Finite Group Databasebeta

← #28339891

Subgroups — #28339891

Order 13696.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812085C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D107C211yesmaximal, characteristic
3C2×#16812068C211yesmaximal, characteristic, complemented
4#20292555C212yesmaximal, complemented
5#20292542C212yesmaximal
6#20292534C211yesmaximal, characteristic, complemented
7#20292523C211yesmaximal, characteristic, complemented
8#20292522C212yesmaximal, complemented
9#20292517C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14#16812085—24no—
15C4×C22⋊D107—24no—
16#16812068—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×C4×Dic107C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C22×Q8×C107C2211yescharacteristic
34C23×C4×C107C2211yescharacteristic
35C22⋊Q8×C107—24no—
36C2×C4⋊C4×C107C2211yescharacteristic
37C2×C4⋊C4×C107C2212yes—
38C2×C22⋊C4×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×C22⋊D107—22no—
41C2×C22⋊D107—22no—
42C2×C22⋊D107—22no—
43C22×Dic107C2312yes—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2312yes—
47C22×Dic107C2311yescharacteristic
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C4⋊Dic107—22no—
52C4⋊Dic107D414yescomplemented
53Dic107⋊C4—22no—
54Dic107⋊C4—24no—
55Dic107⋊C4—24no—
56C22⋊Dic107—22no—
57C22⋊Dic107—24no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—24no—
61C4×Dic107—22no—
62C4×Dic107—24no—
63C24×C107C2311yescharacteristic
64C2×Q8×C107—22no—
65C2×Q8×C107—22no—
66C2×Q8×C107—22no—
67C22×C4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107D414yes—
70C22×C4×C107C2312yes—
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—22no—
75C4⋊C4×C107—24no—
76C22⋊C4×C107—24no—
77C23×C107—21no—
78C23×C107—21no—
79C23×C107—21no—
80C23×C107—21no—
81C23×C107—21no—
82C23×C107—21no—
83C23×C107C2×D412yes—
84C23×C107C2411yescharacteristic
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107#4112yes—
92Q8×C107—44no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—28no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yes—
106C22⋊D107—28no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109C2×Dic107—21no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—21no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—22no—
120C2×Dic107C2×D414yes—
121C2×Dic107#4112yes—
122C4×D107—44no—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—41no—
127C22×C107—41no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107C22×D411yescharacteristic
131C22×C107C22×D411yescharacteristic
132C22×C107C2×#4111yescharacteristic
133C22×C107C2×#4111yescharacteristic
134C22×C107C2×#4111yescharacteristic
135C22×C1072−1+411yescharacteristic
136C22×C1072−1+411yescharacteristic
137C2×D107—22no—
138C2×D107—22no—
139C2×D107—22no—
140C2×D107—41no—
141C2×D107—41no—
142C2×D107—41no—
143C2×D107—41no—
144C4×C107—22no—
145C4×C107—42no—
146C4×C107—42no—
147C4×C107#12114yes—
148Dic107—22no—
149Dic107—24no—
150Dic107—41no—
151Dic107—41no—
152Dic107—42no—
153C2×C107—42no—
154C2×C107C2×#12111yescharacteristic
155C2×C107#52511yescharacteristic
156C2×C107#53611yescharacteristic
157C2×C107#54711yescharacteristic
158C2×C107#54811yescharacteristic
159C2×C107#55211yescharacteristic
160C2×C107#55611yescharacteristic
161D107—42no—
162#2415—1071nomaximal, Sylow
163C107#241511yescharacteristic, complemented, Sylow
164C2×C22⋊Q8—1071no—
165C2×C22⋊Q8D10711yescharacteristic, complemented
166C2×C4×D4—1071no—
167C2×#117—1071no—
168#397—1072no—
169#394—1072no—
170#391—1071no—
171#389—1072no—
172#389—1071no—
173#388—1072no—
174#380—1071no—
175C22×Q8C2×D10711yescharacteristic
176C22×D4—1071no—
177C23×C4—1071no—
178C23×C4C2×D10711yescharacteristic
179C22⋊Q8—24no—
180C22⋊Q8—2144no—
181C4×D4—2144no—
182#117—2144no—
183C2×C4⋊C4—1071no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4—1072no—
187C2×C4⋊C4—1072no—
188C2×C4⋊C4C2×D10711yescharacteristic
189C2×C4⋊C4C2×D10712yes—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1071no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4C2×D10712yes—
196C2×C42—1072no—
197C2×C42—1071no—
198#95—1072no—
199#95—1072no—
200#95—1072no—
201#95—1072no—
202C24—1071no—
203C24C22×D10711yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2142no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—1071no—
213C22×C4—1072no—
214C22×C4—1071no—
215C22×C4—1072no—
216C22×C4—1071no—
217C22×C4—2142no—
218C22×C4—2142no—
219C22×C4—2142no—
220C22×C4C22⋊D10714yes—
221C22×C4C22×D10712yes—
222C22×C4C22×D10712yes—
223C22×C4C22×D10711yescharacteristic
224C22×C4C22×D10711yescharacteristic
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—1074no—
228C4⋊C4—2144no—
229C4⋊C4—2144no—
230C4⋊C4—2142no—
231C4⋊C4—2142no—
232C22⋊C4—24no—
233C22⋊C4—2144no—
234C22⋊C4—2144no—
235C22⋊C4—2144no—
236C22⋊C4—2142no—
237C22⋊C4—2142no—
238C42—2144no—
239C42—2142no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—1072no—
247C23—2141no—
248C23—2141no—
249C23—2141no—
250C23—2141no—
251C23—2141no—
252C23—2141no—
253C23C2×C22⋊D10712yes—
254C23C23×D10711yescharacteristic
255Q8—44no—
256D4—2148no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1074no—
267C2×C4—1072no—
268C2×C4—4284no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2142no—
274C2×C4—2141no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2142no—
278C2×C4—2141no—
279C2×C4—2142no—
280C2×C4—2141no—
281C2×C4—2142no—
282C2×C4D4×D10712yes—
283C2×C4C2×C22⋊D10712yes—
284C2×C4C2×C22⋊D10712yes—
285C2×C4#1313381712yes—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—4281no—
294C22—4281no—
295C22—4281no—
296C22—4281no—
297C22—2142no—
298C22—2142no—
299C22—2142no—
300C22C2×D4×D10711yescharacteristic
301C22C22×C22⋊D10711yescharacteristic
302C22#41×D10711yescharacteristic
303C22#41×D10711yescharacteristic
304C22C2×#1313381711yescharacteristic
305C22#1681215711yescharacteristic
306C22#1681215711yescharacteristic
307C4—22no—
308C4—42no—
309C4—42no—
310C4—4281no—
311C4—4282no—
312C4—4281no—
313C4—2142no—
314C4—2144no—
315C4#1681208114yes—
316C2—42no—
317C2—4282no—
318C2C2×#1681208111yescharacteristic
319C2#2029313411yescharacteristic
320C2#2029320111yescharacteristic
321C2#2029327911yescharacteristic
322C2#2029328711yescharacteristic
323C2#2029331111yescharacteristic
324C2#2029334811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC22×C22⋊D107C22
hypercenterC2×C22⋊Q8D107
hypocenterC107#2415
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812085C2semidirect
3. C2×#16812068C2semidirect
4. #20292555C2semidirect
6. #20292534C2semidirect
7. #20292523C2semidirect
8. #20292522C2semidirect
9. #20292517C2semidirect
10. #20292484C2semidirect
11. C2×C22⋊Q8×C107C2semidirect
52. C4⋊Dic107D4semidirect
163. C107#2415semidirect
165. C2×C22⋊Q8D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C2×C4×C22⋊D1072
C2×#168120682
#202925552
#202925422
#202925342
#202925232
#202925222
#202925172
#202924842
C2×C22⋊Q8×C1072
#2415107