Finite Group Databasebeta

← #28339793

Subgroups — #28339793

Order 13696.

864 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133817C211yesmaximal, characteristic
2C2×#16812092C211yesmaximal, characteristic, complemented
3C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
4#20292547C212yesmaximal, complemented
5#20292542C212yesmaximal, complemented
6#20292507C211yesmaximal, characteristic, complemented
7#20292504C211yesmaximal, characteristic, complemented
8#20292499C212yesmaximal, complemented
9#20292498C212yesmaximal, complemented
10#20292488C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—24no—
13C2×#13133817—24no—
14C2×#13133817—24no—
15C22×Dic214C2211yescharacteristic
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic
20C23×Dic107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812092—24no—
23C2×Dic107⋊C4—24no—
24C2×Dic107⋊C4C2212yes—
25C2×Dic107⋊C4C2212yes—
26C2×Dic107⋊C4C2212yes—
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2212yes—
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C107⋊#95C2212yes—
35C22×D4×C107C2211yescharacteristic
36C23×C4×C107C2211yescharacteristic
37#123×C107—24no—
38C2×C4⋊C4×C107C2212yes—
39C2×C22⋊C4×C107C2211yescharacteristic
40C2×C22⋊C4×C107C2212yes—
41C23×D107C2311yescharacteristic
42#13133817—232no—
43C2×Dic214—22no—
44C2×Dic214—22no—
45C2×Dic214—22no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107D414yes—
52C22×Dic107—22no—
53C22×Dic107—22no—
54C22×Dic107—22no—
55C22×Dic107—22no—
56C22×Dic107—22no—
57C22×Dic107—22no—
58C22×Dic107—24no—
59C22×Dic107D414yes—
60C22×Dic107C2312yes—
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2312yes—
65C2×C4×D107—22no—
66C2×C4×D107—22no—
67C2×C4×D107—22no—
68Dic107⋊C4—24no—
69Dic107⋊C4—24no—
70Dic107⋊C4—24no—
71C22⋊Dic107—22no—
72C22⋊Dic107—24no—
73(C2×D107)⋊C4—24no—
74(C2×D107)⋊C4—24no—
75C24×C107C2311yescharacteristic
76C24×C107C2311yescharacteristic
77C2×D4×C107—22no—
78C2×D4×C107—22no—
79C2×D4×C107D414yes—
80C22×C4×C107—22no—
81C22×C4×C107—22no—
82C22×C4×C107—22no—
83C22×C4×C107C2312yes—
84C22×C4×C107C2312yes—
85C22×C4×C107C2311yescharacteristic
86C4⋊C4×C107—24no—
87C22⋊C4×C107—22no—
88C22⋊C4×C107—24no—
89C23×C107—22no—
90C23×C107—22no—
91C23×C107—21no—
92C23×C107—21no—
93C23×C107—22no—
94C23×C107—22no—
95C23×C107C2×D412yes—
96C23×C107C2×D412yes—
97C23×C107C2×D412yes—
98C23×C107C2×D412yes—
99C23×C107C2411yescharacteristic
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—22no—
103C22×D107—22no—
104C22×D107C2×D412yes—
105D4×C107—28no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—22no—
109C2×C4×C107—22no—
110C2×C4×C107—22no—
111C2×C4×C107—22no—
112C2×C4×C107—22no—
113C2×C4×C107—22no—
114C2×C4×C107—44no—
115C2×C4×C107C2×D412yes—
116C22⋊D107—28no—
117C22⋊D107—28no—
118C2×Dic107—28no—
119C2×Dic107—28no—
120C2×Dic107—28no—
121C2×Dic107—22no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—22no—
129C2×Dic107—22no—
130C2×Dic107—22no—
131C2×Dic107—22no—
132C2×Dic107C2×D412yes—
133C2×Dic107C2×D412yes—
134C2×Dic107C2×D412yes—
135C2×Dic107#4114yescomplemented
136C4×D107—28no—
137Dic214—28no—
138C22×C107—22no—
139C22×C107—22no—
140C22×C107—24no—
141C22×C107—24no—
142C22×C107—22no—
143C22×C107—24no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—42no—
147C22×C107#12014yes—
148C22×C107C22×D411yescharacteristic
149C22×C107C22×D411yescharacteristic
150C22×C107C22×D411yescharacteristic
151C22×C107C2×#4112yes—
152C22×C1072−1+412yes—
153C2×D107—22no—
154C2×D107—24no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—42no—
158C4×C107—22no—
159C4×C107—42no—
160C4×C107—42no—
161Dic107—22no—
162Dic107—22no—
163Dic107—22no—
164Dic107—24no—
165Dic107—42no—
166C2×C107—24no—
167C2×C107—42no—
168C2×C107C2×#12011yescharacteristic
169C2×C107#52511yescharacteristic
170C2×C107#53611yescharacteristic
171C2×C107#54712yes—
172C2×C107#54712yes—
173D107—42no—
174#2406—1071nomaximal, Sylow
175C107#240611yescharacteristic, complemented, Sylow
176C22×#41—1071no—
177C2×#123—1071no—
178C2×#123D10711yescharacteristic, complemented
179C22×C4⋊C4—1071no—
180#395—1072no—
181#394—1072no—
182#386—1071no—
183#386—1071no—
184#385—1072no—
185#385—1072no—
186#381—1071no—
187C2×#41—2144no—
188C2×#41—2144no—
189C2×#41—2144no—
190C22×Q8—1071no—
191C22×D4—1071no—
192C22×D4—1071no—
193C22×D4C2×D10711yescharacteristic
194C23×C4—1071no—
195C23×C4—1071no—
196C23×C4—1071no—
197C23×C4—1071no—
198C23×C4C2×D10711yescharacteristic
199#123—24no—
200#123—2144no—
201C2×C4⋊C4—1072no—
202C2×C4⋊C4—1072no—
203C2×C4⋊C4—1072no—
204C2×C4⋊C4—2144no—
205C2×C4⋊C4C2×D10712yes—
206C2×C22⋊C4—1071no—
207C2×C22⋊C4—1072no—
208C2×C22⋊C4—1072no—
209C2×C22⋊C4—1072no—
210C2×C22⋊C4C2×D10711yescharacteristic
211C2×C22⋊C4C2×D10712yes—
212#95—1072no—
213#95—1072no—
214#95—1072no—
215#95—1072no—
216C24—1071no—
217C24C22×D10711yescharacteristic
218C24C22×D10711yescharacteristic
219#41—21432no—
220C2×Q8—2142no—
221C2×Q8—2142no—
222C2×Q8—2142no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—1074no—
226C2×D4—2142no—
227C2×D4—2142no—
228C2×D4—2142no—
229C2×D4—2142no—
230C2×D4—2142no—
231C2×D4C22⋊D10714yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—1074no—
236C22×C4—1072no—
237C22×C4—1071no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1072no—
241C22×C4—2142no—
242C22×C4—2142no—
243C22×C4—2142no—
244C22×C4—2142no—
245C22×C4—2144no—
246C22×C4—2142no—
247C22×C4—2142no—
248C22×C4—2142no—
249C22×C4—2142no—
250C22×C4—2142no—
251C22×C4C22×D10712yes—
252C22×C4C22×D10712yes—
253C22×C4C22×D10711yescharacteristic
254C4⋊C4—24no—
255C4⋊C4—2144no—
256C4⋊C4—2144no—
257C4⋊C4—2144no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—2144no—
261C22⋊C4—2144no—
262C22⋊C4—2144no—
263C22⋊C4—2142no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—1072no—
271C23—2141no—
272C23—2141no—
273C23—2142no—
274C23—2142no—
275C23D4×D10712yes—
276C23D4×D10712yes—
277C23C2×C22⋊D10712yes—
278C23C2×C22⋊D10712yes—
279C23C23×D10711yescharacteristic
280Q8—2148no—
281D4—28no—
282D4—2148no—
283D4—2148no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—44no—
293C2×C4—1072no—
294C2×C4—1072no—
295C2×C4—1072no—
296C2×C4—1074no—
297C2×C4—2148no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4—2142no—
301C2×C4—2148no—
302C2×C4—2148no—
303C2×C4—2148no—
304C2×C4—2142no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4—2142no—
312C2×C4C2×C22⋊D10712yes—
313C22—22no—
314C22—22no—
315C22—24no—
316C22—24no—
317C22—24no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—4281no—
323C22—4281no—
324C22—4282no—
325C22—2144no—
326C22—2142no—
327C22#1681207714yes—
328C22C2×D4×D10711yescharacteristic
329C22C2×D4×D10711yescharacteristic
330C22C22×C22⋊D10711yescharacteristic
331C22#41×D10712yes—
332C22#1681215612yes—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—4282no—
337C4—2144no—
338C4—2142no—
339C4—2142no—
340C4—2142no—
341C2—24no—
342C2—42no—
343C2—4282no—
344C2C2×#1681207711yescharacteristic
345C2#2029313511yescharacteristic
346C2#2029319911yescharacteristic
347C2#2029327512yes—
348C2#2029327612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
prime radicalC22×#13133817C2
hypercenterC2×#123D107
hypocenterC107#2406
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16812092C2semidirect
3. C22×Dic107⋊C4C2semidirect
4. #20292547C2semidirect
5. #20292542C2semidirect
6. #20292507C2semidirect
7. #20292504C2semidirect
8. #20292499C2semidirect
9. #20292498C2semidirect
10. #20292488C2semidirect
11. C2×#123×C107C2semidirect
135. C2×Dic107#41semidirect
175. C107#2406semidirect
178. C2×#123D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168120922
C22×Dic107⋊C42
#202925472
#202925422
#202925072
#202925042
#202924992
#202924982
#202924882
C2×#123×C1072
#2406107