Finite Group Databasebeta

← #28339955

Subgroups — #28339955

Order 13696.

596 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812083C211yesmaximal, characteristic
2C2×#16812071C211yesmaximal, characteristic, complemented
3#20292554C212yesmaximal, complemented
4#20292543C212yesmaximal, complemented
5#20292528C211yesmaximal, characteristic, complemented
6#20292526C211yesmaximal, characteristic, complemented
7#20292503C212yesmaximal, complemented
8#20292492C211yesmaximal, characteristic, complemented
9#20292489C212yesmaximal, complemented
10C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
11#380×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C4×D107C2212yes—
15#16812083—24no—
16#16812071—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×C4⋊Dic107C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×C22⋊Dic107C2211yescharacteristic
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×C4×Dic107C2211yescharacteristic
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C23×C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2212yes—
37C23×D107C2312yes—
38C2×D428—22no—
39C2×D428—22no—
40C2×D428D414yes—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—42no—
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C2×C4×D107—24no—
50C2×C4×D107—24no—
51C2×C4×D107—42no—
52C4⋊Dic107—22no—
53C4⋊Dic107—42no—
54C22⋊Dic107—22no—
55C22⋊Dic107—22no—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—42no—
59(C2×D107)⋊C4—42no—
60C4×Dic107—22no—
61C4×Dic107—22no—
62C24×C107C2311yescharacteristic
63C22×C4×C107—22no—
64C22×C4×C107—22no—
65C22×C4×C107—22no—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C4⋊C4×C107—22no—
71C4⋊C4×C107—22no—
72C22⋊C4×C107—42no—
73C23×C107—21no—
74C23×C107—21no—
75C23×C107—21no—
76C23×C107—21no—
77C23×C107—21no—
78C23×C107—21no—
79C23×C107#4112yes—
80C23×C107C2411yescharacteristic
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107C2×D414yes—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—44no—
98C2×C4×C107C2×D412yes—
99C2×C4×C107#4112yes—
100C2×C4×C107#4112yes—
101C22⋊D107—48no—
102C2×Dic107—21no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—21no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113C2×Dic107—21no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107#4112yes—
117D428—28no—
118C4×D107—48no—
119C22×C107—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107C22×D411yescharacteristic
127C22×C107C2×#4111yescharacteristic
128C22×C107C2×#4111yescharacteristic
129C22×C107C2×#4111yescharacteristic
130C22×C107C2×#4111yescharacteristic
131C22×C107H(5,𝔽2)11yescharacteristic
132C22×C107H(5,𝔽2)11yescharacteristic
133C2×D107—24no—
134C2×D107—24no—
135C2×D107—24no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—42no—
139C2×D107—42no—
140C4×C107—41no—
141C4×C107—42no—
142C4×C107—42no—
143C4×C107#12314yes—
144Dic107—22no—
145Dic107—41no—
146Dic107—42no—
147Dic107—41no—
148Dic107—41no—
149C2×C107—42no—
150C2×C107C2×#12311yescharacteristic
151C2×C107#52911yescharacteristic
152C2×C107#54011yescharacteristic
153C2×C107#54611yescharacteristic
154C2×C107#54611yescharacteristic
155C2×C107#55311yescharacteristic
156C2×C107#55511yescharacteristic
157D107—44no—
158#2420—1071nomaximal, Sylow
159C107#242011yescharacteristic, complemented, Sylow
160C2×#121—1071no—
161C2×C4×D4—1071no—
162#397—1072no—
163#394—1072no—
164#390—1071no—
165#389—1071no—
166#386—1072no—
167#382—1071no—
168#381—1072no—
169#380D10711yescharacteristic, complemented
170C4×C22⋊C4—1071no—
171C22×D4—1072no—
172C22×D4—1071no—
173C23×C4—1072no—
174C23×C4C2×D10711yescharacteristic
175#121—2144no—
176C4×D4—2144no—
177C2×C4⋊C4—1072no—
178C2×C4⋊C4—1071no—
179C2×C4⋊C4C2×D10711yescharacteristic
180C2×C4⋊C4C2×D10711yescharacteristic
181C2×C22⋊C4—1072no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1071no—
184C2×C22⋊C4—1071no—
185C2×C22⋊C4—1072no—
186C2×C22⋊C4—1072no—
187C2×C22⋊C4C2×D10712yes—
188C2×C42—1071no—
189C2×C42—1071no—
190#95—1071no—
191#95—1071no—
192#95—1072no—
193#95—1072no—
194#95—1072no—
195#95C2×D10712yes—
196C24—1072no—
197C24C22×D10711yescharacteristic
198C2×D4—1074no—
199C2×D4—4282no—
200C2×D4—2144no—
201C2×D4—2144no—
202C2×D4—2142no—
203C2×D4—2142no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—1071no—
208C22×C4—1071no—
209C22×C4—1071no—
210C22×C4—1072no—
211C22×C4—1071no—
212C22×C4—4282no—
213C22×C4—2144no—
214C22×C4—2144no—
215C22×C4C22×D10711yescharacteristic
216C22×C4C22×D10712yes—
217C22×C4C22×D10712yes—
218C22×C4C22×D10711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—4282no—
222C4⋊C4—2142no—
223C22⋊C4—42no—
224C22⋊C4—4282no—
225C22⋊C4—4282no—
226C22⋊C4—2144no—
227C22⋊C4—2144no—
228C22⋊C4—2142no—
229C22⋊C4—2142no—
230C42—2142no—
231C42—2142no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—1074no—
239C23—2142no—
240C23—2142no—
241C23—2142no—
242C23—2142no—
243C23—2142no—
244C23—2142no—
245C23#1313381912yes—
246C23C23×D10711yescharacteristic
247D4—4288no—
248D4—2148no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—44no—
259C2×C4—1072no—
260C2×C4—4288no—
261C2×C4—2141no—
262C2×C4—2142no—
263C2×C4—2141no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4—2141no—
267C2×C4—2141no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4—2141no—
275C2×C4D4×D10712yes—
276C2×C4#1313381712yes—
277C2×C4#1313381812yes—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—4282no—
286C22—4282no—
287C22—4282no—
288C22—4282no—
289C22—2144no—
290C22—2144no—
291C22—2144no—
292C22C2×D4×D10711yescharacteristic
293C22#41×D10711yescharacteristic
294C22C2×#1313381711yescharacteristic
295C22C2×#1313381811yescharacteristic
296C22C2×#1313381911yescharacteristic
297C22#1681215411yescharacteristic
298C22#1681215511yescharacteristic
299C4—41no—
300C4—42no—
301C4—42no—
302C4—4282no—
303C4—4281no—
304C4—4281no—
305C4—4281no—
306C4—2142no—
307C4#1681209114yes—
308C2—42no—
309C2—4284no—
310C2C2×#1681209111yescharacteristic
311C2#2029316311yescharacteristic
312C2#2029323011yescharacteristic
313C2#2029326811yescharacteristic
314C2#2029327411yescharacteristic
315C2#2029332511yescharacteristic
316C2#2029334011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#380×C107C2
prime radicalC2×#16812083C2
hypercenter#380D107
hypocenterC107#2420
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16812071C2semidirect
3. #20292554C2semidirect
4. #20292543C2semidirect
5. #20292528C2semidirect
6. #20292526C2semidirect
7. #20292503C2semidirect
8. #20292492C2semidirect
9. #20292489C2semidirect
10. C4×C22⋊Dic107C2semidirect
11. #380×C107C2semidirect
159. C107#2420semidirect
169. #380D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120712
#202925542
#202925432
#202925282
#202925262
#202925032
#202924922
#202924892
C4×C22⋊Dic1072
#380×C1072
#2420107