Finite Group Databasebeta

← #28339979

Subgroups — #28339979

Order 13696.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812086C211yesmaximal, characteristic
2C2×#16812074C211yesmaximal, characteristic, complemented
3#20292563C211yesmaximal, characteristic
4#20292560C211yesmaximal, characteristic, complemented
5#20292558C211yesmaximal, characteristic, complemented
6#20292531C211yesmaximal, characteristic, complemented
7#20292522C211yesmaximal, characteristic, complemented
8#20292521C211yesmaximal, characteristic
9#20292515C211yesmaximal, characteristic
10#20292513C211yesmaximal, characteristic, complemented
11#20292512C211yesmaximal, characteristic, complemented
12#20292496C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic
14C22⋊C4×Dic107C211yesmaximal, characteristic
15#399×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18#16812086—24no—
19#16812074—24no—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C4×Dic107C2211yescharacteristic
34C2×C4×Dic107C2211yescharacteristic
35C2×C4×Dic107C2211yescharacteristic
36C2×C4×Dic107C2211yescharacteristic
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×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53C2×Dic214—22no—
54C2×Dic214—22no—
55C2×Dic214D414yes—
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—
73Dic107⋊C4—22no—
74Dic107⋊C4—41no—
75Dic107⋊C4—41no—
76Dic107⋊C4—41no—
77C22⋊Dic107—22no—
78C22⋊Dic107—41no—
79C22⋊Dic107Q814yescomplemented
80C4×Dic107—22no—
81C4×Dic107—22no—
82C4×Dic107—22no—
83C4×Dic107—22no—
84C24×C107C2311yescharacteristic
85C22×C4×C107C2311yescharacteristic
86C22×C4×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C4⋊C4×C107—22no—
92C4⋊C4×C107—22no—
93C4⋊C4×C107—41no—
94C22⋊C4×C107—22no—
95C22⋊C4×C107—22no—
96C22⋊C4×C107—41no—
97C23×C107—21no—
98C23×C107—21no—
99C23×C107—21no—
100C23×C107—21no—
101C23×C107—21no—
102C23×C107—21no—
103C23×C107C2×Q812yes—
104C23×C107C2411yescharacteristic
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×C107C2×D412yes—
122C2×C4×C107#4112yes—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
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—44no—
142C2×Dic107C2×D412yes—
143C2×Dic107C2×D412yes—
144C2×Dic107C2×Q812yes—
145C2×Dic107C2×Q812yes—
146C2×Dic107#4112yes—
147C2×Dic107#4112yes—
148Dic214—28no—
149C22×C107—22no—
150C22×C107—22no—
151C22×C107—22no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107—41no—
156C22×C107C22×D411yescharacteristic
157C22×C107C22×Q811yescharacteristic
158C22×C107C2×#4111yescharacteristic
159C22×C107C2×#4111yescharacteristic
160C22×C107C2×#4111yescharacteristic
161C22×C107H(5,𝔽2)11yescharacteristic
162C22×C1072−1+411yescharacteristic
163C4×C107—22no—
164C4×C107—22no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107—41no—
169Dic107—22no—
170Dic107—22no—
171Dic107—22no—
172Dic107—41no—
173Dic107—41no—
174Dic107—41no—
175Dic107—41no—
176Dic107#12314yes—
177C2×C107—42no—
178C2×C107C2×#12311yescharacteristic
179C2×C107#53311yescharacteristic
180C2×C107#54211yescharacteristic
181C2×C107D4×Q811yescharacteristic
182C2×C107#55011yescharacteristic
183C2×C107#55211yescharacteristic
184C2×C107#55411yescharacteristic
185#2423—1071nomaximal, Sylow
186C107#242311yescharacteristic, complemented, Sylow
187C2×C22⋊Q8—1071no—
188C2×C4×Q8—1071no—
189#399—1071no—
190#399D10711yescharacteristic
191#398—1071no—
192#398—1071no—
193#391—1071no—
194#389—1071no—
195#388—1071no—
196#388—1071no—
197#387—1071no—
198#387—1071no—
199#384—1071no—
200#380—1071no—
201C4×C22⋊C4—1071no—
202C22×Q8—1071no—
203C23×C4—1071no—
204C22⋊Q8—2144no—
205C4×Q8—2144no—
206C2×C4⋊C4—1071no—
207C2×C4⋊C4—1071no—
208C2×C4⋊C4—1071no—
209C2×C4⋊C4—1071no—
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⋊C4C2×D10711yescharacteristic
217C2×C4⋊C4C2×D10711yescharacteristic
218C2×C4⋊C4C2×D10711yescharacteristic
219C2×C22⋊C4—1071no—
220C2×C22⋊C4—1071no—
221C2×C22⋊C4—1071no—
222C2×C22⋊C4C2×D10711yescharacteristic
223C2×C22⋊C4C2×D10711yescharacteristic
224C2×C22⋊C4C2×D10711yescharacteristic
225C2×C42—1071no—
226C2×C42—1071no—
227C2×C42—1071no—
228C2×C42—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#95C2×D10711yescharacteristic
239C24C22×D10711yescharacteristic
240C2×Q8—1074no—
241C2×Q8—2142no—
242C2×Q8—2142no—
243C22×C4—1071no—
244C22×C4—1071no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1071no—
249C22×C4—1071no—
250C22×C4—1071no—
251C22×C4—2142no—
252C22×C4—2142no—
253C22×C4—2142no—
254C22×C4C22×D10711yescharacteristic
255C22×C4C22×D10711yescharacteristic
256C22×C4C22×D10711yescharacteristic
257C22×C4C22×D10711yescharacteristic
258C22×C4C22×D10711yescharacteristic
259C22×C4C22×D10711yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—4281no—
264C4⋊C4—4281no—
265C4⋊C4—4281no—
266C4⋊C4—2142no—
267C4⋊C4—2142no—
268C4⋊C4—2142no—
269C4⋊C4—2142no—
270C4⋊C4—2142no—
271C4⋊C4—2142no—
272C4⋊C4—2142no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—1074no—
277C22⋊C4—4281no—
278C22⋊C4—2142no—
279C42—2142no—
280C42—2142no—
281C42—2142no—
282C42—2142no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D10712yes—
290C23C23×D10711yescharacteristic
291Q8—2148no—
292C2×C4—21no—
293C2×C4—21no—
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—1072no—
309C2×C4—1072no—
310C2×C4—1072no—
311C2×C4—1072no—
312C2×C4—1072no—
313C2×C4—1072no—
314C2×C4—4284no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
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×C4D4×D10712yes—
334C2×C4#1313381712yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D10711yescharacteristic
343C22C2×Q8×D10711yescharacteristic
344C22#41×D10711yescharacteristic
345C22#41×D10711yescharacteristic
346C22C2×#1313381711yescharacteristic
347C22#1681215411yescharacteristic
348C22#1681215611yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—1074no—
356C4—4281no—
357C4—4281no—
358C4—4281no—
359C4—4281no—
360C4—2142no—
361C4—2142no—
362C4—2142no—
363C2—42no—
364C2#123×D10711yescharacteristic
365C2#2029317911yescharacteristic
366C2#2029324511yescharacteristic
367C2#2029329011yescharacteristic
368C2#2029329711yescharacteristic
369C2#2029331411yescharacteristic
370C2#2029333111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radicalC24×C107C23
hypercenter#399D107
hypocenterC107#2423
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#16812074C2semidirect
4. #20292560C2semidirect
5. #20292558C2semidirect
6. #20292531C2semidirect
7. #20292522C2semidirect
10. #20292513C2semidirect
11. #20292512C2semidirect
12. #20292496C2semidirect
79. C22⋊Dic107Q8semidirect
186. C107#2423semidirect

Maximal subgroups

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

subgroupindex
C2×#168120862
C2×#168120742
#202925632
#202925602
#202925582
#202925312
#202925222
#202925212
#202925152
#202925132
#202925122
#202924962
#202924852
C22⋊C4×Dic1072
#399×C1072
#2423107