Finite Group Databasebeta

← #28339148

Subgroups — #28339148

Order 13696.

668 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×#16812080C211yesmaximal, characteristic
2C2×#16812066C211yesmaximal, characteristic, complemented
3#20292530C211yesmaximal, characteristic, complemented
4#20292527C212yesmaximal, complemented
5#20292523C211yesmaximal, characteristic, complemented
6#20292521C212yesmaximal, complemented
7#20292506C212yesmaximal, complemented
8#20292500C212yesmaximal, complemented
9#20292484C211yesmaximal, characteristic, complemented
10C4×C4⋊Dic107C211yesmaximal, characteristic, complemented
11C2×C4×D4×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812080C418yescomplemented
15#16812066—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×C4×Dic107C2211yescharacteristic
28C2×C4×Dic107C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C23×C4×C107C2212yes—
34C4×D4×C107—24no—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37C2×C42×C107C2211yescharacteristic
38C23×D107C2311yescharacteristic
39C2×C22⋊D107—42no—
40C2×C22⋊D107—42no—
41C2×C22⋊D107C2×C418yescomplemented
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107C2×C414yescomplemented
50C4⋊Dic107—22no—
51C4⋊Dic107C2×C414yescomplemented
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—42no—
55C22⋊Dic107—42no—
56C22⋊Dic107—42no—
57C22⋊Dic107C2×C418yescomplemented
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—42no—
61C4×Dic107—22no—
62C4×Dic107—42no—
63C24×C107C2312yes—
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107C2×C414yes—
67C22×C4×C107—24no—
68C22×C4×C107—42no—
69C22×C4×C107—42no—
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—22no—
75C22⋊C4×C107—24no—
76C42×C107D414yescomplemented
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107C22×C414yes—
84C23×C107C2411yescharacteristic
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107C22×C412yes—
92D4×C107—28no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—48no—
101C2×C4×C107C22×C412yes—
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yes—
106C22⋊D107—48no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107C22×C412yes—
121C2×Dic107C22×C414yes—
122C4×D107—28no—
123C22×C107—24no—
124C22×C107—24no—
125C22×C107—24no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107C23×C411yescharacteristic
131C22×C107C22×D411yescharacteristic
132C22×C107C2×#4111yescharacteristic
133C22×C107H(5,𝔽2)11yescharacteristic
134C22×C107H(5,𝔽2)11yescharacteristic
135C22×C107H(5,𝔽2)11yescharacteristic
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—22no—
146C4×C107—42no—
147C4×C107C4×D414yes—
148Dic107—22no—
149Dic107—24no—
150Dic107—41no—
151Dic107—41no—
152Dic107—42no—
153C2×C107—44no—
154C2×C107C2×C4×D411yescharacteristic
155C2×C107#51811yescharacteristic
156C2×C107#52011yescharacteristic
157C2×C107#53511yescharacteristic
158C2×C107#53711yescharacteristic
159C2×C107#54011yescharacteristic
160C2×C107#54211yescharacteristic
161D107—42no—
162#2324—1071nomaximal, Sylow
163C107#232411yescharacteristic, complemented, Sylow
164C2×#121—1071no—
165C2×C4×D4D10711yescharacteristic, complemented
166C2×#117—1071no—
167#390—1071no—
168#390—1072no—
169#389—1071no—
170#388—1072no—
171#386—1072no—
172#385—1072no—
173#380—1071no—
174C4×C4⋊C4—1071no—
175C22×D4—1072no—
176C22×D4C2×D10711yescharacteristic
177C23×C4—1071no—
178C23×C4C2×D10712yes—
179#121—1078no—
180C4×D4—24no—
181#117—2144no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4—1072no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4—1071no—
187C2×C4⋊C4C2×D10711yescharacteristic
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1072no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1071no—
194C2×C22⋊C4C2×D10712yes—
195C2×C42—1072no—
196C2×C42—1071no—
197C2×C42C2×D10711yescharacteristic
198#95—1072no—
199#95—1072no—
200#95—1072no—
201C24—1071no—
202C24C22×D10712yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—1078no—
206C2×D4—4282no—
207C2×D4—4282no—
208C2×D4C4×D10714yes—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—1072no—
213C22×C4—1071no—
214C22×C4—1074no—
215C22×C4—1071no—
216C22×C4—1072no—
217C22×C4—1071no—
218C22×C4—2142no—
219C22×C4—2142no—
220C22×C4C22×D10712yes—
221C22×C4C22×D10711yescharacteristic
222C22×C4C22×D10711yescharacteristic
223C22×C4C22×D10711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1074no—
226C4⋊C4—4282no—
227C4⋊C4—2142no—
228C4⋊C4—2142no—
229C4⋊C4—2142no—
230C22⋊C4—24no—
231C22⋊C4—1078no—
232C22⋊C4—4282no—
233C22⋊C4—4282no—
234C22⋊C4—4282no—
235C22⋊C4—2142no—
236C22⋊C4—2142no—
237C42—4282no—
238C42—2142no—
239C42C22⋊D10714yescomplemented
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—1072no—
247C23—2141no—
248C23—2141no—
249C23—2141no—
250C23—2141no—
251C23—2141no—
252C23—2141no—
253C23C2×C4×D10714yes—
254C23C23×D10711yescharacteristic
255D4—28no—
256D4—4288no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—48no—
265C2×C4—1072no—
266C2×C4—1074no—
267C2×C4—2141no—
268C2×C4—2141no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4—2142no—
272C2×C4—2141no—
273C2×C4—2142no—
274C2×C4—2141no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2142no—
279C2×C4—2142no—
280C2×C4—2148no—
281C2×C4C2×C4×D10712yes—
282C2×C4C2×C22⋊D10712yes—
283C2×C4C2×C22⋊D10712yes—
284C2×C4C2×C22⋊D10712yes—
285C2×C4#1313381812yes—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—4281no—
294C22—4281no—
295C22—4281no—
296C22—4281no—
297C22—2142no—
298C22—2142no—
299C22—2142no—
300C22C22×C4×D10711yescharacteristic
301C22C22×C22⋊D10711yescharacteristic
302C22C2×#1313381811yescharacteristic
303C22#1681215411yescharacteristic
304C22#1681215411yescharacteristic
305C22#1681215511yescharacteristic
306C22#1681215611yescharacteristic
307C4—22no—
308C4—22no—
309C4—42no—
310C4—4281no—
311C4—4281no—
312C4—4282no—
313C4—2142no—
314C4—2144no—
315C4C4×C22⋊D10714yes—
316C2—44no—
317C2—4282no—
318C2C2×C4×C22⋊D10711yescharacteristic
319C2#2029309911yescharacteristic
320C2#2029310611yescharacteristic
321C2#2029319411yescharacteristic
322C2#2029320511yescharacteristic
323C2#2029322811yescharacteristic
324C2#2029324211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC2×C4×D4×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812080C2
hypercenterC2×C4×D4D107
hypocenterC107#2324
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16812066C2semidirect
3. #20292530C2semidirect
4. #20292527C2semidirect
5. #20292523C2semidirect
6. #20292521C2semidirect
7. #20292506C2semidirect
8. #20292500C2semidirect
9. #20292484C2semidirect
10. C4×C4⋊Dic107C2semidirect
11. C2×C4×D4×C107C2semidirect
14. #16812080C4semidirect
41. C2×C22⋊D107C2×C4semidirect
49. C2×C4×D107C2×C4semidirect
51. C4⋊Dic107C2×C4semidirect
57. C22⋊Dic107C2×C4semidirect
76. C42×C107D4semidirect
163. C107#2324semidirect
165. C2×C4×D4D107semidirect
239. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120802
C2×#168120662
#202925302
#202925272
#202925232
#202925212
#202925062
#202925002
#202924842
C4×C4⋊Dic1072
C2×C4×D4×C1072
#2324107