Finite Group Databasebeta

← #28339909

Subgroups — #28339909

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×#16812086C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic, complemented
3C2×D4×Dic107C211yesmaximal, characteristic
4C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
5#20292561C211yesmaximal, characteristic, complemented
6#20292559C211yesmaximal, characteristic, complemented
7#20292550C211yesmaximal, characteristic, complemented
8#20292550C211yesmaximal, characteristic, complemented
9#20292531C211yesmaximal, characteristic, complemented
10#20292524C211yesmaximal, characteristic, complemented
11#20292507C211yesmaximal, characteristic
12#20292487C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic, complemented
14#20292485C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic
20#16812086—24no—
21#16812086—24no—
22D4×Dic107—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×C4⋊Dic107C2211yescharacteristic
26C2×C4⋊Dic107C2211yescharacteristic
27C2×Dic107⋊C4—24no—
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×Dic107⋊C4C2211yescharacteristic
34C2×Dic107⋊C4C2211yescharacteristic
35C2×Dic107⋊C4C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C22⋊Dic107C2211yescharacteristic
38C2×C22⋊Dic107C2211yescharacteristic
39C2×C22⋊Dic107C2211yescharacteristic
40C2×C22⋊Dic107C2211yescharacteristic
41C2×C4×Dic107C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C22×D4×C107C2211yescharacteristic
49C23×C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54#95×C107C2211yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—22no—
57C2×Dic214—22no—
58C22×Dic107—22no—
59C22×Dic107—22no—
60C22×Dic107—22no—
61C22×Dic107—22no—
62C22×Dic107—22no—
63C22×Dic107—22no—
64C22×Dic107—22no—
65C22×Dic107—41no—
66C22×Dic107D414yes—
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C4⋊Dic107—22no—
76C4⋊Dic107—22no—
77C4⋊Dic107—22no—
78C4⋊Dic107—41no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4—22no—
82Dic107⋊C4—22no—
83Dic107⋊C4—22no—
84Dic107⋊C4—22no—
85Dic107⋊C4—41no—
86Dic107⋊C4D414yescomplemented
87C22⋊Dic107—22no—
88C22⋊Dic107—22no—
89C22⋊Dic107—22no—
90C22⋊Dic107—22no—
91C22⋊Dic107—41no—
92C4×Dic107—22no—
93C24×C107C2311yescharacteristic
94C24×C107C2311yescharacteristic
95C2×D4×C107—22no—
96C2×D4×C107—22no—
97C2×D4×C107Q814yescomplemented
98C22×C4×C107—22no—
99C22×C4×C107—22no—
100C22×C4×C107—41no—
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C22×C4×C107C2311yescharacteristic
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×C107—21no—
116C23×C107—21no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107C2×D412yes—
120C23×C107C2×Q812yes—
121C23×C107C2×Q812yes—
122C23×C107#4112yes—
123C23×C107C2411yescharacteristic
124D4×C107—28no—
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—44no—
139C2×C4×C107C2×D412yes—
140C2×C4×C107C2×Q812yes—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—28no—
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×Dic107—44no—
162C2×Dic107—44no—
163C2×Dic107C2×D412yes—
164C2×Dic107C2×D412yes—
165C2×Dic107C2×D412yes—
166C2×Dic107C2×D412yes—
167C2×Dic107#4112yes—
168Dic214—44no—
169C22×C107—22no—
170C22×C107—22no—
171C22×C107—22no—
172C22×C107—22no—
173C22×C107—22no—
174C22×C107—22no—
175C22×C107—22no—
176C22×C107—22no—
177C22×C107—22no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107C22⋊Q814yescomplemented
183C22×C107C22×D411yescharacteristic
184C22×C107C22×D411yescharacteristic
185C22×C107C22×Q811yescharacteristic
186C22×C107C2×#4111yescharacteristic
187C22×C107C2×#4111yescharacteristic
188C22×C107H(5,𝔽2)11yescharacteristic
189C22×C1072−1+411yescharacteristic
190C4×C107—22no—
191C4×C107—22no—
192C4×C107—41no—
193C4×C107—41no—
194C4×C107—41no—
195Dic107—22no—
196Dic107—22no—
197Dic107—41no—
198Dic107—41no—
199Dic107—41no—
200Dic107—41no—
201Dic107—41no—
202Dic107#12114yes—
203C2×C107—24no—
204C2×C107—42no—
205C2×C107C2×#12111yescharacteristic
206C2×C107C2×C22⋊Q811yescharacteristic
207C2×C107#53711yescharacteristic
208C2×C107#54611yescharacteristic
209C2×C107D4×Q811yescharacteristic
210C2×C107#55211yescharacteristic
211C2×C107#55411yescharacteristic
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#396—1071no—
222#391—1071no—
223#389—1071no—
224#386—1071no—
225#386D10711yescharacteristic
226#380—1071no—
227#380—1071no—
228#380—1071no—
229C22×Q8—1071no—
230C22×D4C2×D10711yescharacteristic
231C23×C4—1071no—
232C23×C4—1071no—
233C23×C4—1071no—
234C23×C4C2×D10711yescharacteristic
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—1071no—
248C2×C4⋊C4—1071no—
249C2×C4⋊C4—1071no—
250C2×C4⋊C4—2144no—
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#95C2×D10711yescharacteristic
267#95C2×D10711yescharacteristic
268C24C22×D10711yescharacteristic
269C24C22×D10711yescharacteristic
270C2×Q8—2142no—
271C2×Q8—2142no—
272C2×Q8—2142no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4Dic21414yescomplemented
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—1071no—
280C22×C4—1074no—
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×C4—2142no—
292C22×C4—2142no—
293C22×C4—2142no—
294C22×C4—2142no—
295C22×C4—2142no—
296C22×C4C22×D10711yescharacteristic
297C22×C4C22×D10711yescharacteristic
298C22×C4C22×D10711yescharacteristic
299C22×C4C22×D10711yescharacteristic
300C22×C4C22×D10711yescharacteristic
301C4⋊C4—1074no—
302C4⋊C4—4281no—
303C4⋊C4—4281no—
304C4⋊C4—2142no—
305C4⋊C4—2142no—
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—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23D4×D10712yes—
333C23C2×Dic21412yes—
334C23C2×Dic21412yes—
335C23#1313381712yes—
336C23C23×D10711yescharacteristic
337Q8—4284no—
338D4—28no—
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—44no—
353C2×C4—1072no—
354C2×C4—1072no—
355C2×C4—1072no—
356C2×C4—1072no—
357C2×C4—1072no—
358C2×C4—4284no—
359C2×C4—4284no—
360C2×C4—2148no—
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×C4—2141no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4D4×D10712yes—
381C2×C4C2×Dic21412yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#1681208614yescomplemented
396C22C2×D4×D10711yescharacteristic
397C22C2×D4×D10711yescharacteristic
398C22C22×Dic21411yescharacteristic
399C22#41×D10711yescharacteristic
400C22C2×#1313381711yescharacteristic
401C22#1681215411yescharacteristic
402C22#1681215611yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—1074no—
409C4—4281no—
410C4—4281no—
411C4—4281no—
412C4—4281no—
413C4—4281no—
414C4—2142no—
415C4—2142no—
416C2—24no—
417C2—42no—
418C2#121×D10711yescharacteristic
419C2C2×#1681208611yescharacteristic
420C2#2029320411yescharacteristic
421C2#2029326411yescharacteristic
422C2D4×Dic21411yescharacteristic
423C2#2029331411yescharacteristic
424C2#2029333611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC22×D4×C107C22
hypercenter#386D107
hypocenterC107#2417
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#16812086C2semidirect
2. C2×#16812086C2semidirect
4. C22×Dic107⋊C4C2semidirect
5. #20292561C2semidirect
6. #20292559C2semidirect
7. #20292550C2semidirect
8. #20292550C2semidirect
9. #20292531C2semidirect
10. #20292524C2semidirect
12. #20292487C2semidirect
13. #20292485C2semidirect
14. #20292485C2semidirect
86. Dic107⋊C4D4semidirect
97. C2×D4×C107Q8semidirect
182. C22×C107C22⋊Q8semidirect
213. C107#2417semidirect
275. C2×D4Dic214semidirect
395. C22#16812086semidirect

Maximal subgroups

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

subgroupindex
C2×#168120862
C2×D4×Dic1072
C22×Dic107⋊C42
#202925612
#202925592
#202925502
#202925312
#202925242
#202925072
#202924872
#202924852
#386×C1072
#2417107