Finite Group Databasebeta

← #28339863

Subgroups — #28339863

Order 13696.

610 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812091C211yesmaximal, characteristic
2C2×#16812091C211yesmaximal, characteristic
3C2×#16812067C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292560C211yesmaximal, characteristic, complemented
6#20292557C211yesmaximal, characteristic, complemented
7#20292550C211yesmaximal, characteristic, complemented
8#20292526C211yesmaximal, characteristic, complemented
9#20292524C211yesmaximal, characteristic, complemented
10#20292520C211yesmaximal, characteristic, complemented
11#20292515C211yesmaximal, characteristic, complemented
12#20292503C211yesmaximal, characteristic
13#20292498C211yesmaximal, characteristic, complemented
14#20292489C211yesmaximal, characteristic, complemented
15#396×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812091—24no—
21#16812091—24no—
22#16812067—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×C4⋊Dic107C2211yescharacteristic
26C2×C4⋊Dic107C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C4⋊C4×D107—24no—
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C4⋊C4×C107C2211yescharacteristic
50C2×C4⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54#95×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C2×C22⋊D107—22no—
57C2×C22⋊D107—22no—
58C2×C22⋊D107—22no—
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C2×C4×D107—22no—
67C2×C4×D107—22no—
68C2×C4×D107—22no—
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C2×C4×D107—22no—
72C2×C4×D107—22no—
73C2×C4×D107—22no—
74C2×C4×D107D414yes—
75C4⋊Dic107—22no—
76C4⋊Dic107—22no—
77C4⋊Dic107—22no—
78C4⋊Dic107—22no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4—22no—
82Dic107⋊C4—22no—
83Dic107⋊C4—22no—
84C22⋊Dic107—22no—
85C22⋊Dic107—22no—
86C22⋊Dic107—22no—
87(C2×D107)⋊C4—22no—
88(C2×D107)⋊C4—22no—
89(C2×D107)⋊C4D414yes—
90C4×Dic107—22no—
91C4×Dic107—22no—
92C24×C107C2311yescharacteristic
93C22×C4×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96C22×C4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C4⋊C4×C107—22no—
100C4⋊C4×C107—22no—
101C4⋊C4×C107—22no—
102C22⋊C4×C107—22no—
103C22⋊C4×C107—22no—
104C22⋊C4×C107—22no—
105C23×C107—21no—
106C23×C107—21no—
107C23×C107—21no—
108C23×C107—21no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107—21no—
112C23×C107C2411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—21no—
117C22×D107C2×D412yes—
118C22×D107C2×D412yes—
119C22×D107#4112yes—
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×C107—21no—
135C2×C4×C107C2×D412yes—
136C2×C4×C107C2×D412yes—
137C2×C4×C107#4112yes—
138C22⋊D107—44no—
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×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157C2×Dic107C2×D412yes—
158C2×Dic107C2×D412yes—
159C2×Dic107#4112yes—
160C4×D107—28no—
161C4×D107—28no—
162C4×D107—44no—
163C22×C107—41no—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—41no—
170C22×C107C22×D411yescharacteristic
171C22×C107C22×D411yescharacteristic
172C22×C107C2×#4111yescharacteristic
173C22×C107C2×#4111yescharacteristic
174C22×C107C2×#4111yescharacteristic
175C22×C1072−1+411yescharacteristic
176C22×C1072−1+411yescharacteristic
177C2×D107—22no—
178C2×D107—22no—
179C2×D107—22no—
180C2×D107—22no—
181C2×D107—22no—
182C2×D107—22no—
183C2×D107#12114yes—
184C4×C107—22no—
185C4×C107—22no—
186C4×C107—22no—
187C4×C107—41no—
188C4×C107—41no—
189C4×C107—41no—
190Dic107—22no—
191Dic107—22no—
192Dic107—22no—
193Dic107—41no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—81no—
198C2×C107C2×#12111yescharacteristic
199C2×C107#52511yescharacteristic
200C2×C107#53611yescharacteristic
201C2×C107#54711yescharacteristic
202C2×C107#54711yescharacteristic
203C2×C107#55211yescharacteristic
204C2×C107#55211yescharacteristic
205D107—24no—
206#2413—1071nomaximal, Sylow
207C107#241311yescharacteristic, complemented, Sylow
208C2×#123—1071no—
209C2×#123—1071no—
210C2×#117—1071no—
211C22×C4⋊C4—1071no—
212#398—1071no—
213#398—1071no—
214#396—1071no—
215#396D10711yescharacteristic, complemented
216#389—1071no—
217#389—1071no—
218#388—1071no—
219#388—1071no—
220#386—1071no—
221#385—1071no—
222#381—1071no—
223C22×D4—1071no—
224C23×C4—1071no—
225C23×C4—1071no—
226C23×C4—1071no—
227#123—2144no—
228#123—2144no—
229#117—2144no—
230C2×C4⋊C4—1071no—
231C2×C4⋊C4—1071no—
232C2×C4⋊C4—1071no—
233C2×C4⋊C4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C4⋊C4—1071no—
237C2×C4⋊C4—1071no—
238C2×C4⋊C4—1071no—
239C2×C4⋊C4—2144no—
240C2×C4⋊C4C2×D10711yescharacteristic
241C2×C4⋊C4C2×D10711yescharacteristic
242C2×C4⋊C4C2×D10711yescharacteristic
243C2×C22⋊C4—1071no—
244C2×C22⋊C4—1071no—
245C2×C22⋊C4—1071no—
246C2×C22⋊C4—1071no—
247C2×C22⋊C4—1071no—
248C2×C22⋊C4—1071no—
249C2×C22⋊C4C2×D10711yescharacteristic
250C2×C22⋊C4C2×D10711yescharacteristic
251C2×C22⋊C4C2×D10711yescharacteristic
252C2×C42—1071no—
253C2×C42—1071no—
254#95—1071no—
255#95—1071no—
256#95—1071no—
257#95—1071no—
258#95—1071no—
259#95—1071no—
260#95—1071no—
261#95C2×D10711yescharacteristic
262C24—1071no—
263C24C22×D10711yescharacteristic
264C2×D4—2142no—
265C2×D4—2142no—
266C2×D4—2142no—
267C22×C4—1071no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—1074no—
271C22×C4—1071no—
272C22×C4—1071no—
273C22×C4—1071no—
274C22×C4—1071no—
275C22×C4—2142no—
276C22×C4—2142no—
277C22×C4—2142no—
278C22×C4—2142no—
279C22×C4—2142no—
280C22×C4—2142no—
281C22×C4—2142no—
282C22×C4—2142no—
283C22×C4C22×D10711yescharacteristic
284C22×C4C22×D10711yescharacteristic
285C22×C4C22×D10711yescharacteristic
286C22×C4C22×D10711yescharacteristic
287C22×C4C22×D10711yescharacteristic
288C22×C4C22×D10711yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—2142no—
293C4⋊C4—2142no—
294C4⋊C4—2142no—
295C4⋊C4—2142no—
296C4⋊C4—2142no—
297C4⋊C4—2142no—
298C4⋊C4—2142no—
299C4⋊C4—2142no—
300C4⋊C4—2142no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1074no—
305C22⋊C4—2142no—
306C22⋊C4—2142no—
307C22⋊C4—2142no—
308C22⋊C4—2142no—
309C22⋊C4—2142no—
310C42—2142no—
311C42—2142no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—1072no—
320C23—1072no—
321C23—1072no—
322C23—2141no—
323C23—2141no—
324C23—2141no—
325C23—2141no—
326C23C23×D10711yescharacteristic
327D4—4284no—
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—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—1072no—
344C2×C4—1072no—
345C2×C4—1072no—
346C2×C4—4284no—
347C2×C4—2148no—
348C2×C4—2148no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4—2141no—
358C2×C4—2141no—
359C2×C4—2141no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4D4×D10712yes—
368C2×C4D4×D10712yes—
369C2×C4#1313381712yes—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1074no—
378C22—2142no—
379C22—2142no—
380C22—2142no—
381C22—2142no—
382C22—2142no—
383C22—2142no—
384C22C2×D4×D10711yescharacteristic
385C22C2×D4×D10711yescharacteristic
386C22#41×D10711yescharacteristic
387C22#41×D10711yescharacteristic
388C22C2×#1313381711yescharacteristic
389C22#1681215611yescharacteristic
390C22#1681215711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—4281no—
398C4—4281no—
399C4—4281no—
400C4—4281no—
401C4—2142no—
402C4—2142no—
403C4—2142no—
404C2—81no—
405C2—2144no—
406C2#121×D10711yescharacteristic
407C2#2029313711yescharacteristic
408C2#2029320011yescharacteristic
409C2#2029327711yescharacteristic
410C2#2029327911yescharacteristic
411C2#2029331111yescharacteristic
412C2#2029331311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#396×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#396D107
hypocenterC107#2413
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
3. C2×#16812067C2semidirect
4. C2×C4⋊C4×D107C2semidirect
5. #20292560C2semidirect
6. #20292557C2semidirect
7. #20292550C2semidirect
8. #20292526C2semidirect
9. #20292524C2semidirect
10. #20292520C2semidirect
11. #20292515C2semidirect
13. #20292498C2semidirect
14. #20292489C2semidirect
15. #396×C107C2semidirect
207. C107#2413semidirect
215. #396D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120912
C2×#168120672
C2×C4⋊C4×D1072
#202925602
#202925572
#202925502
#202925262
#202925242
#202925202
#202925152
#202925032
#202924982
#202924892
#396×C1072
#2413107