Finite Group Databasebeta

← #28350078

Subgroups — #28350078

Order 13696.

1094 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
1#20293353C211yesmaximal, characteristic, complemented
2#20293349C213yesmaximal, complemented
3#20293318C213yesmaximal, complemented
4#20293312C213yesmaximal, complemented
5Q8×C22⋊D107C211yesmaximal, characteristic, complemented
6#20293289C211yesmaximal, characteristic, complemented
7#20293288C211yesmaximal, characteristic, complemented
8#20293278C213yesmaximal, complemented
9#20293181C213yesmaximal, complemented
10#20293164C213yesmaximal, complemented
11#20293161C213yesmaximal, complemented
12C4×#13133819C211yesmaximal, characteristic, complemented
13C4×#13133818C213yesmaximal, complemented
14C4×Q8×D107C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2213yes—
18C2×Q8×D107C2211yescharacteristic
19#16812108C2213yes—
20#16812107C2213yes—
21#16812104C2216yes—
22#16812103C2216yes—
23#16812101C2213yes—
24#16812100C2213yes—
25#16812099C2216yes—
26#16812098C2213yes—
27#16812097C2213yes—
28#16812094C2216yes—
29#16812092C2216yes—
30#16812089C2213yes—
31#16812088C2213yes—
32#16812087C2216yes—
33#16812085C2213yes—
34#16812085C2213yes—
35#16812084C2213yes—
36#16812083C2213yes—
37Q8×Dic107C2211yescharacteristic
38Q8×Dic107C2211yescharacteristic
39C4×Dic214C2213yes—
40C4×Dic214C2213yes—
41C4×Dic214C2213yes—
42#16812074C2213yes—
43C4×D428C2213yes—
44C4×D428C2213yes—
45C4×D428C2213yes—
46#16812071C2213yes—
47C4×C22⋊D107C2213yes—
48C4×C22⋊D107C2213yes—
49C4×C22⋊D107C2213yes—
50C4×C22⋊D107C2211yescharacteristic
51#16812068C2213yes—
52#16812067C2213yes—
53#16812066C2216yes—
54#16812066C2213yes—
55#16812066C2213yes—
56C4⋊C4×D107C2213yes—
57C42×D107C2213yes—
58C42×D107C2213yes—
59C22×Q8×C107C2211yescharacteristic
60C4×Q8×C107C2213yes—
61C4×Q8×C107C2211yescharacteristic
62C4×Q8×C107C2213yes—
63C4×Q8×C107C2211yescharacteristic
64C2×C4⋊C4×C107C2213yes—
65C2×C42×C107C2213yes—
66#13133819—24no—
67#13133818—212no—
68C2×Dic214C2313yes—
69C2×Dic214C2313yes—
70Q8×D107—24no—
71C2×D428C2313yes—
72C2×D428C2313yes—
73C2×C22⋊D107C2313yes—
74C2×C22⋊D107C2311yescharacteristic
75C2×C4×D107C2311yescharacteristic
76C2×C4×D107C2313yes—
77C2×C4×D107C2313yes—
78C2×C4×D107C2313yes—
79C2×C4×D107C2313yes—
80C2×C4×D107C2313yes—
81C4⋊Dic107C2313yes—
82C4⋊Dic107C2313yes—
83C4⋊Dic107C2313yes—
84C4⋊Dic107C2313yes—
85Dic107⋊C4C2316yes—
86Dic107⋊C4C2311yescharacteristic
87Dic107⋊C4C2313yes—
88Dic107⋊C4C2313yes—
89Dic107⋊C4C2313yes—
90Dic107⋊C4C2316yes—
91Dic107⋊C4C2316yes—
92C22⋊Dic107C2311yescharacteristic
93C22⋊Dic107C2313yes—
94(C2×D107)⋊C4C2311yescharacteristic
95(C2×D107)⋊C4C2313yes—
96(C2×D107)⋊C4C2313yes—
97(C2×D107)⋊C4C2313yes—
98(C2×D107)⋊C4C2316yes—
99(C2×D107)⋊C4C2316yes—
100(C2×D107)⋊C4C2316yes—
101C4×Dic107C2311yescharacteristic
102C4×Dic107C2313yes—
103C4×Dic107C2313yes—
104C4×Dic107C2313yes—
105C4×Dic107C2313yes—
106C4×Dic107C2313yes—
107C2×Q8×C107—24no—
108C2×Q8×C107C2313yes—
109C2×Q8×C107C2311yescharacteristic
110C22×C4×C107C2313yes—
111C22×C4×C107C2311yescharacteristic
112C22×C4×C107C2313yes—
113C4⋊C4×C107C2313yes—
114C4⋊C4×C107C2313yes—
115C4⋊C4×C107C2316yes—
116C42×C107C2313yes—
117C42×C107C2313yes—
118C42×C107C2313yes—
119C42×C107C2313yes—
120C23×C107C2411yescharacteristic
121C22×D107C2413yes—
122C22×D107C2411yescharacteristic
123Q8×C107—26no—
124Q8×C107#4114yescomplemented
125C2×C4×C107—26no—
126C2×C4×C107—26no—
127C2×C4×C107—22no—
128C2×C4×C107C2411yescharacteristic
129C2×C4×C107C2413yes—
130C2×C4×C107C2413yes—
131C2×C4×C107C2413yes—
132C2×C4×C107C2413yes—
133C2×C4×C107C2411yescharacteristic
134C22⋊D107—26no—
135C22⋊D107#4114yescomplemented
136C2×Dic107C2411yescharacteristic
137C2×Dic107C2411yescharacteristic
138C2×Dic107C2413yes—
139C2×Dic107C2413yes—
140C2×Dic107C2413yes—
141C2×Dic107C2411yescharacteristic
142D428—26no—
143D428—26no—
144C4×D107—26no—
145C4×D107—26no—
146C4×D107—26no—
147C4×D107—22no—
148C4×D107—26no—
149C4×D107—26no—
150Dic214—26no—
151Dic214—26no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107C2×#4112yes—
155C22×C107C2511yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—23no—
159C2×D107—23no—
160C2×D107—23no—
161C2×D107C2×#4112yes—
162C4×C107—23no—
163C4×C107—23no—
164C4×C107—23no—
165C4×C107—21no—
166C4×C107C2×#4116yes—
167C4×C1072−1+412yes—
168Dic107—23no—
169Dic107—23no—
170Dic107—21no—
171Dic107—21no—
172Dic107—23no—
173Dic107C2×#4112yes—
174C2×C107—22no—
175C2×C107C22×#4111yescharacteristic
176C2×C107C22×#4111yescharacteristic
177C2×C107C2×2−1+411yescharacteristic
178D107—22no—
179D107—43no—
180#3482—1071nomaximal, Sylow
181C107#348211yescharacteristic, complemented, Sylow
182#557—1071no—
183#556—1073no—
184#552—1073no—
185#552—1073no—
186D4×Q8—1071no—
187#548—1071no—
188#548—1071no—
189#547—1073no—
190#533—1073no—
191#529—1073no—
192#529—1073no—
193C4×#41—1071no—
194C4×#41—1073no—
195C2×C4×Q8—1071no—
196C2×C4×Q8D10711yescharacteristic, complemented
197C2×#41—1071no—
198C2×#41—1073no—
199C22×Q8—1071no—
200C22×Q8C2×D10711yescharacteristic
201C4⋊Q8—1073no—
202C4⋊Q8—1073no—
203#126—1076no—
204#126—1076no—
205#125—1073no—
206#125—1076no—
207#125—1073no—
208#124—1073no—
209#124—1073no—
210#123—1076no—
211#123—1076no—
212C22⋊Q8—1073no—
213C22⋊Q8—1073no—
214C22⋊Q8—1073no—
215C22⋊Q8—1076no—
216C22⋊Q8—1073no—
217#121—1073no—
218#121—1073no—
219C4×Q8—1071no—
220C4×Q8—1071no—
221C4×Q8—1073no—
222C4×Q8—1073no—
223C4×Q8—1073no—
224C4×Q8—1073no—
225C4×Q8C2×D10711yescharacteristic
226C4×Q8C2×D10713yes—
227C4×Q8C2×D10711yescharacteristic
228C4×Q8C2×D10713yes—
229C4×D4—1071no—
230C4×D4—1073no—
231C4×D4—1073no—
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4—1073no—
235C4×D4—1073no—
236C4×D4—1073no—
237#117—1073no—
238#117—1073no—
239#117—1073no—
240#117—1073no—
241#117—1076no—
242C2×C4⋊C4—1073no—
243C2×C4⋊C4C2×D10713yes—
244C2×C42—1073no—
245C2×C42—1073no—
246C2×C42C2×D10713yes—
247#41—21412no—
248#41—2144no—
249C2×Q8—24no—
250C2×Q8—1073no—
251C2×Q8—1073no—
252C2×Q8—2144no—
253C2×Q8C22×D10713yes—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—1071no—
256C2×D4—1073no—
257C2×D4—1073no—
258C2×D4—1073no—
259C22×C4—1073no—
260C22×C4—1073no—
261C22×C4—1073no—
262C22×C4—1073no—
263C22×C4—1071no—
264C22×C4—1073no—
265C22×C4C22×D10713yes—
266C22×C4C22×D10713yes—
267C22×C4C22×D10711yescharacteristic
268C4⋊C4—1073no—
269C4⋊C4—1076no—
270C4⋊C4—1071no—
271C4⋊C4—1073no—
272C4⋊C4—1073no—
273C4⋊C4—1073no—
274C4⋊C4—1073no—
275C4⋊C4—1073no—
276C4⋊C4—1073no—
277C4⋊C4—1076no—
278C4⋊C4—1076no—
279C4⋊C4C22×D10713yes—
280C4⋊C4C22×D10713yes—
281C4⋊C4C22×D10716yes—
282C22⋊C4—1071no—
283C22⋊C4—1073no—
284C22⋊C4—1071no—
285C22⋊C4—1073no—
286C22⋊C4—1076no—
287C22⋊C4—1073no—
288C22⋊C4—1073no—
289C22⋊C4—1076no—
290C22⋊C4—1076no—
291C42—1073no—
292C42—1073no—
293C42—1073no—
294C42—1073no—
295C42—1071no—
296C42—1073no—
297C42C22×D10713yes—
298C42C22×D10713yes—
299C42C22×D10713yes—
300C42C22×D10713yes—
301C23—1071no—
302C23—1073no—
303C23C23×D10711yescharacteristic
304Q8—26no—
305Q8—2146no—
306Q8—2146no—
307Q8#1313381814yescomplemented
308D4—1074no—
309D4—2146no—
310D4—2146no—
311D4—2146no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—1071no—
316C2×C4—1071no—
317C2×C4—1071no—
318C2×C4—1073no—
319C2×C4—1073no—
320C2×C4—1073no—
321C2×C4—2146no—
322C2×C4—2146no—
323C2×C4—2146no—
324C2×C4—2146no—
325C2×C4—2146no—
326C2×C4—2142no—
327C2×C4C23×D10713yes—
328C2×C4C23×D10711yescharacteristic
329C2×C4C23×D10713yes—
330C2×C4C23×D10713yes—
331C2×C4C23×D10713yes—
332C2×C4C23×D10711yescharacteristic
333C22—21no—
334C22—21no—
335C22—1072no—
336C22—2143no—
337C22—2143no—
338C22—2143no—
339C22—2141no—
340C22—2141no—
341C22#41×D10712yes—
342C22C24×D10711yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—1072no—
348C4—2143no—
349C4—2141no—
350C4—2143no—
351C4—2141no—
352C4—2143no—
353C4C2×#1313381816yes—
354C4#1681215712yes—
355C2—22no—
356C2—4283no—
357C2—2142no—
358C2C2×#41×D10711yescharacteristic
359C2C22×#1313381811yescharacteristic
360C2C2×#1681215711yescharacteristic
subgroupG/H
centerC2×C4C23×D107
commutator, socle, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×C4×Q8×C107C2
hypercenterC2×C4×Q8D107
hypocenterC107#3482
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20293353C2semidirect
2. #20293349C2semidirect
3. #20293318C2semidirect
4. #20293312C2semidirect
5. Q8×C22⋊D107C2semidirect
6. #20293289C2semidirect
7. #20293288C2semidirect
8. #20293278C2semidirect
9. #20293181C2semidirect
10. #20293164C2semidirect
11. #20293161C2semidirect
12. C4×#13133819C2semidirect
13. C4×#13133818C2semidirect
14. C4×Q8×D107C2semidirect
15. C2×C4×Q8×C107C2semidirect
124. Q8×C107#41semidirect
135. C22⋊D107#41semidirect
181. C107#3482semidirect
196. C2×C4×Q8D107semidirect
307. Q8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202933532
#202933492
#202933182
#202933122
Q8×C22⋊D1072
#202932892
#202932882
#202932782
#202931812
#202931642
#202931612
C4×#131338192
C4×#131338182
C4×Q8×D1072
C2×C4×Q8×C1072
#3482107