Finite Group Databasebeta

← #28339905

Subgroups — #28339905

Order 13696.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812087C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic, complemented
3C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
4C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292560C211yesmaximal, characteristic, complemented
6#20292557C211yesmaximal, characteristic, complemented
7#20292550C211yesmaximal, characteristic, complemented
8#20292531C211yesmaximal, characteristic, complemented
9#20292522C211yesmaximal, characteristic, complemented
10#20292503C211yesmaximal, characteristic
11#20292503C211yesmaximal, characteristic
12#20292486C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic, complemented
14#20292484C211yesmaximal, characteristic, complemented
15#396×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812087—24no—
23#16812086—24no—
24(C22⋊D107)⋊C4—24no—
25C2×C4⋊Dic107C2211yescharacteristic
26C2×C4⋊Dic107C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×Dic107⋊C4C2211yescharacteristic
34C4⋊C4×D107—24no—
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
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×Dic214—22no—
57C2×Dic214—22no—
58C2×Dic214—22no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107Q814yescomplemented
62C22×Dic107—22no—
63C22×Dic107—22no—
64C22×Dic107—22no—
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C2×C4×D107—22no—
73C2×C4×D107—22no—
74C2×C4×D107—22no—
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—22no—
78C2×C4×D107—41no—
79C2×C4×D107—41no—
80C2×C4×D107D414yes—
81C4⋊Dic107—22no—
82C4⋊Dic107—22no—
83Dic107⋊C4—22no—
84Dic107⋊C4—22no—
85Dic107⋊C4—22no—
86Dic107⋊C4—22no—
87Dic107⋊C4—22no—
88Dic107⋊C4—41no—
89Dic107⋊C4D414yescomplemented
90C22⋊Dic107—22no—
91C22⋊Dic107—41no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—22no—
94(C2×D107)⋊C4—22no—
95C4×Dic107—22no—
96C24×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C4⋊C4×C107—22no—
104C4⋊C4×C107—22no—
105C4⋊C4×C107—41no—
106C22⋊C4×C107—22no—
107C22⋊C4×C107—22no—
108C22⋊C4×C107—41no—
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×D107C2×D412yes—
122C22×D107C2×Q812yes—
123C22×D107#4112yes—
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×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107C2×D412yes—
141C2×C4×C107C2×D412yes—
142C22⋊D107—28no—
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×Dic107—21no—
158C2×Dic107—21no—
159C2×Dic107—44no—
160C2×Dic107C2×D412yes—
161C2×Dic107C2×D412yes—
162C2×Dic107C2×D412yes—
163C2×Dic107C2×Q812yes—
164C2×Dic107#4112yes—
165C4×D107—28no—
166C4×D107—44no—
167C4×D107—44no—
168Dic214—44no—
169C22×C107—22no—
170C22×C107—22no—
171C22×C107—22no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107C22×D411yescharacteristic
177C22×C107C22×D411yescharacteristic
178C22×C107C22×Q811yescharacteristic
179C22×C107C2×#4111yescharacteristic
180C22×C107C2×#4111yescharacteristic
181C22×C107H(5,𝔽2)11yescharacteristic
182C22×C1072−1+411yescharacteristic
183C2×D107—22no—
184C2×D107—22no—
185C2×D107—22no—
186C2×D107—22no—
187C2×D107—22no—
188C2×D107—22no—
189C2×D107C22⋊Q814yescomplemented
190C4×C107—22no—
191C4×C107—22no—
192C4×C107—41no—
193C4×C107—41no—
194C4×C107—41no—
195C4×C107—41no—
196Dic107—22no—
197Dic107—22no—
198Dic107—41no—
199Dic107—41no—
200Dic107—41no—
201Dic107—41no—
202Dic107#12114yes—
203C2×C107—42no—
204C2×C107C2×#12111yescharacteristic
205C2×C107C2×C22⋊Q811yescharacteristic
206C2×C107#53711yescharacteristic
207C2×C107#54611yescharacteristic
208C2×C107D4×Q811yescharacteristic
209C2×C107#55211yescharacteristic
210C2×C107#55411yescharacteristic
211D107—24no—
212#2417—1071nomaximal, Sylow
213C107#241711yescharacteristic, complemented, Sylow
214C2×C22⋊Q8—1071no—
215C2×C22⋊Q8—1071no—
216C2×C4×D4—1071no—
217C22×C4⋊C4—1071no—
218#398—1071no—
219#398—1071no—
220#396—1071no—
221#396D10711yescharacteristic, complemented
222#391—1071no—
223#389—1071no—
224#386—1071no—
225#386—1071no—
226#380—1071no—
227#380—1071no—
228#380—1071no—
229C22×Q8—1071no—
230C22×D4—1071no—
231C23×C4—1071no—
232C23×C4—1071no—
233C23×C4—1071no—
234C23×C4—1071no—
235C22⋊Q8—2144no—
236C22⋊Q8—2144no—
237C4×D4—2144no—
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⋊C4—1071no—
246C2×C4⋊C4—1071no—
247C2×C4⋊C4—2144no—
248C2×C4⋊C4C2×D10711yescharacteristic
249C2×C4⋊C4C2×D10711yescharacteristic
250C2×C4⋊C4C2×D10711yescharacteristic
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×C22⋊C4C2×D10711yescharacteristic
259C2×C42—1071no—
260#95—1071no—
261#95—1071no—
262#95—1071no—
263#95—1071no—
264#95—1071no—
265#95—1071no—
266#95—1071no—
267#95C2×D10711yescharacteristic
268C24—1071no—
269C24C22×D10711yescharacteristic
270C2×Q8—2142no—
271C2×Q8—2142no—
272C2×Q8—2142no—
273C2×D4—1074no—
274C2×D4—2142no—
275C2×D4—2142no—
276C22×C4—1071no—
277C22×C4—1071no—
278C22×C4—1071no—
279C22×C4—1074no—
280C22×C4—1071no—
281C22×C4—1071no—
282C22×C4—1071no—
283C22×C4—1071no—
284C22×C4—4281no—
285C22×C4—4281no—
286C22×C4—2142no—
287C22×C4—2142no—
288C22×C4—2142no—
289C22×C4—2142no—
290C22×C4—2142no—
291C22×C4—2142no—
292C22×C4—2142no—
293C22×C4—2142no—
294C22×C4—2142no—
295C22×C4C22×D10711yescharacteristic
296C22×C4C22×D10711yescharacteristic
297C22×C4C22×D10711yescharacteristic
298C22×C4C22×D10711yescharacteristic
299C22×C4C22×D10711yescharacteristic
300C22×C4C22×D10711yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—1074no—
305C4⋊C4—4281no—
306C4⋊C4—2142no—
307C4⋊C4—2142no—
308C4⋊C4—2142no—
309C4⋊C4—2142no—
310C4⋊C4—2142no—
311C4⋊C4—2142no—
312C4⋊C4—2142no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—4281no—
317C22⋊C4—2142no—
318C22⋊C4—2142no—
319C22⋊C4—2142no—
320C22⋊C4—2142no—
321C42—2142no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—1072no—
329C23—1072no—
330C23—1072no—
331C23—2141no—
332C23—2141no—
333C23—2141no—
334C23—2141no—
335C23Q8×D10712yes—
336C23C23×D10711yescharacteristic
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—21no—
354C2×C4—21no—
355C2×C4—1072no—
356C2×C4—1072no—
357C2×C4—1072no—
358C2×C4—1072no—
359C2×C4—1072no—
360C2×C4—4284no—
361C2×C4—4284no—
362C2×C4—4284no—
363C2×C4—2148no—
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×C4—2141no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4D4×D10712yes—
381C2×C4D4×D10712yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—1074no—
390C22—2142no—
391C22—2142no—
392C22—2142no—
393C22—2142no—
394C22—2142no—
395C22—2142no—
396C22C2×D4×D10711yescharacteristic
397C22C2×D4×D10711yescharacteristic
398C22C2×Q8×D10711yescharacteristic
399C22#41×D10711yescharacteristic
400C22#41×D10711yescharacteristic
401C22#1681215411yescharacteristic
402C22#1681215611yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—1074no—
410C4—4281no—
411C4—4281no—
412C4—4281no—
413C4—4281no—
414C4—2142no—
415C4—2142no—
416C2—42no—
417C2—2144no—
418C2#121×D10711yescharacteristic
419C2C22⋊Q8×D10711yescharacteristic
420C2#2029320611yescharacteristic
421C2#2029326611yescharacteristic
422C2#2029329011yescharacteristic
423C2#2029331311yescharacteristic
424C2#2029333111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#396×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#396D107
hypocenterC107#2417
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#16812087C2semidirect
2. C2×#16812086C2semidirect
4. C2×C4⋊C4×D107C2semidirect
5. #20292560C2semidirect
6. #20292557C2semidirect
7. #20292550C2semidirect
8. #20292531C2semidirect
9. #20292522C2semidirect
12. #20292486C2semidirect
13. #20292485C2semidirect
14. #20292484C2semidirect
15. #396×C107C2semidirect
61. C2×C22⋊D107Q8semidirect
89. Dic107⋊C4D4semidirect
189. C2×D107C22⋊Q8semidirect
213. C107#2417semidirect
221. #396D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120872
C2×#168120862
C2×(C22⋊D107)⋊C42
C2×C4⋊C4×D1072
#202925602
#202925572
#202925502
#202925312
#202925222
#202925032
#202924862
#202924852
#202924842
#396×C1072
#2417107