Finite Group Databasebeta

← #28348333

Subgroups — #28348333

Order 13696.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293332C211yesmaximal, characteristic
2Q8×C22⋊D107C211yesmaximal, characteristic
3#20292976C211yesmaximal, characteristic, complemented
4#20292962C211yesmaximal, characteristic, complemented
5#20292961C211yesmaximal, characteristic, complemented
6#20292920C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292867C211yesmaximal, characteristic, complemented
9#20292836C211yesmaximal, characteristic, complemented
10#20292784C211yesmaximal, characteristic, complemented
11#20292783C211yesmaximal, characteristic, complemented
12#20292719C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic
14#103×D107C211yesmaximal, characteristic, complemented
15C2×#103×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107—21no—
23#16812107C2211yescharacteristic
24#16812099—21no—
25#16812089C2211yescharacteristic
26#16812088—21no—
27#16812088—21no—
28#16812088C2211yescharacteristic
29#16812088C2211yescharacteristic
30#16812087—21no—
31#16812085—21no—
32#16812085C2211yescharacteristic
33Q8×Dic107C2211yescharacteristic
34#16812074C2211yescharacteristic
35C4×C22⋊D107—21no—
36C4×C22⋊D107—21no—
37C4×C22⋊D107C2211yescharacteristic
38C4⋊C4×D107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812042C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22×Q8×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107—41no—
65C2×C22⋊D107D411yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69Dic428—41no—
70#13133808—22no—
71#13133808—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C8×D107—41no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4D411yescharacteristic
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107D411yescharacteristic
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C2×Q8×C107—21no—
98C2×Q8×C107—22no—
99C2×Q8×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107—21no—
104C22×C4×C107C2311yescharacteristic
105C2×C8×C107—41no—
106C2×C8×C107C2311yescharacteristic
107C2×C8×C107C2311yescharacteristic
108C4⋊C4×C107—21no—
109C4⋊C4×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C22×D107C2×D411yescharacteristic
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107—42no—
117Q8×C107#4112yes—
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—42no—
125C2×C4×C107—81no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2411yescharacteristic
128C22⋊D107Q1614yescomplemented
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C4×D107—21no—
137C4×D107—21no—
138C4×D107—42no—
139C4×D107—81no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—21no—
144Dic214—42no—
145Dic214—81no—
146C8×C107—21no—
147C8×C107—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107C2×Q1612yes—
153C22×C107C22×D411yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107C2×Q1612yes—
157C4×C107—22no—
158C4×C107—21no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C2×#4111yescharacteristic
163C4×C107H(5,𝔽2)11yescharacteristic
164Dic107—21no—
165Dic107—21no—
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107C2×Q1612yes—
171C2×C107—22no—
172C2×C107#54611yescharacteristic
173C2×C107C22×Q1611yescharacteristic
174C2×C107#57711yescharacteristic
175D107—22no—
176#3299—1071nomaximal, Sylow
177C107#329911yescharacteristic, complemented, Sylow
178#554—1071no—
179D4×Q8—1071no—
180#487—1071no—
181#484—1071no—
182#484—1071no—
183#477—1071no—
184#467—1071no—
185#467—1071no—
186#462—1071no—
187#451—1071no—
188#451—1071no—
189C4×Q16—1071no—
190C8×D4—1071no—
191C2×#103—1071no—
192C2×#103D10711yescharacteristic, complemented
193C22×Q8—1071no—
194C22×Q8C2×D10711yescharacteristic
195C2×Q16—1071no—
196C2×Q16—1071no—
197C2×Q16—1071no—
198C22×C8—1071no—
199C22×C8C2×D10711yescharacteristic
200C4⋊Q8—1071no—
201C4⋊Q8—1071no—
202C4⋊Q8—2141no—
203#125—2141no—
204C22⋊Q8—1071no—
205C22⋊Q8—1071no—
206C22⋊Q8—1071no—
207C22⋊Q8—1071no—
208C22⋊Q8—2141no—
209C22⋊Q8—2141no—
210C22⋊Q8—2141no—
211C22⋊Q8—2141no—
212C4×Q8—1071no—
213C4×Q8—1071no—
214C4×D4—1071no—
215C4×D4—2141no—
216C4×D4—2141no—
217C2×C4⋊C4—1071no—
218C2×C4⋊C4C2×D10711yescharacteristic
219#107—1071no—
220#107—1071no—
221#107—1071no—
222C4⋊C8—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103C2×D10711yescharacteristic
230#103C2×D10711yescharacteristic
231#103C2×D10711yescharacteristic
232#103C2×D10711yescharacteristic
233C22⋊C8—1071no—
234C22⋊C8—1071no—
235C4×C8—1071no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—1071no—
240C2×Q8—1071no—
241C2×Q8—4281no—
242C2×Q8—2141no—
243C2×Q8—2141no—
244C2×Q8—2142no—
245C2×Q8C22×D10711yescharacteristic
246C2×Q8C22×D10711yescharacteristic
247C2×D4—1071no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1071no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4C22×D10711yescharacteristic
254Q16—4281no—
255Q16—4281no—
256Q16—2142no—
257C2×C8—41no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8—4281no—
261C2×C8C22×D10711yescharacteristic
262C2×C8C22×D10711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4—2141no—
276C4⋊C4—2141no—
277C4⋊C4C22×D10711yescharacteristic
278C4⋊C4C22×D10711yescharacteristic
279C22⋊C4—1071no—
280C22⋊C4—1071no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C42—1071no—
286C42—2141no—
287C42—2141no—
288C23—1071no—
289C23D4×D10711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—8561no—
295Q8—4282no—
296Q8—2141no—
297Q8—2141no—
298Q8—2141no—
299Q8—2141no—
300Q8#1313381812yes—
301D4—1074no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—8561no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—1071no—
314C2×C4—4282no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4D4×D10711yescharacteristic
322C2×C4C23×D10711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2141no—
326C8—2141no—
327C22—21no—
328C22—21no—
329C22—1072no—
330C22—2141no—
331C22—2141no—
332C22Q16×D10712yes—
333C22C2×D4×D10711yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1072no—
340C4—4281no—
341C4—4281no—
342C4—4281no—
343C4—4281no—
344C4—2141no—
345C4—2141no—
346C4C2×#1313381811yescharacteristic
347C4#1681215411yescharacteristic
348C2—22no—
349C2—2142no—
350C2#2029326711yescharacteristic
351C2C2×Q16×D10711yescharacteristic
352C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
prime radicalC2×C22⋊D107D4
hypercenterC2×#103D107
hypocenterC107#3299
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #20292976C2semidirect
4. #20292962C2semidirect
5. #20292961C2semidirect
6. #20292920C2semidirect
7. #20292868C2semidirect
8. #20292867C2semidirect
9. #20292836C2semidirect
10. #20292784C2semidirect
11. #20292783C2semidirect
12. #20292719C2semidirect
14. #103×D107C2semidirect
15. C2×#103×C107C2semidirect
128. C22⋊D107Q16semidirect
177. C107#3299semidirect
192. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
#202933322
Q8×C22⋊D1072
#202929762
#202929622
#202929612
#202929202
#202928682
#202928672
#202928362
#202927842
#202927832
#202927192
C8×C22⋊D1072
#103×D1072
C2×#103×C1072
#3299107