Finite Group Databasebeta

← #28339969

Subgroups — #28339969

Order 13696.

652 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812092C212yesmaximal, complemented
2C2×#16812080C211yesmaximal, characteristic
3#20292555C212yesmaximal, complemented
4#20292542C212yesmaximal, complemented
5#20292530C211yesmaximal, characteristic, complemented
6#20292525C211yesmaximal, characteristic, complemented
7#20292500C212yesmaximal, complemented
8#20292496C211yesmaximal, characteristic, complemented
9#20292484C211yesmaximal, characteristic, complemented
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
11C2×C4×D4×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812092—28no—
15#16812080—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×C4×Dic107C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2211yescharacteristic
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—24no—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—42no—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107—22no—
52Dic107⋊C4—24no—
53Dic107⋊C4—24no—
54Dic107⋊C4—42no—
55C22⋊Dic107—24no—
56C22⋊Dic107—24no—
57C22⋊Dic107—24no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4D414yescomplemented
61C4×Dic107—22no—
62C24×C107C2312yes—
63C2×D4×C107—22no—
64C2×D4×C107—22no—
65C2×D4×C107D414yes—
66C22×C4×C107—24no—
67C22×C4×C107—24no—
68C22×C4×C107—42no—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C4⋊C4×C107—22no—
74C22⋊C4×C107—24no—
75C42×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107C2×D414yes—
83C23×C107C2411yescharacteristic
84C22×D107—21no—
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107C2×D412yes—
91D4×C107—28no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—48no—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D412yes—
103C2×C4×C107#4112yes—
104C2×C4×C107#4112yes—
105C22⋊D107—48no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—21no—
110C2×Dic107—22no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107C2×D412yes—
120C2×Dic107#4112yes—
121C4×D107—44no—
122C22×C107—24no—
123C22×C107—24no—
124C22×C107—24no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107C22×D411yescharacteristic
130C22×C107C22×D411yescharacteristic
131C22×C107C2×#4111yescharacteristic
132C22×C107C2×#4111yescharacteristic
133C22×C107C2×#4111yescharacteristic
134C22×C107H(5,𝔽2)11yescharacteristic
135C22×C1072−1+411yescharacteristic
136C2×D107—22no—
137C2×D107—22no—
138C2×D107—22no—
139C2×D107—41no—
140C2×D107—41no—
141C2×D107—41no—
142C2×D107—41no—
143C4×C107—22no—
144C4×C107—42no—
145C4×C107—41no—
146C4×C107#12314yescomplemented
147Dic107—22no—
148Dic107—22no—
149Dic107—41no—
150Dic107—42no—
151Dic107—42no—
152C2×C107—44no—
153C2×C107C2×#12311yescharacteristic
154C2×C107#53211yescharacteristic
155C2×C107#54211yescharacteristic
156C2×C107D4211yescharacteristic
157C2×C107#54711yescharacteristic
158C2×C107#54711yescharacteristic
159C2×C107#55311yescharacteristic
160D107—42no—
161#2422—1071nomaximal, Sylow
162C107#242211yescharacteristic, complemented, Sylow
163C2×#123—1072no—
164C2×#121—1071no—
165C2×C4×D4D10711yescharacteristic, complemented
166#397—1072no—
167#394—1072no—
168#390—1071no—
169#389—1071no—
170#385—1072no—
171#384—1071no—
172#380—1071no—
173C4×C22⋊C4—1071no—
174C22×D4—1072no—
175C22×D4C2×D10711yescharacteristic
176C23×C4—1071no—
177C23×C4C2×D10712yes—
178#123—2148no—
179#121—2144no—
180C4×D4—24no—
181C2×C4⋊C4—1072no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4—1072no—
184C2×C4⋊C4—1072no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4C2×D10711yescharacteristic
187C2×C22⋊C4—1071no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1072no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4C2×D10712yes—
194C2×C42—1071no—
195C2×C42C2×D10711yescharacteristic
196#95—1071no—
197#95—1071no—
198#95—1072no—
199#95—1072no—
200C24—1071no—
201C24C22×D10712yes—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—4282no—
205C2×D4—2144no—
206C2×D4—2144no—
207C2×D4C22⋊D10714yes—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—1072no—
215C22×C4—1072no—
216C22×C4—2142no—
217C22×C4—2142no—
218C22×C4—2142no—
219C22×C4C22×D10711yescharacteristic
220C22×C4C22×D10712yes—
221C22×C4C22×D10711yescharacteristic
222C22×C4C22×D10711yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—4282no—
225C4⋊C4—2144no—
226C4⋊C4—2144no—
227C4⋊C4—2142no—
228C4⋊C4—2142no—
229C22⋊C4—24no—
230C22⋊C4—1074no—
231C22⋊C4—2142no—
232C22⋊C4—2144no—
233C22⋊C4—2144no—
234C22⋊C4—2144no—
235C22⋊C4—2144no—
236C42—22no—
237C42—2142no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—1072no—
245C23—2141no—
246C23—2141no—
247C23—2141no—
248C23—2141no—
249C23—2141no—
250C23—2141no—
251C23C2×C22⋊D10714yes—
252C23C23×D10711yescharacteristic
253D4—28no—
254D4—4288no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—48no—
264C2×C4—1072no—
265C2×C4—1072no—
266C2×C4—4284no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4—2142no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4—2141no—
275C2×C4—2142no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2141no—
279C2×C4—2141no—
280C2×C4D4×D10712yes—
281C2×C4C2×C22⋊D10712yes—
282C2×C4#1313381812yes—
283C2×C4#1313381812yes—
284C22—24no—
285C22—24no—
286C22—24no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—4281no—
292C22—4281no—
293C22—4281no—
294C22—4281no—
295C22—2142no—
296C22—2142no—
297C22—2142no—
298C22C2×D4×D10711yescharacteristic
299C22C22×C22⋊D10711yescharacteristic
300C22#41×D10711yescharacteristic
301C22C2×#1313381811yescharacteristic
302C22C2×#1313381811yescharacteristic
303C22#1681215411yescharacteristic
304C22#1681215611yescharacteristic
305C4—22no—
306C4—42no—
307C4—41no—
308C4—4281no—
309C4—4282no—
310C4—4282no—
311C4—2142no—
312C4—2142no—
313C4#1681209214yescomplemented
314C2—44no—
315C2—4282no—
316C2C2×#1681209211yescharacteristic
317C2#2029317411yescharacteristic
318C2#2029324211yescharacteristic
319C2D4×C22⋊D10711yescharacteristic
320C2#2029327611yescharacteristic
321C2#2029328211yescharacteristic
322C2#2029332711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4×D4×C107C2
prime radicalC2×#16812080C2
hypercenterC2×C4×D4D107
hypocenterC107#2422
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812092C2semidirect
3. #20292555C2semidirect
4. #20292542C2semidirect
5. #20292530C2semidirect
6. #20292525C2semidirect
7. #20292500C2semidirect
8. #20292496C2semidirect
9. #20292484C2semidirect
10. C4×(C2×D107)⋊C4C2semidirect
11. C2×C4×D4×C107C2semidirect
60. (C2×D107)⋊C4D4semidirect
146. C4×C107#123semidirect
162. C107#2422semidirect
165. C2×C4×D4D107semidirect
313. C4#16812092semidirect

Maximal subgroups

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

subgroupindex
C2×#168120922
C2×#168120802
#202925552
#202925422
#202925302
#202925252
#202925002
#202924962
#202924842
C4×(C2×D107)⋊C42
C2×C4×D4×C1072
#2422107