Finite Group Databasebeta

← #28349495

Subgroups — #28349495

Order 13696.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293355C211yesmaximal, characteristic, complemented
2#20293346C211yesmaximal, characteristic
3#20293022C211yesmaximal, characteristic
4#20293018C211yesmaximal, characteristic
5#20293005C211yesmaximal, characteristic, complemented
6#20292848C211yesmaximal, characteristic
7#20292838C211yesmaximal, characteristic, complemented
8#20292836C211yesmaximal, characteristic, complemented
9#20292835C211yesmaximal, characteristic, complemented
10#20292830C211yesmaximal, characteristic
11#20292756C211yesmaximal, characteristic, complemented
12#20292732C211yesmaximal, characteristic, complemented
13#20292731C211yesmaximal, characteristic, complemented
14#20292718C211yesmaximal, characteristic
15#464×C107C211yesmaximal, characteristic
16C2×Dic428C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133808C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133806C2211yescharacteristic
23#16812109C2211yescharacteristic
24#16812107C2211yescharacteristic
25#16812107C2211yescharacteristic
26#16812107C2211yescharacteristic
27#16812102—21no—
28#16812102—21no—
29#16812101—21no—
30#16812099—21no—
31#16812099—21no—
32#16812096C2211yescharacteristic
33#16812086—21no—
34Q8×Dic107C2211yescharacteristic
35C4×Dic214—21no—
36C4×Dic214C2211yescharacteristic
37C4×Dic214C2211yescharacteristic
38#16812074C2211yescharacteristic
39#16812074C2211yescharacteristic
40D4×Dic107C2211yescharacteristic
41#16812065—21no—
42#16812040C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812020C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214D411yescharacteristic
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C22×Dic107—21no—
64Dic428—41no—
65Dic428—41no—
66#13133808—22no—
67#13133808—22no—
68#13133808—41no—
69#13133806—22no—
70#13133806—41no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107D411yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
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
88Dic107⋊C4D411yescharacteristic
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107C2311yescharacteristic
101Q16×C107—41no—
102SD16×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116D4×C107—21no—
117D4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—42no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133Dic214—22no—
134Dic214—21no—
135Dic214—21no—
136Dic214—21no—
137Dic214—21no—
138Dic214—21no—
139Dic214—21no—
140C8×C107—21no—
141C8×C107—21no—
142C107⋊C8—21no—
143C107⋊C8#4112yes—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107C22×D411yescharacteristic
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C2×#4111yescharacteristic
154C4×C107H(5,𝔽2)11yescharacteristic
155Dic107—21no—
156Dic107—21no—
157Dic107—21no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161Dic107—41no—
162Dic107#13712yes—
163C2×C107—81no—
164C2×C107#55011yescharacteristic
165C2×C107C2×#13711yescharacteristic
166C2×C107#57811yescharacteristic
167#3398—1071nomaximal, Sylow
168C107#339811yescharacteristic, complemented, Sylow
169#557—1071no—
170#556—1071no—
171#497—1071no—
172#497—1071no—
173#494—1071no—
174#464—1071no—
175#464D10711yescharacteristic
176#462—1071no—
177#462—1071no—
178#462—1071no—
179#461—1071no—
180#446—1071no—
181#441—1071no—
182#441—1071no—
183C4×SD16—1071no—
184C2×Q16—1071no—
185C2×Q16—1071no—
186C2×Q16—1071no—
187C2×Q16—1071no—
188C2×Q16—1071no—
189C2×Q16C2×D10711yescharacteristic
190C2×SD16—1071no—
191C2×SD16—1071no—
192C2×SD16C2×D10711yescharacteristic
193C4⋊Q8—1071no—
194C4⋊Q8—1071no—
195C4⋊Q8—1071no—
196C4⋊Q8—1071no—
197#126—2141no—
198#126—2141no—
199#125—2141no—
200#125—2141no—
201#125—2141no—
202#124—1071no—
203#124C2×D10711yescharacteristic
204C22⋊Q8—2141no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×Q8—1071no—
209C4×Q8—1071no—
210C4×Q8—2141no—
211C4×Q8C2×D10711yescharacteristic
212C4×D4—1071no—
213#117—2141no—
214#106—1071no—
215C4⋊C8—1071no—
216C4⋊C8—1071no—
217C4⋊C8C2×D10711yescharacteristic
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#103C2×D10711yescharacteristic
223#102C2×D10711yescharacteristic
224#97—1071no—
225#97—1071no—
226C4×C8—1071no—
227C2×Q8—1071no—
228C2×Q8—1071no—
229C2×Q8—1071no—
230C2×Q8—1071no—
231C2×Q8C22×D10711yescharacteristic
232C2×Q8C22×D10711yescharacteristic
233C2×D4C22×D10711yescharacteristic
234C22×C4—2141no—
235Q16—41no—
236Q16—4281no—
237Q16—4281no—
238Q16—4281no—
239Q16—2142no—
240Q16—2142no—
241SD16—41no—
242SD16—4281no—
243SD16—2142no—
244C2×C8—1071no—
245C2×C8—1071no—
246C2×C8C22×D10711yescharacteristic
247C2×C8C22×D10711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—1071no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—1071no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4C22×D10711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C42—21no—
272C42—1071no—
273C42—1071no—
274C42—2141no—
275C42—2141no—
276C42—2141no—
277C42—2141no—
278C42C22×D10711yescharacteristic
279C23—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—2142no—
285Q8—2141no—
286Q8—2141no—
287Q8—2141no—
288Q8—2141no—
289Q8—2141no—
290Q8—2141no—
291D4—21no—
292D4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—1071no—
299C2×C4—1071no—
300C2×C4—4282no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4D4×D10711yescharacteristic
306C2×C4D4×D10711yescharacteristic
307C2×C4C23×D10711yescharacteristic
308C8—21no—
309C8—21no—
310C8—1072no—
311C8—2141no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C2×D4×D10711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—1072no—
322C4—4281no—
323C4—4281no—
324C4—4281no—
325C4—4281no—
326C4—2141no—
327C4—2141no—
328C4—2141no—
329C4#41×D10711yescharacteristic
330C4#1681215411yescharacteristic
331C2—81no—
332C2#2029329511yescharacteristic
333C2#137×D10711yescharacteristic
334C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×C107C23
hypercenter#464D107
hypocenterC107#3398
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #20293355C2semidirect
5. #20293005C2semidirect
7. #20292838C2semidirect
8. #20292836C2semidirect
9. #20292835C2semidirect
11. #20292756C2semidirect
12. #20292732C2semidirect
13. #20292731C2semidirect
168. C107#3398semidirect

Maximal subgroups

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

subgroupindex
#202933552
#202933462
#202930222
#202930182
#202930052
#202928482
#202928382
#202928362
#202928352
#202928302
#202927562
#202927322
#202927312
#202927182
#464×C1072
#3398107