Finite Group Databasebeta

← #28347874

Subgroups — #28347874

Order 13696.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293248C211yesmaximal, characteristic, complemented
2#20293232C211yesmaximal, characteristic, complemented
3#20292935C211yesmaximal, characteristic, complemented
4#20292928C211yesmaximal, characteristic, complemented
5#20292918C211yesmaximal, characteristic, complemented
6#20292908C211yesmaximal, characteristic, complemented
7#20292880C211yesmaximal, characteristic, complemented
8#20292873C211yesmaximal, characteristic, complemented
9#20292863C211yesmaximal, characteristic, complemented
10#20292856C211yesmaximal, characteristic, complemented
11#20292843C211yesmaximal, characteristic, complemented
12#20292843C211yesmaximal, characteristic, complemented
13#20292828C211yesmaximal, characteristic, complemented
14#20292823C211yesmaximal, characteristic, complemented
15#424×C107C211yesmaximal, characteristic, complemented
16C2×#13133807C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×D856C2211yescharacteristic
19C2×D856C2211yescharacteristic
20#16812109C2211yescharacteristic
21#16812106C2211yescharacteristic
22#16812104—21no—
23#16812101C2211yescharacteristic
24#16812097C2211yescharacteristic
25#16812095—21no—
26#16812094—21no—
27#16812094—21no—
28#16812091—21no—
29#16812088C2211yescharacteristic
30#16812087—21no—
31#16812083C2211yescharacteristic
32#16812083C2211yescharacteristic
33#16812083C2211yescharacteristic
34#16812082—21no—
35#16812082—21no—
36C4×Dic214C2211yescharacteristic
37C4×D428C2211yescharacteristic
38C4×D428C2211yescharacteristic
39C4×D428C2211yescharacteristic
40#16812043C2211yescharacteristic
41#16812043C2211yescharacteristic
42#16812041C2211yescharacteristic
43#16812041C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C2×M16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54#117×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428—21no—
62C2×D428—21no—
63C2×D428C2311yescharacteristic
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72#13133807—41no—
73#13133807—41no—
74D856—41no—
75D856—41no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85C22⋊Dic107—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94C4×Dic107—21no—
95C22×C4×C107C2311yescharacteristic
96M16×C107—41no—
97C2×C8×C107—41no—
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107D411yescharacteristic
103C4⋊C4×C107D411yescharacteristic
104C22⋊C4×C107D411yescharacteristic
105C22⋊C4×C107D411yescharacteristic
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107C2×D411yescharacteristic
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2411yescharacteristic
120C22⋊D107—81no—
121C22⋊D107—81no—
122C22⋊D107—81no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128D428—21no—
129D428—21no—
130D428—21no—
131D428—21no—
132D428—21no—
133D428—21no—
134D428—81no—
135D428—81no—
136C4×D107—81no—
137C4×D107—81no—
138C4×D107—81no—
139Dic214—21no—
140Dic214—21no—
141Dic214—81no—
142C8×C107—21no—
143C8×C107—21no—
144C8×C107—21no—
145C8×C107—21no—
146C22×C107—21no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107C22×D411yescharacteristic
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107—21no—
163C4×C107—21no—
164C4×C107H(5,𝔽2)11yescharacteristic
165C4×C1072−1+411yescharacteristic
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171C2×C107—41no—
172C2×C107#53711yescharacteristic
173C2×C107#57611yescharacteristic
174C2×C107#57711yescharacteristic
175D107—81no—
176D107—81no—
177D107—81no—
178#3257—1071nomaximal, Sylow
179C107#325711yescharacteristic, complemented, Sylow
180#542—1071no—
181#540—1071no—
182#479—1071no—
183#478—1071no—
184#476—1071no—
185#474—1071no—
186#469—1071no—
187#468—1071no—
188#466—1071no—
189#465—1071no—
190#463—1071no—
191#463—1071no—
192#460—1071no—
193#459—1071no—
194#424D10711yescharacteristic, complemented
195C2×SD16—1071no—
196C2×SD16—1071no—
197C2×D8—1071no—
198C2×D8—1071no—
199C2×M16C2×D10711yescharacteristic
200C22×C8C2×D10711yescharacteristic
201C4⋊Q8—1071no—
202#127—1071no—
203#126—2141no—
204#125—1071no—
205#124—1071no—
206#124—2141no—
207#123—2141no—
208#123—2141no—
209#123—2141no—
210C22⋊Q8—1071no—
211C22⋊Q8—2141no—
212#121—1071no—
213#121—1071no—
214#121—1071no—
215#121—2141no—
216#121—2141no—
217C4×Q8—1071no—
218C4×D4—1071no—
219C4×D4—1071no—
220C4×D4—1071no—
221#117C2×D10711yescharacteristic
222#107—1071no—
223#107—1071no—
224#106—1071no—
225#106—1071no—
226C4⋊C8C2×D10711yescharacteristic
227C4⋊C8C2×D10711yescharacteristic
228C4⋊C8C2×D10711yescharacteristic
229C4⋊C8C2×D10711yescharacteristic
230#103—1071no—
231#103—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235#102—1071no—
236#102—1071no—
237#102—1071no—
238C2×Q8—1071no—
239C2×Q8—2141no—
240C2×D4—1071no—
241C2×D4—1071no—
242C2×D4—1071no—
243C2×D4—2141no—
244C2×D4—2141no—
245C2×D4—2141no—
246C2×D4—2141no—
247C2×D4—2141no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252SD16—4281no—
253SD16—4281no—
254D8—4281no—
255D8—4281no—
256M16—41no—
257C2×C8—41no—
258C2×C8C22×D10711yescharacteristic
259C2×C8C22×D10711yescharacteristic
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—1071no—
263C4⋊C4—1071no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4D42811yescharacteristic
272C4⋊C4D42811yescharacteristic
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4D42811yescharacteristic
283C22⋊C4D42811yescharacteristic
284C42—2141no—
285C42C22×D10711yescharacteristic
286C42C22×D10711yescharacteristic
287C23—2141no—
288C23—2141no—
289C23—2141no—
290C23C2×D42811yescharacteristic
291Q8—8561no—
292Q8—2141no—
293Q8—2141no—
294D4—8561no—
295D4—8561no—
296D4—8561no—
297D4—8561no—
298D4—8561no—
299D4—2141no—
300D4—2141no—
301D4—2141no—
302D4—2141no—
303D4—2141no—
304D4—2141no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—8561no—
308C2×C4—8561no—
309C2×C4—8561no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4C2×D42811yescharacteristic
316C2×C4C2×D42811yescharacteristic
317C2×C4C2×D42811yescharacteristic
318C2×C4C2×D42811yescharacteristic
319C2×C4C2×D42811yescharacteristic
320C2×C4C23×D10711yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—4281no—
329C22—4281no—
330C22—4281no—
331C22—4281no—
332C22—4281no—
333C22—4281no—
334C22—4281no—
335C22—4281no—
336C22—4281no—
337C22C22×D42811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4—4281no—
348C4#1681215411yescharacteristic
349C4#1681215711yescharacteristic
350C2—41no—
351C2—8561no—
352C2—8561no—
353C2—8561no—
354C2#2029320811yescharacteristic
355C2#2029343211yescharacteristic
356C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
hypercenter#424D107
hypocenterC107#3257
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20293248C2semidirect
2. #20293232C2semidirect
3. #20292935C2semidirect
4. #20292928C2semidirect
5. #20292918C2semidirect
6. #20292908C2semidirect
7. #20292880C2semidirect
8. #20292873C2semidirect
9. #20292863C2semidirect
10. #20292856C2semidirect
11. #20292843C2semidirect
12. #20292843C2semidirect
13. #20292828C2semidirect
14. #20292823C2semidirect
15. #424×C107C2semidirect
179. C107#3257semidirect
194. #424D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202932322
#202929352
#202929282
#202929182
#202929082
#202928802
#202928732
#202928632
#202928562
#202928432
#202928282
#202928232
#424×C1072
#3257107