Finite Group Databasebeta

← #28344758

Subgroups — #28344758

Order 13696.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812095C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic, complemented
3C2×#16812081C211yesmaximal, characteristic
4C22×C4×Dic107C211yesmaximal, characteristic, complemented
5#20292531C211yesmaximal, characteristic, complemented
6#20292527C211yesmaximal, characteristic, complemented
7#20292520C211yesmaximal, characteristic, complemented
8#20292519C211yesmaximal, characteristic, complemented
9#20292516C211yesmaximal, characteristic, complemented
10#20292515C211yesmaximal, characteristic, complemented
11#20292507C211yesmaximal, characteristic, complemented
12#20292506C211yesmaximal, characteristic, complemented
13#20292504C211yesmaximal, characteristic, complemented
14#20292491C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic
20C23×Dic107C2211yescharacteristic
21#16812095—24no—
22#16812086—24no—
23#16812081—24no—
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×C4×Dic107—24no—
38C2×C4×Dic107C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×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
55C23×D107C2311yescharacteristic
56C2×Dic214—22no—
57C2×Dic214—22no—
58C2×Dic214—22no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C22×Dic107—22no—
66C22×Dic107—22no—
67C22×Dic107—22no—
68C22×Dic107—22no—
69C22×Dic107—22no—
70C22×Dic107D414yes—
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C22×Dic107C2311yescharacteristic
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C4⋊Dic107—22no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4—22no—
82C22⋊Dic107—22no—
83C22⋊Dic107—22no—
84C22⋊Dic107—22no—
85C22⋊Dic107—22no—
86(C2×D107)⋊C4—22no—
87(C2×D107)⋊C4—22no—
88(C2×D107)⋊C4—22no—
89(C2×D107)⋊C4—22no—
90(C2×D107)⋊C4—22no—
91C4×Dic107—22no—
92C4×Dic107—22no—
93C4×Dic107—22no—
94C4×Dic107D414yescomplemented
95C24×C107C2311yescharacteristic
96C24×C107C2311yescharacteristic
97C2×D4×C107—22no—
98C2×D4×C107—22no—
99C2×D4×C107—22no—
100C22×C4×C107—22no—
101C22×C4×C107—22no—
102C22×C4×C107—22no—
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C22×C4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108C22⋊C4×C107—22no—
109C22⋊C4×C107—22no—
110C22⋊C4×C107—22no—
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×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107C2×D412yes—
123C23×C107#4112yes—
124C23×C107#4112yes—
125C23×C107C2411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—21no—
133D4×C107—44no—
134C2×C4×C107—28no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107C2×D412yes—
149C2×C4×C107#4112yes—
150C22⋊D107—44no—
151C22⋊D107—44no—
152C2×Dic107—28no—
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107—21no—
159C2×Dic107—28no—
160C2×Dic107—21no—
161C2×Dic107—21no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—21no—
166C2×Dic107—21no—
167C2×Dic107—21no—
168C2×Dic107—21no—
169C2×Dic107C2×D412yes—
170C2×Dic107C2×D412yes—
171C2×Dic107C2×D412yes—
172C2×Dic107C2×D412yes—
173C2×Dic107#4112yes—
174C2×Dic107#4112yes—
175Dic214—44no—
176C22×C107—22no—
177C22×C107—22no—
178C22×C107—22no—
179C22×C107—22no—
180C22×C107—22no—
181C22×C107—22no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107#12414yescomplemented
190C22×C107C22×D411yescharacteristic
191C22×C107C22×D411yescharacteristic
192C22×C107C2×#4111yescharacteristic
193C22×C107C2×#4111yescharacteristic
194C22×C107C2×#4111yescharacteristic
195C22×C107C2×#4111yescharacteristic
196C22×C107C2×#4111yescharacteristic
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—41no—
204C4×C107—22no—
205C4×C107—22no—
206C4×C107—41no—
207C4×C107—41no—
208C4×C107—41no—
209Dic107—22no—
210Dic107—22no—
211Dic107—22no—
212Dic107—41no—
213Dic107—41no—
214Dic107—41no—
215Dic107#12114yes—
216C2×C107—24no—
217C2×C107—81no—
218C2×C107C2×#12111yescharacteristic
219C2×C107#52511yescharacteristic
220C2×C107#52511yescharacteristic
221C2×C107C2×#12411yescharacteristic
222C2×C107#52911yescharacteristic
223C2×C107#52911yescharacteristic
224C2×C107#53211yescharacteristic
225D107—81no—
226#2850—1071nomaximal, Sylow
227C107#285011yescharacteristic, complemented, Sylow
228C2×#124—1071no—
229C2×C22⋊Q8—1071no—
230C2×#121—1071no—
231C22×C42—1071no—
232#391—1071no—
233#390—1071no—
234#388—1071no—
235#388—1071no—
236#388—1071no—
237#388—1071no—
238#386—1071no—
239#386—1071no—
240#386—1071no—
241#386D10711yescharacteristic, complemented
242#382—1071no—
243C22×Q8—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4C2×D10711yescharacteristic
247C23×C4—1071no—
248C23×C4—1071no—
249C23×C4C2×D10711yescharacteristic
250#124—2144no—
251C22⋊Q8—2144no—
252#121—2144no—
253C2×C4⋊C4—1071no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4—1071no—
257C2×C22⋊C4—1071no—
258C2×C22⋊C4—1071no—
259C2×C22⋊C4—1071no—
260C2×C22⋊C4—1071no—
261C2×C22⋊C4—1071no—
262C2×C22⋊C4—1071no—
263C2×C22⋊C4—1071no—
264C2×C22⋊C4—1071no—
265C2×C22⋊C4—1071no—
266C2×C22⋊C4C2×D10711yescharacteristic
267C2×C22⋊C4C2×D10711yescharacteristic
268C2×C22⋊C4C2×D10711yescharacteristic
269C2×C42—1071no—
270C2×C42—1071no—
271C2×C42—1071no—
272C2×C42—1071no—
273C2×C42—2144no—
274#95—1071no—
275#95—1071no—
276#95—1071no—
277#95—1071no—
278#95—1071no—
279#95—1071no—
280#95C2×D10711yescharacteristic
281#95C2×D10711yescharacteristic
282C24—1071no—
283C24C22×D10711yescharacteristic
284C24C22×D10711yescharacteristic
285C2×Q8—2142no—
286C2×Q8—2142no—
287C2×Q8—2142no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—2142no—
292C2×D4—2142no—
293C2×D4—2142no—
294C2×D4—2142no—
295C2×D4—2142no—
296C2×D4—2142no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—1071no—
301C22×C4—1074no—
302C22×C4—1071no—
303C22×C4—1071no—
304C22×C4—1071no—
305C22×C4—1071no—
306C22×C4—1071no—
307C22×C4—1071no—
308C22×C4—2142no—
309C22×C4—2142no—
310C22×C4—2142no—
311C22×C4—2142no—
312C22×C4—2142no—
313C22×C4C22×D10711yescharacteristic
314C22×C4C22×D10711yescharacteristic
315C22×C4C22×D10711yescharacteristic
316C22×C4C22×D10711yescharacteristic
317C22×C4C22×D10711yescharacteristic
318C4⋊C4—2142no—
319C4⋊C4—2142no—
320C4⋊C4—2142no—
321C4⋊C4—2142no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—2142no—
326C22⋊C4—2142no—
327C22⋊C4—2142no—
328C22⋊C4—2142no—
329C22⋊C4—2142no—
330C22⋊C4—2142no—
331C22⋊C4—2142no—
332C22⋊C4—2142no—
333C22⋊C4—2142no—
334C42—1074no—
335C42—2142no—
336C42—2142no—
337C42—2142no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—2141no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355C23—2141no—
356C23D4×D10712yes—
357C23#1313381712yes—
358C23#1313381812yes—
359C23C23×D10711yescharacteristic
360Q8—4284no—
361D4—44no—
362D4—4284no—
363D4—4284no—
364C2×C4—28no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—1072no—
379C2×C4—1072no—
380C2×C4—1072no—
381C2×C4—1072no—
382C2×C4—1072no—
383C2×C4—1072no—
384C2×C4—2148no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4—2141no—
391C2×C4—2141no—
392C2×C4—2141no—
393C2×C4—2141no—
394C2×C4—2148no—
395C2×C4—2141no—
396C2×C4—2141no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4D4×D10712yes—
402C2×C4#1313381712yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—4281no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22#1681209514yescomplemented
424C22C2×D4×D10711yescharacteristic
425C22C2×D4×D10711yescharacteristic
426C22#41×D10711yescharacteristic
427C22#41×D10711yescharacteristic
428C22C2×#1313381711yescharacteristic
429C22C2×#1313381711yescharacteristic
430C22C2×#1313381811yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—1074no—
437C4—4281no—
438C4—4281no—
439C4—4281no—
440C4—2142no—
441C4—2142no—
442C4—2142no—
443C2—24no—
444C2—81no—
445C2—8561no—
446C2#121×D10711yescharacteristic
447C2#2029313311yescharacteristic
448C2#2029313611yescharacteristic
449C2C2×#1681209511yescharacteristic
450C2#2029315411yescharacteristic
451C2#2029315711yescharacteristic
452C2#2029317211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC2×#16812081C2
hypercenter#386D107
hypocenterC107#2850
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#16812095C2semidirect
2. C2×#16812086C2semidirect
4. C22×C4×Dic107C2semidirect
5. #20292531C2semidirect
6. #20292527C2semidirect
7. #20292520C2semidirect
8. #20292519C2semidirect
9. #20292516C2semidirect
10. #20292515C2semidirect
11. #20292507C2semidirect
12. #20292506C2semidirect
13. #20292504C2semidirect
14. #20292491C2semidirect
15. #386×C107C2semidirect
94. C4×Dic107D4semidirect
189. C22×C107#124semidirect
227. C107#2850semidirect
241. #386D107semidirect
423. C22#16812095semidirect

Maximal subgroups

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

subgroupindex
C2×#168120952
C2×#168120862
C2×#168120812
C22×C4×Dic1072
#202925312
#202925272
#202925202
#202925192
#202925162
#202925152
#202925072
#202925062
#202925042
#202924912
#386×C1072
#2850107