Finite Group Databasebeta

← #28344481

Subgroups — #28344481

Order 13696.

478 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812094C211yesmaximal, characteristic
2C2×#16812087C211yesmaximal, characteristic, complemented
3#20292561C211yesmaximal, characteristic, complemented
4#20292560C211yesmaximal, characteristic, complemented
5#20292556C211yesmaximal, characteristic, complemented
6#20292556C211yesmaximal, characteristic, complemented
7#20292554C211yesmaximal, characteristic, complemented
8#20292549C211yesmaximal, characteristic, complemented
9#20292549C211yesmaximal, characteristic, complemented
10#20292547C211yesmaximal, characteristic, complemented
11#20292545C211yesmaximal, characteristic
12#20292544C211yesmaximal, characteristic
13#20292540C211yesmaximal, characteristic, complemented
14#20292484C211yesmaximal, characteristic, complemented
15#398×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19#16812094—24no—
20#16812087—24no—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×C4⋊Dic107C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
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
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53C23×D107C2311yescharacteristic
54C23×D107C2311yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—41no—
57C2×Dic214—41no—
58C2×D428—22no—
59C2×D428—41no—
60C2×D428—41no—
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C2×C4×D107—41no—
68C2×C4×D107—41no—
69C2×C4×D107D414yes—
70C4⋊Dic107—41no—
71C4⋊Dic107—41no—
72Dic107⋊C4—22no—
73Dic107⋊C4—22no—
74Dic107⋊C4—22no—
75Dic107⋊C4—41no—
76Dic107⋊C4—41no—
77(C2×D107)⋊C4—22no—
78(C2×D107)⋊C4—22no—
79(C2×D107)⋊C4—22no—
80(C2×D107)⋊C4—22no—
81(C2×D107)⋊C4—22no—
82(C2×D107)⋊C4—41no—
83(C2×D107)⋊C4—41no—
84(C2×D107)⋊C4—41no—
85(C2×D107)⋊C4—41no—
86(C2×D107)⋊C4—41no—
87(C2×D107)⋊C4—41no—
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C4⋊C4×C107—22no—
96C4⋊C4×C107—22no—
97C4⋊C4×C107—41no—
98C4⋊C4×C107—41no—
99C23×C107C2411yescharacteristic
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—21no—
103C22×D107—21no—
104C22×D107—21no—
105C22×D107—21no—
106C22×D107—21no—
107C22×D107—21no—
108C22×D107—21no—
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—21no—
112C22×D107—21no—
113C22×D107C2×D412yes—
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×C107—21no—
124C2×C4×C107—21no—
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×C107C2×D412yes—
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×Dic107C2×D412yes—
153D428—44no—
154C4×D107—44no—
155Dic214—44no—
156C22×C107C22×D411yescharacteristic
157C22×C107H(5,𝔽2)11yescharacteristic
158C22×C107H(5,𝔽2)11yescharacteristic
159C22×C107H(5,𝔽2)11yescharacteristic
160C22×C1072−1+411yescharacteristic
161C22×C1072−1+411yescharacteristic
162C22×C1072−1+411yescharacteristic
163C2×D107—22no—
164C2×D107—22no—
165C2×D107—22no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—41no—
175C2×D107—41no—
176C2×D107—41no—
177C4×C107—22no—
178C4×C107—41no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107—41no—
182C4×C107—41no—
183C4×C107—41no—
184Dic107—22no—
185Dic107—41no—
186Dic107—41no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190C2×C107#53411yescharacteristic
191C2×C107#53611yescharacteristic
192C2×C107#53711yescharacteristic
193C2×C107#56211yescharacteristic
194C2×C107#56211yescharacteristic
195C2×C107#56311yescharacteristic
196C2×C107#56311yescharacteristic
197D107—42no—
198D107—81no—
199#2814—1071nomaximal, Sylow
200C107#281411yescharacteristic, complemented, Sylow
201C2×#123—1071no—
202C2×C22⋊Q8—1071no—
203#398—1071no—
204#398—1071no—
205#398D10711yescharacteristic, complemented
206#397—1071no—
207#397—1071no—
208#397—1071no—
209#396—1071no—
210#396—1071no—
211#395—1071no—
212#394—1071no—
213#394—1071no—
214#393—1071no—
215#380—1071no—
216C22×Q8—1071no—
217C22×D4—1071no—
218C23×C4—1071no—
219#123—2144no—
220C22⋊Q8—2144no—
221C2×C4⋊C4—1071no—
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×C4⋊C4C2×D10711yescharacteristic
229C2×C4⋊C4C2×D10711yescharacteristic
230C2×C4⋊C4C2×D10711yescharacteristic
231C2×C4⋊C4C2×D10711yescharacteristic
232C2×C22⋊C4—1071no—
233C2×C22⋊C4—1071no—
234C2×C22⋊C4—1071no—
235C2×C22⋊C4—1071no—
236C2×C22⋊C4—1071no—
237C2×C22⋊C4—1071no—
238C2×C22⋊C4—1071no—
239C2×C22⋊C4—1071no—
240C2×C22⋊C4—1071no—
241C2×C22⋊C4—1071no—
242C2×C22⋊C4—1071no—
243#95—1071no—
244#95—1071no—
245#95—1071no—
246#95—1071no—
247#95—1071no—
248#95—1071no—
249#95—1071no—
250#95C2×D10711yescharacteristic
251#95C2×D10711yescharacteristic
252#95C2×D10711yescharacteristic
253C24—1071no—
254C24—1071no—
255C2×Q8—4281no—
256C2×Q8—4281no—
257C2×Q8—2142no—
258C2×D4—4281no—
259C2×D4—4281no—
260C2×D4—2142no—
261C22×C4—1071no—
262C22×C4—1074no—
263C22×C4—1071no—
264C22×C4—1071no—
265C22×C4—1071no—
266C22×C4—1071no—
267C22×C4—1071no—
268C22×C4—4281no—
269C22×C4—4281no—
270C22×C4C22×D10711yescharacteristic
271C22×C4C22×D10711yescharacteristic
272C22×C4C22×D10711yescharacteristic
273C22×C4C22×D10711yescharacteristic
274C22×C4C22×D10711yescharacteristic
275C22×C4C22×D10711yescharacteristic
276C22×C4C22×D10711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—4281no—
282C4⋊C4—4281no—
283C4⋊C4—4281no—
284C4⋊C4—4281no—
285C4⋊C4—2142no—
286C4⋊C4—2142no—
287C4⋊C4—2142no—
288C22⋊C4—4281no—
289C22⋊C4—4281no—
290C22⋊C4—4281no—
291C22⋊C4—4281no—
292C22⋊C4—4281no—
293C22⋊C4—4281no—
294C22⋊C4—2142no—
295C22⋊C4—2142no—
296C22⋊C4—2142no—
297C22⋊C4—2142no—
298C22⋊C4—2142no—
299C23—1072no—
300C23—2141no—
301C23—2141no—
302C23—2141no—
303C23—2141no—
304C23—2141no—
305C23—2141no—
306C23—2141no—
307C23—2141no—
308C23—2141no—
309C23—2141no—
310C23—2141no—
311C23—2141no—
312C23—2141no—
313C23C23×D10711yescharacteristic
314Q8—4284no—
315D4—4284no—
316C2×C4—21no—
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—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—1072no—
337C2×C4—4284no—
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—2141no—
347C2×C4—2141no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4D4×D10712yes—
356C22—4281no—
357C22—4281no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—4281no—
365C22—4281no—
366C22—4281no—
367C22—2142no—
368C22—2142no—
369C22—2142no—
370C22C2×D4×D10711yescharacteristic
371C22#1681215411yescharacteristic
372C22#1681215511yescharacteristic
373C22#1681215511yescharacteristic
374C22#1681215611yescharacteristic
375C22#1681215711yescharacteristic
376C22#1681215711yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—4281no—
385C4—4281no—
386C4—4281no—
387C4—4281no—
388C4—4281no—
389C4—2142no—
390C2—8561no—
391C2—4282no—
392C2#2029318511yescharacteristic
393C2#2029320011yescharacteristic
394C2#2029320711yescharacteristic
395C2#2029337411yescharacteristic
396C2#2029337511yescharacteristic
397C2#2029337611yescharacteristic
398C2#2029338111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radicalC22×D428C22
hypercenter#398D107
hypocenterC107#2814
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#16812087C2semidirect
3. #20292561C2semidirect
4. #20292560C2semidirect
5. #20292556C2semidirect
6. #20292556C2semidirect
7. #20292554C2semidirect
8. #20292549C2semidirect
9. #20292549C2semidirect
10. #20292547C2semidirect
13. #20292540C2semidirect
14. #20292484C2semidirect
15. #398×C107C2semidirect
200. C107#2814semidirect
205. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120872
#202925612
#202925602
#202925562
#202925542
#202925492
#202925472
#202925452
#202925442
#202925402
#202924842
#398×C1072
#2814107