Finite Group Databasebeta

← #28340073

Subgroups — #28340073

Order 13696.

914 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812081C212yesmaximal, complemented
3#20292547C211yesmaximal, characteristic, complemented
4#20292544C211yesmaximal, characteristic, complemented
5#20292539C211yesmaximal, characteristic, complemented
6#20292536C211yesmaximal, characteristic, complemented
7#20292504C212yesmaximal, complemented
8#20292503C212yesmaximal, complemented
9#20292498C212yesmaximal, complemented
10C2×C107⋊#95C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—28no—
13C2×#13133818—24no—
14C22×Dic214C2211yescharacteristic
15C22×D428C2211yescharacteristic
16C22×C22⋊D107C2212yes—
17C22×C22⋊D107C2212yes—
18C23×Dic107C2212yes—
19C22×C4×D107C2212yes—
20#16812081—28no—
21C2×Dic107⋊C4C2212yes—
22C2×C22⋊Dic107C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C107⋊#95C2212yes—
27C107⋊#95C2211yescharacteristic
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C22×D4×C107C2211yescharacteristic
32C23×C4×C107C2211yescharacteristic
33#123×C107—24no—
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2211yescharacteristic
38C2×C22⋊C4×C107C2211yescharacteristic
39C23×D107C2312yes—
40#13133818—232no—
41C2×Dic214—22no—
42C2×Dic214—24no—
43C2×D428—24no—
44C2×D428—22no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—24no—
48C2×C22⋊D107—24no—
49C2×C22⋊D107—42no—
50C2×C22⋊D107D418yescomplemented
51C22×Dic107—24no—
52C22×Dic107—24no—
53C22×Dic107—42no—
54C22×Dic107C2312yes—
55C22×Dic107C2312yes—
56C22×Dic107C2312yes—
57C2×C4×D107—24no—
58C2×C4×D107—24no—
59C2×C4×D107—24no—
60Dic107⋊C4—24no—
61C22⋊Dic107—24no—
62(C2×D107)⋊C4—24no—
63(C2×D107)⋊C4—24no—
64(C2×D107)⋊C4—42no—
65C24×C107C2311yescharacteristic
66C24×C107C2311yescharacteristic
67C2×D4×C107—24no—
68C2×D4×C107—22no—
69C22×C4×C107—24no—
70C22×C4×C107D414yescomplemented
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C4⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—22no—
81C23×C107—22no—
82C23×C107—21no—
83C23×C107—21no—
84C23×C107—21no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—21no—
88C23×C107—21no—
89C23×C107C2×D414yes—
90C23×C107C2×D412yes—
91C23×C107C2411yescharacteristic
92C22×D107—22no—
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—22no—
97C22×D107—22no—
98C22×D107C2×D414yes—
99D4×C107—44no—
100C2×C4×C107—28no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107—22no—
107C2×C4×C107—22no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2×D412yes—
112C22⋊D107—216no—
113C22⋊D107—48no—
114C2×Dic107—22no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—22no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107—22no—
121C2×Dic107—48no—
122C2×Dic107C2×D414yes—
123C2×Dic107#4114yescomplemented
124D428—28no—
125C4×D107—216no—
126Dic214—28no—
127C22×C107—22no—
128C22×C107—22no—
129C22×C107—24no—
130C22×C107—24no—
131C22×C107—41no—
132C22×C107—41no—
133C22×C107—41no—
134C22×C107—42no—
135C22×C107—42no—
136C22×C107#12014yes—
137C22×C107C22×D412yes—
138C22×C107C22×D411yescharacteristic
139C22×C107C2×#4112yes—
140C22×C107H(5,𝔽2)11yescharacteristic
141C22×C1072−1+411yescharacteristic
142C2×D107—24no—
143C2×D107—24no—
144C2×D107—24no—
145C2×D107—42no—
146C2×D107—42no—
147C2×D107—42no—
148C2×D107—42no—
149C4×C107—22no—
150C4×C107—22no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107—41no—
154Dic107—24no—
155Dic107—24no—
156Dic107—42no—
157C2×C107—24no—
158C2×C107—81no—
159C2×C107C2×#12011yescharacteristic
160C2×C107#53211yescharacteristic
161C2×C107#53711yescharacteristic
162C2×C107#54612yes—
163C2×C107#54712yes—
164D107—44no—
165#2430—1071nomaximal, Sylow
166C107#243011yescharacteristic, complemented, Sylow
167C22×#41—1071no—
168C2×#123D10711yescharacteristic, complemented
169C2×#121—1072no—
170#395—1071no—
171#394—1071no—
172#393—1071no—
173#392—1071no—
174#386—1072no—
175#386—1072no—
176#385—1072no—
177C2×#95—1071no—
178C2×#41—2148no—
179C2×#41—2144no—
180C22×Q8—1071no—
181C22×D4—1072no—
182C22×D4—1072no—
183C22×D4—1071no—
184C22×D4C2×D10711yescharacteristic
185C23×C4—1072no—
186C23×C4—1072no—
187C23×C4C2×D10711yescharacteristic
188#123—24no—
189#121—2148no—
190C2×C4⋊C4—1072no—
191C2×C4⋊C4C2×D10711yescharacteristic
192C2×C4⋊C4C2×D10711yescharacteristic
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1072no—
197C2×C22⋊C4C2×D10711yescharacteristic
198C2×C22⋊C4C2×D10711yescharacteristic
199C2×C22⋊C4C2×D10711yescharacteristic
200#95—1072no—
201#95—1071no—
202#95—1071no—
203#95—1072no—
204#95—1072no—
205C24—1072no—
206C24C22×D10711yescharacteristic
207C24C22×D10711yescharacteristic
208#41—21432no—
209C2×Q8—2144no—
210C2×Q8—2142no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—1078no—
214C2×D4—4282no—
215C2×D4—2144no—
216C2×D4—2144no—
217C2×D4—2144no—
218C2×D4—2144no—
219C2×D4—2144no—
220C2×D4—2142no—
221C22×C4—24no—
222C22×C4—1072no—
223C22×C4—1072no—
224C22×C4—1072no—
225C22×C4—4282no—
226C22×C4—2144no—
227C22×C4—2144no—
228C22×C4—2144no—
229C22×C4—2144no—
230C22×C4—2144no—
231C22×C4C22⋊D10714yescomplemented
232C22×C4C22×D10711yescharacteristic
233C22×C4C22×D10711yescharacteristic
234C22×C4C22×D10711yescharacteristic
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10711yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—2144no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—4282no—
244C22⋊C4—2144no—
245C22⋊C4—2144no—
246C22⋊C4—2144no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—1074no—
256C23—2142no—
257C23—2142no—
258C23—2142no—
259C23—2142no—
260C23—2142no—
261C23—2142no—
262C23D4×D10714yes—
263C23C2×C22⋊D10712yes—
264C23C23×D10711yescharacteristic
265Q8—2148no—
266D4—44no—
267D4—4288no—
268D4—2148no—
269D4—21416no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—1074no—
281C2×C4—1074no—
282C2×C4—4288no—
283C2×C4—21416no—
284C2×C4—2142no—
285C2×C4—2142no—
286C2×C4—2142no—
287C2×C4—2142no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4—2142no—
291C2×C4C2×C22⋊D10712yes—
292C2×C4C2×C22⋊D10712yes—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—4282no—
303C22—4282no—
304C22—4282no—
305C22—4282no—
306C22—2144no—
307C22—2144no—
308C22—2144no—
309C22#1681207714yes—
310C22C2×D4×D10712yes—
311C22C22×C22⋊D10711yescharacteristic
312C22#41×D10712yes—
313C22#1681215511yescharacteristic
314C22#1681215611yescharacteristic
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—4282no—
321C4—2144no—
322C4—2144no—
323C2—24no—
324C2—81no—
325C2—4284no—
326C2C2×#1681207711yescharacteristic
327C2#2029317311yescharacteristic
328C2#2029320511yescharacteristic
329C2#2029326912yes—
330C2#2029327512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
hypercenterC2×#123D107
hypocenterC107#2430
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812081C2semidirect
3. #20292547C2semidirect
4. #20292544C2semidirect
5. #20292539C2semidirect
6. #20292536C2semidirect
7. #20292504C2semidirect
8. #20292503C2semidirect
9. #20292498C2semidirect
10. C2×C107⋊#95C2semidirect
11. C2×#123×C107C2semidirect
50. C2×C22⋊D107D4semidirect
70. C22×C4×C107D4semidirect
123. C2×Dic107#41semidirect
166. C107#2430semidirect
168. C2×#123D107semidirect
231. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168120812
#202925472
#202925442
#202925392
#202925362
#202925042
#202925032
#202924982
C2×C107⋊#952
C2×#123×C1072
#2430107