Finite Group Databasebeta

← #28344774

Subgroups — #28344774

Order 13696.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812102C211yesmaximal, characteristic, complemented
2C22×C4×Dic107C211yesmaximal, characteristic, complemented
3#20292521C211yesmaximal, characteristic, complemented
4#20292520C211yesmaximal, characteristic, complemented
5#20292515C211yesmaximal, characteristic, complemented
6#20292514C211yesmaximal, characteristic, complemented
7#20292512C211yesmaximal, characteristic, complemented
8#20292509C211yesmaximal, characteristic, complemented
9#20292507C211yesmaximal, characteristic
10#20292507C211yesmaximal, characteristic
11#20292502C211yesmaximal, characteristic, complemented
12#20292500C211yesmaximal, characteristic, complemented
13#20292499C211yesmaximal, characteristic, complemented
14#20292494C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic
16C23×Dic107C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18#16812102—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21C2×Dic107⋊C4C2211yescharacteristic
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C4×Dic107—24no—
32C2×C4×Dic107C2211yescharacteristic
33C2×C4×Dic107C2211yescharacteristic
34C2×C4×Dic107C2211yescharacteristic
35C2×C4×Dic107C2211yescharacteristic
36C107⋊#95C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C22×D4×C107C2211yescharacteristic
47C23×C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53C22×Dic107—22no—
54C22×Dic107—22no—
55C22×Dic107—22no—
56C22×Dic107—22no—
57C22×Dic107—22no—
58C22×Dic107—22no—
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C4⋊Dic107—22no—
68C4⋊Dic107—22no—
69Dic107⋊C4—22no—
70Dic107⋊C4—22no—
71Dic107⋊C4—22no—
72Dic107⋊C4—22no—
73C22⋊Dic107—22no—
74C22⋊Dic107—22no—
75C22⋊Dic107—22no—
76C22⋊Dic107—22no—
77C22⋊Dic107—22no—
78C22⋊Dic107—22no—
79C4×Dic107—22no—
80C4×Dic107—22no—
81C4×Dic107—22no—
82C4×Dic107D414yescomplemented
83C24×C107C2311yescharacteristic
84C24×C107C2311yescharacteristic
85C2×D4×C107—22no—
86C2×D4×C107—22no—
87C2×D4×C107—22no—
88C22×C4×C107—22no—
89C22×C4×C107—22no—
90C22×C4×C107—22no—
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96C22⋊C4×C107—22no—
97C22⋊C4×C107—22no—
98C22⋊C4×C107—22no—
99C23×C107—21no—
100C23×C107—21no—
101C23×C107—21no—
102C23×C107—21no—
103C23×C107—21no—
104C23×C107—21no—
105C23×C107—21no—
106C23×C107—21no—
107C23×C107—21no—
108C23×C107—21no—
109C23×C107—21no—
110C23×C107#4112yes—
111C23×C107#4112yes—
112C23×C107#4112yes—
113C23×C107C2411yescharacteristic
114D4×C107—44no—
115C2×C4×C107—28no—
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×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107C2×D412yes—
130C2×C4×C107#4112yes—
131C2×Dic107—28no—
132C2×Dic107—28no—
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×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107C2×D412yes—
154C2×Dic107C2×D412yes—
155C2×Dic107#4112yes—
156C2×Dic107#4112yes—
157C22×C107—22no—
158C22×C107—22no—
159C22×C107—22no—
160C22×C107—22no—
161C22×C107—22no—
162C22×C107—22no—
163C22×C107—41no—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—41no—
170C22×C107#12614yescomplemented
171C22×C107C22×D411yescharacteristic
172C22×C107C2×#4111yescharacteristic
173C22×C107C2×#4111yescharacteristic
174C22×C107C2×#4111yescharacteristic
175C22×C107C2×#4111yescharacteristic
176C22×C107C2×#4111yescharacteristic
177C22×C107C2×#4111yescharacteristic
178C4×C107—22no—
179C4×C107—22no—
180C4×C107—41no—
181C4×C107—41no—
182C4×C107—41no—
183Dic107—22no—
184Dic107—22no—
185Dic107—22no—
186Dic107—22no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190Dic107—41no—
191C2×C107—24no—
192C2×C107—81no—
193C2×C107#52511yescharacteristic
194C2×C107#52511yescharacteristic
195C2×C107#52511yescharacteristic
196C2×C107C2×#12611yescharacteristic
197C2×C107#52911yescharacteristic
198C2×C107#52911yescharacteristic
199C2×C107#52911yescharacteristic
200#2852—1071nomaximal, Sylow
201C107#285211yescharacteristic, complemented, Sylow
202C2×#126—1071no—
203C22×C42—1071no—
204#388—1071no—
205#388—1071no—
206#388—1071no—
207#387—1071no—
208#387—1071no—
209#387—1071no—
210#386—1071no—
211#386—1071no—
212#386D10711yescharacteristic
213#385—1071no—
214#385—1071no—
215#385—1071no—
216#383—1071no—
217C22×D4C2×D10711yescharacteristic
218C23×C4—1071no—
219C23×C4—1071no—
220C23×C4C2×D10711yescharacteristic
221#126—2144no—
222C2×C4⋊C4—1071no—
223C2×C4⋊C4—1071no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1071no—
226C2×C4⋊C4—1071no—
227C2×C4⋊C4—1071no—
228C2×C22⋊C4—1071no—
229C2×C22⋊C4—1071no—
230C2×C22⋊C4—1071no—
231C2×C22⋊C4—1071no—
232C2×C22⋊C4—1071no—
233C2×C22⋊C4—1071no—
234C2×C22⋊C4C2×D10711yescharacteristic
235C2×C22⋊C4C2×D10711yescharacteristic
236C2×C22⋊C4C2×D10711yescharacteristic
237C2×C42—1071no—
238C2×C42—1071no—
239C2×C42—1071no—
240C2×C42—1071no—
241C2×C42—2144no—
242#95—1071no—
243#95—1071no—
244#95—1071no—
245#95—1071no—
246#95—1071no—
247#95—1071no—
248#95—1071no—
249#95—1071no—
250#95—1071no—
251#95—1071no—
252#95C2×D10711yescharacteristic
253#95C2×D10711yescharacteristic
254C24C22×D10711yescharacteristic
255C24C22×D10711yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—1071no—
263C22×C4—1071no—
264C22×C4—1071no—
265C22×C4—1071no—
266C22×C4—1071no—
267C22×C4—1071no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—2142no—
271C22×C4—2142no—
272C22×C4—2142no—
273C22×C4—2142no—
274C22×C4—2142no—
275C22×C4—2142no—
276C22×C4C22×D10711yescharacteristic
277C22×C4C22×D10711yescharacteristic
278C22×C4C22×D10711yescharacteristic
279C22×C4C22×D10711yescharacteristic
280C22×C4C22×D10711yescharacteristic
281C4⋊C4—2142no—
282C4⋊C4—2142no—
283C4⋊C4—2142no—
284C4⋊C4—2142no—
285C4⋊C4—2142no—
286C4⋊C4—2142no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—2142no—
291C22⋊C4—2142no—
292C22⋊C4—2142no—
293C22⋊C4—2142no—
294C22⋊C4—2142no—
295C22⋊C4—2142no—
296C42—1074no—
297C42—2142no—
298C42—2142no—
299C42—2142no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23#1313381712yes—
312C23#1313381712yes—
313C23#1313381812yes—
314C23C23×D10711yescharacteristic
315D4—44no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—1072no—
331C2×C4—1072no—
332C2×C4—1072no—
333C2×C4—1072no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2148no—
347C2×C4—2148no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4D4×D10712yes—
357C2×C4#1313381712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#1681210214yescomplemented
372C22C2×D4×D10711yescharacteristic
373C22#41×D10711yescharacteristic
374C22#41×D10711yescharacteristic
375C22C2×#1313381711yescharacteristic
376C22C2×#1313381711yescharacteristic
377C22C2×#1313381711yescharacteristic
378C22C2×#1313381811yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—4281no—
385C4—4281no—
386C4—4281no—
387C4—4281no—
388C4—2142no—
389C4—2142no—
390C4—2142no—
391C4—2142no—
392C2—24no—
393C2—81no—
394C2#2029313311yescharacteristic
395C2#2029313611yescharacteristic
396C2#2029313711yescharacteristic
397C2C2×#1681210211yescharacteristic
398C2#2029315411yescharacteristic
399C2#2029315711yescharacteristic
400C2#2029315811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC22×D4×C107C22
hypercenter#386D107
hypocenterC107#2852
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#16812102C2semidirect
2. C22×C4×Dic107C2semidirect
3. #20292521C2semidirect
4. #20292520C2semidirect
5. #20292515C2semidirect
6. #20292514C2semidirect
7. #20292512C2semidirect
8. #20292509C2semidirect
11. #20292502C2semidirect
12. #20292500C2semidirect
13. #20292499C2semidirect
14. #20292494C2semidirect
82. C4×Dic107D4semidirect
170. C22×C107#126semidirect
201. C107#2852semidirect
371. C22#16812102semidirect

Maximal subgroups

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

subgroupindex
C2×#168121022
C22×C4×Dic1072
#202925212
#202925202
#202925152
#202925142
#202925122
#202925092
#202925072
#202925022
#202925002
#202924992
#202924942
#386×C1072
#2852107