Finite Group Databasebeta

← #28340310

Subgroups — #28340310

Order 13696.

620 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812107C211yesmaximal, characteristic, complemented
2C2×#16812087C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D107C211yesmaximal, characteristic
4#20292550C211yesmaximal, characteristic, complemented
5#20292549C211yesmaximal, characteristic, complemented
6#20292539C211yesmaximal, characteristic, complemented
7#20292538C211yesmaximal, characteristic, complemented
8#20292527C211yesmaximal, characteristic
9#20292527C211yesmaximal, characteristic
10#20292522C211yesmaximal, characteristic, complemented
11#20292522C211yesmaximal, characteristic, complemented
12#20292500C211yesmaximal, characteristic, complemented
13#20292498C211yesmaximal, characteristic, complemented
14C4⋊C4×Dic107C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19#16812107—24no—
20#16812087—24no—
21C4×C22⋊D107—24no—
22C2×C4⋊Dic107C2211yescharacteristic
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×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×C4×Dic107C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C22×Q8×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49C22⋊Q8×C107—24no—
50C2×C4⋊C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic
52C2×C4⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C2×Dic214—22no—
57C2×Dic214—22no—
58C2×Dic214—41no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107Q814yescomplemented
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C2×C4×D107—41no—
72C4⋊Dic107—22no—
73Dic107⋊C4—22no—
74Dic107⋊C4—22no—
75Dic107⋊C4—22no—
76Dic107⋊C4—22no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4D414yes—
80C22⋊Dic107—22no—
81C22⋊Dic107—22no—
82C22⋊Dic107—41no—
83(C2×D107)⋊C4—22no—
84(C2×D107)⋊C4—22no—
85(C2×D107)⋊C4—22no—
86(C2×D107)⋊C4—22no—
87(C2×D107)⋊C4—41no—
88C4×Dic107—22no—
89C4×Dic107—22no—
90C4×Dic107—22no—
91C24×C107C2311yescharacteristic
92C2×Q8×C107—22no—
93C2×Q8×C107—22no—
94C2×Q8×C107—41no—
95C22×C4×C107—22no—
96C22×C4×C107—22no—
97C22×C4×C107—41no—
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C4⋊C4×C107—22no—
105C4⋊C4×C107—22no—
106C4⋊C4×C107D414yescomplemented
107C22⋊C4×C107—22no—
108C22⋊C4×C107—22no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—21no—
115C23×C107C2×Q812yes—
116C23×C107C2411yescharacteristic
117C22×D107—21no—
118C22×D107—21no—
119C22×D107—21no—
120C22×D107—21no—
121C22×D107—21no—
122C22×D107—21no—
123C22×D107C2×Q812yes—
124Q8×C107—44no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—44no—
140C2×C4×C107C2×D412yes—
141C2×C4×C107C2×D412yes—
142C2×C4×C107C2×D412yes—
143C2×C4×C107C2×D412yes—
144C22⋊D107—28no—
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×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107—21no—
159C2×Dic107—21no—
160C2×Dic107—21no—
161C2×Dic107C2×D412yes—
162C2×Dic107C2×D412yes—
163C2×Dic107C2×Q812yes—
164C2×Dic107#4112yes—
165C2×Dic107#4112yes—
166C4×D107—44no—
167Dic214—44no—
168C22×C107—22no—
169C22×C107—22no—
170C22×C107—22no—
171C22×C107—41no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107C22×D411yescharacteristic
176C22×C107C22×D411yescharacteristic
177C22×C107C22×Q811yescharacteristic
178C22×C107C2×#4111yescharacteristic
179C22×C107C2×#4111yescharacteristic
180C22×C107H(5,𝔽2)11yescharacteristic
181C22×C1072−1+411yescharacteristic
182C2×D107—22no—
183C2×D107—22no—
184C2×D107—22no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C4×C107—22no—
190C4×C107—22no—
191C4×C107—22no—
192C4×C107—22no—
193C4×C107—41no—
194C4×C107—41no—
195Dic107—22no—
196Dic107—22no—
197Dic107—41no—
198Dic107—41no—
199Dic107—41no—
200Dic107—41no—
201Dic107C22⋊Q814yescomplemented
202C2×C107—42no—
203C2×C107C2×C22⋊Q811yescharacteristic
204C2×C107#53211yescharacteristic
205C2×C107#54211yescharacteristic
206C2×C107D4211yescharacteristic
207C2×C107#54811yescharacteristic
208C2×C107D4×Q811yescharacteristic
209C2×C107#55411yescharacteristic
210D107—42no—
211#2452—1071nomaximal, Sylow
212C107#245211yescharacteristic, complemented, Sylow
213C2×C4⋊Q8—1071no—
214C2×C22⋊Q8—1071no—
215C2×C22⋊Q8D10711yescharacteristic, complemented
216C2×C4×D4—1071no—
217#396—1071no—
218#396—1071no—
219#393—1071no—
220#393—1071no—
221#390—1071no—
222#390—1071no—
223#389—1071no—
224#389—1071no—
225#385—1071no—
226#385—1071no—
227C4×C4⋊C4—1071no—
228C22×Q8—1071no—
229C22×Q8C2×D10711yescharacteristic
230C22×D4—1071no—
231C23×C4—1071no—
232C23×C4C2×D10711yescharacteristic
233C4⋊Q8—2144no—
234C22⋊Q8—24no—
235C22⋊Q8—2144no—
236C4×D4—2144no—
237C2×C4⋊C4—1071no—
238C2×C4⋊C4—1071no—
239C2×C4⋊C4—1071no—
240C2×C4⋊C4—1071no—
241C2×C4⋊C4—1071no—
242C2×C4⋊C4—1071no—
243C2×C4⋊C4—1071no—
244C2×C4⋊C4—1071no—
245C2×C4⋊C4C2×D10711yescharacteristic
246C2×C4⋊C4C2×D10711yescharacteristic
247C2×C4⋊C4C2×D10711yescharacteristic
248C2×C22⋊C4—1071no—
249C2×C22⋊C4—1071no—
250C2×C22⋊C4—1071no—
251C2×C22⋊C4—1071no—
252C2×C22⋊C4—1071no—
253C2×C22⋊C4—1071no—
254C2×C22⋊C4—1071no—
255C2×C22⋊C4—1071no—
256C2×C22⋊C4C2×D10711yescharacteristic
257C2×C22⋊C4C2×D10711yescharacteristic
258C2×C42—1071no—
259C2×C42—1071no—
260C2×C42—1071no—
261#95—1071no—
262#95—1071no—
263#95—1071no—
264#95—1071no—
265#95—1071no—
266#95—1071no—
267C24—1071no—
268C24C22×D10711yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—4281no—
273C2×Q8—2142no—
274C2×Q8—2142no—
275C2×D4—1074no—
276C2×D4—2142no—
277C2×D4—2142no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—1071no—
282C22×C4—1071no—
283C22×C4—1071no—
284C22×C4—1071no—
285C22×C4—1071no—
286C22×C4—1071no—
287C22×C4—1071no—
288C22×C4—4281no—
289C22×C4—2142no—
290C22×C4—2142no—
291C22×C4C22×D10711yescharacteristic
292C22×C4C22×D10711yescharacteristic
293C22×C4C22×D10711yescharacteristic
294C22×C4C22×D10711yescharacteristic
295C22×C4C22×D10711yescharacteristic
296C22×C4C22×D10711yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—1074no—
300C4⋊C4—2142no—
301C4⋊C4—2142no—
302C4⋊C4—2142no—
303C4⋊C4—2142no—
304C4⋊C4—2142no—
305C4⋊C4—2142no—
306C4⋊C4—2142no—
307C4⋊C4C22⋊D10714yescomplemented
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—4281no—
311C22⋊C4—4281no—
312C22⋊C4—2142no—
313C22⋊C4—2142no—
314C22⋊C4—2142no—
315C22⋊C4—2142no—
316C22⋊C4—2142no—
317C22⋊C4—2142no—
318C42—2142no—
319C42—2142no—
320C42—2142no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—1072no—
328C23—2141no—
329C23—2141no—
330C23—2141no—
331C23—2141no—
332C23—2141no—
333C23—2141no—
334C23Q8×D10712yes—
335C23C23×D10711yescharacteristic
336Q8—44no—
337Q8—4284no—
338D4—2148no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—1072no—
355C2×C4—1072no—
356C2×C4—1072no—
357C2×C4—1072no—
358C2×C4—1072no—
359C2×C4—4284no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4—2141no—
376C2×C4D4×D10712yes—
377C2×C4C2×C22⋊D10712yes—
378C2×C4C2×C22⋊D10712yes—
379C2×C4C2×C22⋊D10712yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—4281no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—2142no—
392C22—2142no—
393C22—2142no—
394C22C2×D4×D10711yescharacteristic
395C22C22×C22⋊D10711yescharacteristic
396C22C2×Q8×D10711yescharacteristic
397C22#41×D10711yescharacteristic
398C22#41×D10711yescharacteristic
399C22#1681215411yescharacteristic
400C22#1681215711yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—1074no—
408C4—4281no—
409C4—4281no—
410C4—4281no—
411C4—4281no—
412C4—2142no—
413C4—2142no—
414C2—42no—
415C2—4282no—
416C2C22⋊Q8×D10711yescharacteristic
417C2#2029317311yescharacteristic
418C2#2029324711yescharacteristic
419C2D4×C22⋊D10711yescharacteristic
420C2#2029328711yescharacteristic
421C2Q8×C22⋊D10711yescharacteristic
422C2#2029333111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC22×C22⋊D107C22
hypercenterC2×C22⋊Q8D107
hypocenterC107#2452
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812107C2semidirect
2. C2×#16812087C2semidirect
4. #20292550C2semidirect
5. #20292549C2semidirect
6. #20292539C2semidirect
7. #20292538C2semidirect
10. #20292522C2semidirect
11. #20292522C2semidirect
12. #20292500C2semidirect
13. #20292498C2semidirect
14. C4⋊C4×Dic107C2semidirect
15. C2×C22⋊Q8×C107C2semidirect
61. C2×C22⋊D107Q8semidirect
106. C4⋊C4×C107D4semidirect
201. Dic107C22⋊Q8semidirect
212. C107#2452semidirect
215. C2×C22⋊Q8D107semidirect
307. C4⋊C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121072
C2×#168120872
C2×C4×C22⋊D1072
#202925502
#202925492
#202925392
#202925382
#202925272
#202925222
#202925002
#202924982
C4⋊C4×Dic1072
C2×C22⋊Q8×C1072
#2452107