Finite Group Databasebeta

← #28340144

Subgroups — #28340144

Order 13696.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812087C211yesmaximal, characteristic
2C2×#16812066C211yesmaximal, characteristic
3#20292565C211yesmaximal, characteristic, complemented
4#20292558C211yesmaximal, characteristic, complemented
5#20292557C211yesmaximal, characteristic, complemented
6#20292547C211yesmaximal, characteristic, complemented
7#20292531C211yesmaximal, characteristic, complemented
8#20292511C211yesmaximal, characteristic, complemented
9#20292510C211yesmaximal, characteristic, complemented
10#20292498C211yesmaximal, characteristic
11#20292498C211yesmaximal, characteristic
12#20292489C211yesmaximal, characteristic
13#20292484C211yesmaximal, characteristic
14#95×D107C211yesmaximal, characteristic
15#389×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812087—24no—
21#16812066—24no—
22C2×C4⋊Dic107C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×C4×Dic107C2211yescharacteristic
34C107⋊#95C2211yescharacteristic
35C107⋊#95C2211yescharacteristic
36C107⋊#95C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C2×C4⋊C4×C107C2211yescharacteristic
47C2×C4⋊C4×C107C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C4⋊C4×C107C2211yescharacteristic
50C2×C42×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53C23×D107C2311yescharacteristic
54C2×Dic214—22no—
55C2×Dic214—22no—
56C2×Dic214—41no—
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C2×C4×D107—22no—
65C2×C4×D107—22no—
66C2×C4×D107—22no—
67C2×C4×D107—22no—
68C2×C4×D107—22no—
69C2×C4×D107—22no—
70C2×C4×D107—41no—
71C2×C4×D107—41no—
72C2×C4×D107Q814yescomplemented
73C4⋊Dic107—22no—
74Dic107⋊C4—22no—
75Dic107⋊C4—22no—
76Dic107⋊C4—22no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—41no—
80(C2×D107)⋊C4—22no—
81(C2×D107)⋊C4—22no—
82(C2×D107)⋊C4—22no—
83(C2×D107)⋊C4D414yes—
84C4×Dic107—22no—
85C22×C4×C107C2311yescharacteristic
86C22×C4×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C4⋊C4×C107—22no—
93C4⋊C4×C107—22no—
94C4⋊C4×C107—41no—
95C4⋊C4×C107—41no—
96C42×C107—22no—
97C23×C107C2411yescharacteristic
98C22×D107—21no—
99C22×D107—21no—
100C22×D107—21no—
101C22×D107—21no—
102C22×D107C2×D412yes—
103C22×D107C2×Q812yes—
104C22×D107#4112yes—
105C2×C4×C107—21no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D412yes—
124C2×C4×C107C2×Q812yes—
125C2×C4×C107#4112yes—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107C2×D412yes—
145C2×Dic107C2×Q812yes—
146C2×Dic107#4112yes—
147C4×D107—28no—
148C4×D107—44no—
149C4×D107—44no—
150Dic214—44no—
151C22×C107C22×D411yescharacteristic
152C22×C107C22×Q811yescharacteristic
153C22×C107C2×#4111yescharacteristic
154C22×C107C2×#4111yescharacteristic
155C22×C107C2×#4111yescharacteristic
156C22×C107H(5,𝔽2)11yescharacteristic
157C22×C1072−1+411yescharacteristic
158C2×D107—22no—
159C2×D107—22no—
160C2×D107—22no—
161C2×D107—22no—
162C2×D107—22no—
163C2×D107—22no—
164C2×D107C22⋊Q814yes—
165C4×C107—22no—
166C4×C107—22no—
167C4×C107—22no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107—41no—
172Dic107—22no—
173Dic107—22no—
174Dic107—22no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178Dic107—41no—
179C2×C107C2×C22⋊Q811yescharacteristic
180C2×C107#53311yescharacteristic
181C2×C107#54411yescharacteristic
182C2×C107#54611yescharacteristic
183C2×C107#54711yescharacteristic
184C2×C107#55111yescharacteristic
185C2×C107#55211yescharacteristic
186D107—24no—
187#2438—1071nomaximal, Sylow
188C107#243811yescharacteristic, complemented, Sylow
189C2×C22⋊Q8—1071no—
190C2×#117—1071no—
191#400—1071no—
192#398—1071no—
193#398—1071no—
194#395—1071no—
195#391—1071no—
196#389D10711yescharacteristic, complemented
197#387—1071no—
198#387—1071no—
199#385—1071no—
200#385—1071no—
201#381—1071no—
202#380—1071no—
203C2×#95—1071no—
204C22×Q8—1071no—
205C23×C4—1071no—
206C23×C4—1071no—
207C23×C4—1071no—
208C22⋊Q8—2144no—
209#117—2144no—
210C2×C4⋊C4—1071no—
211C2×C4⋊C4—1071no—
212C2×C4⋊C4—1071no—
213C2×C4⋊C4—1071no—
214C2×C4⋊C4—1071no—
215C2×C4⋊C4—1071no—
216C2×C4⋊C4—1071no—
217C2×C4⋊C4C2×D10711yescharacteristic
218C2×C4⋊C4C2×D10711yescharacteristic
219C2×C4⋊C4C2×D10711yescharacteristic
220C2×C4⋊C4C2×D10711yescharacteristic
221C2×C22⋊C4—1071no—
222C2×C22⋊C4—1071no—
223C2×C22⋊C4—1071no—
224C2×C22⋊C4—1071no—
225C2×C42—1071no—
226C2×C42C2×D10711yescharacteristic
227#95—1071no—
228#95—1071no—
229#95—1071no—
230#95—1071no—
231#95—1071no—
232#95—1071no—
233#95—1071no—
234#95—1071no—
235#95—1071no—
236#95—1071no—
237#95—1071no—
238#95—1071no—
239#95C2×D10711yescharacteristic
240#95C2×D10711yescharacteristic
241C24—1071no—
242C2×Q8—4281no—
243C2×Q8—2142no—
244C2×Q8—2142no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1074no—
249C22×C4—1071no—
250C22×C4—1071no—
251C22×C4—1071no—
252C22×C4—1071no—
253C22×C4—4281no—
254C22×C4—4281no—
255C22×C4—2142no—
256C22×C4—2142no—
257C22×C4—2142no—
258C22×C4—2142no—
259C22×C4—2142no—
260C22×C4—2142no—
261C22×C4C22×D10711yescharacteristic
262C22×C4C22×D10711yescharacteristic
263C22×C4C22×D10711yescharacteristic
264C22×C4C22×D10711yescharacteristic
265C22×C4C22×D10711yescharacteristic
266C22×C4C22×D10711yescharacteristic
267C22×C4C22×D10711yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—4281no—
273C4⋊C4—2142no—
274C4⋊C4—2142no—
275C4⋊C4—2142no—
276C4⋊C4—2142no—
277C4⋊C4—2142no—
278C4⋊C4—2142no—
279C22⋊C4—1074no—
280C22⋊C4—2142no—
281C22⋊C4—2142no—
282C22⋊C4—2142no—
283C42—22no—
284C42—2142no—
285C23—1072no—
286C23—1072no—
287C23—1072no—
288C23—2141no—
289C23—2141no—
290C23—2141no—
291C23—2141no—
292C23C23×D10711yescharacteristic
293Q8—4284no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—1072no—
313C2×C4—1072no—
314C2×C4—1072no—
315C2×C4—4284no—
316C2×C4—4284no—
317C2×C4—2148no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4—2141no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4D4×D10712yes—
337C2×C4Q8×D10712yes—
338C2×C4#1313381812yes—
339C22—1074no—
340C22—2142no—
341C22—2142no—
342C22—2142no—
343C22—2142no—
344C22—2142no—
345C22—2142no—
346C22C2×D4×D10711yescharacteristic
347C22C2×Q8×D10711yescharacteristic
348C22#41×D10711yescharacteristic
349C22#41×D10711yescharacteristic
350C22C2×#1313381811yescharacteristic
351C22#1681215411yescharacteristic
352C22#1681215711yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—4281no—
361C4—4281no—
362C4—4281no—
363C4—4281no—
364C4—2142no—
365C4—2142no—
366C4—2142no—
367C2—2144no—
368C2C22⋊Q8×D10711yescharacteristic
369C2#2029318111yescharacteristic
370C2#2029325411yescharacteristic
371C2#2029327411yescharacteristic
372C2#2029327811yescharacteristic
373C2#2029330611yescharacteristic
374C2#2029330911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#389×C107C2
prime radicalC23×D107C23
hypercenter#389D107
hypocenterC107#2438
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20292565C2semidirect
4. #20292558C2semidirect
5. #20292557C2semidirect
6. #20292547C2semidirect
7. #20292531C2semidirect
8. #20292511C2semidirect
9. #20292510C2semidirect
15. #389×C107C2semidirect
72. C2×C4×D107Q8semidirect
188. C107#2438semidirect
196. #389D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120872
C2×#168120662
#202925652
#202925582
#202925572
#202925472
#202925312
#202925112
#202925102
#202924982
#202924892
#202924842
#95×D1072
#389×C1072
#2438107