semidirect
200. #387D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120872
C2×C4×Dic2142
C2×#168120672
#202925312
#202925262
#202925242
#202925192
#202925182
#202925162
#202925082
#202924862
#202924782
#387×C1072
#2323107