Finite Group Databasebeta

← #28344788

Subgroups — #28344788

Order 13696.

688 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812094C211yesmaximal, characteristic, complemented
2C2×#16812091C211yesmaximal, characteristic, complemented
3C2×#16812082C211yesmaximal, characteristic
4C2×C42×D107C211yesmaximal, characteristic, complemented
5#20292527C211yesmaximal, characteristic, complemented
6#20292527C211yesmaximal, characteristic, complemented
7#20292524C211yesmaximal, characteristic, complemented
8#20292522C211yesmaximal, characteristic, complemented
9#20292519C211yesmaximal, characteristic, complemented
10#20292518C211yesmaximal, characteristic, complemented
11#20292515C211yesmaximal, characteristic, complemented
12#20292505C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292486C211yesmaximal, characteristic, complemented
15#388×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812094—24no—
23#16812091—24no—
24#16812082—24no—
25C2×C4⋊Dic107C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C2×C4×Dic107C2211yescharacteristic
42C2×C4×Dic107C2211yescharacteristic
43C42×D107—24no—
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C107⋊#95C2211yescharacteristic
48C2×C4⋊C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52C2×C42×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54#95×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C23×D107C2311yescharacteristic
57C2×D428—22no—
58C2×D428—22no—
59C2×D428—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
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—22no—
79C2×C4×D107D414yes—
80C2×C4×D107D414yescomplemented
81C4⋊Dic107—22no—
82Dic107⋊C4—22no—
83Dic107⋊C4—22no—
84Dic107⋊C4—22no—
85Dic107⋊C4—22no—
86Dic107⋊C4—22no—
87C22⋊Dic107—22no—
88C22⋊Dic107—22no—
89(C2×D107)⋊C4—22no—
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—22no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—22no—
94(C2×D107)⋊C4—22no—
95(C2×D107)⋊C4—22no—
96C4×Dic107—22no—
97C4×Dic107—22no—
98C4×Dic107—22no—
99C24×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C22×C4×C107C2311yescharacteristic
106C4⋊C4×C107—22no—
107C22⋊C4×C107—22no—
108C22⋊C4×C107—22no—
109C22⋊C4×C107—22no—
110C42×C107—22no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—21no—
115C23×C107—21no—
116C23×C107—21no—
117C23×C107—21no—
118C23×C107C2411yescharacteristic
119C22×D107—21no—
120C22×D107—21no—
121C22×D107—21no—
122C22×D107—21no—
123C22×D107—21no—
124C22×D107—21no—
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107C2×D412yes—
131C22×D107C2×D412yes—
132C22×D107#4112yes—
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×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×C107C2×D412yes—
150C2×C4×C107#4112yes—
151C22⋊D107—44no—
152C22⋊D107—44no—
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—21no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—21no—
166C2×Dic107C2×D412yes—
167C2×Dic107C2×D412yes—
168C2×Dic107#4112yes—
169C2×Dic107#4112yes—
170C2×Dic107#4112yes—
171D428—44no—
172C4×D107—28no—
173C4×D107—28no—
174C4×D107—28no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107C22×D411yescharacteristic
183C22×C107C22×D411yescharacteristic
184C22×C107C2×#4111yescharacteristic
185C22×C107C2×#4111yescharacteristic
186C22×C107C2×#4111yescharacteristic
187C22×C107C2×#4111yescharacteristic
188C22×C107C2×#4111yescharacteristic
189C2×D107—22no—
190C2×D107—22no—
191C2×D107—22no—
192C2×D107—22no—
193C2×D107—22no—
194C2×D107—22no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107#12114yes—
203C4×C107—22no—
204C4×C107—22no—
205C4×C107—22no—
206C4×C107—41no—
207C4×C107—41no—
208C4×C107—41no—
209Dic107—22no—
210Dic107—22no—
211Dic107—41no—
212Dic107—41no—
213Dic107—41no—
214Dic107#12314yescomplemented
215C2×C107—81no—
216C2×C107C2×#12111yescharacteristic
217C2×C107C2×#12311yescharacteristic
218C2×C107#52511yescharacteristic
219C2×C107#52911yescharacteristic
220C2×C107#52911yescharacteristic
221C2×C107#53211yescharacteristic
222C2×C107#53211yescharacteristic
223D107—24no—
224D107—81no—
225#2854—1071nomaximal, Sylow
226C107#285411yescharacteristic, complemented, Sylow
227C2×#123—1071no—
228C2×#123—1071no—
229C2×#121—1071no—
230C22×C42—1071no—
231#390—1071no—
232#390—1071no—
233#389—1071no—
234#389—1071no—
235#388—1071no—
236#388—1071no—
237#388—1071no—
238#388D10711yescharacteristic, complemented
239#386—1071no—
240#386—1071no—
241#380—1071no—
242C22×D4—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C23×C4—1071no—
246C23×C4—1071no—
247C23×C4—1071no—
248#123—2144no—
249#123—2144no—
250#121—2144no—
251C2×C4⋊C4—1071no—
252C2×C4⋊C4—1071no—
253C2×C4⋊C4—1071no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4—1071no—
257C2×C4⋊C4C2×D10711yescharacteristic
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⋊C4—1071no—
267C2×C22⋊C4C2×D10711yescharacteristic
268C2×C22⋊C4C2×D10711yescharacteristic
269C2×C22⋊C4C2×D10711yescharacteristic
270C2×C42—1071no—
271C2×C42—1071no—
272C2×C42—1071no—
273C2×C42—2144no—
274C2×C42C2×D10711yescharacteristic
275#95—1071no—
276#95—1071no—
277#95—1071no—
278#95—1071no—
279#95C2×D10711yescharacteristic
280#95C2×D10711yescharacteristic
281C24—1071no—
282C24—1071no—
283C24C22×D10711yescharacteristic
284C2×D4—2142no—
285C2×D4—2142no—
286C2×D4—2142no—
287C2×D4—2142no—
288C2×D4—2142no—
289C2×D4—2142no—
290C2×D4—2142no—
291C2×D4—2142no—
292C2×D4—2142no—
293C22×C4—1071no—
294C22×C4—1071no—
295C22×C4—1071no—
296C22×C4—1074no—
297C22×C4—1074no—
298C22×C4—1071no—
299C22×C4—1071no—
300C22×C4—1071no—
301C22×C4—2142no—
302C22×C4—2142no—
303C22×C4—2142no—
304C22×C4—2142no—
305C22×C4—2142no—
306C22×C4—2142no—
307C22×C4—2142no—
308C22×C4C22×D10711yescharacteristic
309C22×C4C22×D10711yescharacteristic
310C22×C4C22×D10711yescharacteristic
311C22×C4C22×D10711yescharacteristic
312C22×C4C22×D10711yescharacteristic
313C22×C4C22×D10711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—2142no—
316C4⋊C4—2142no—
317C4⋊C4—2142no—
318C4⋊C4—2142no—
319C4⋊C4—2142no—
320C4⋊C4—2142no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—2142no—
325C22⋊C4—2142no—
326C22⋊C4—2142no—
327C22⋊C4—2142no—
328C22⋊C4—2142no—
329C22⋊C4—2142no—
330C22⋊C4—2142no—
331C22⋊C4—2142no—
332C22⋊C4—2142no—
333C42—22no—
334C42—2142no—
335C42—2142no—
336C42—2142no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—1072no—
345C23—1072no—
346C23—1072no—
347C23—2141no—
348C23—2141no—
349C23—2141no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355C23—2141no—
356C23—2141no—
357C23—2141no—
358C23C23×D10711yescharacteristic
359D4—4284no—
360D4—4284no—
361D4—4284no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
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—1072no—
378C2×C4—1072no—
379C2×C4—1072no—
380C2×C4—1072no—
381C2×C4—1072no—
382C2×C4—2148no—
383C2×C4—2148no—
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—2141no—
395C2×C4—2141no—
396C2×C4—2141no—
397C2×C4—2141no—
398C2×C4D4×D10712yes—
399C2×C4D4×D10712yes—
400C2×C4#1313381812yes—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—1074no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—4281no—
415C22—4281no—
416C22—2142no—
417C22—2142no—
418C22—2142no—
419C22—2142no—
420C22—2142no—
421C22—2142no—
422C22C2×D4×D10711yescharacteristic
423C22C2×D4×D10711yescharacteristic
424C22#41×D10711yescharacteristic
425C22#41×D10711yescharacteristic
426C22#41×D10711yescharacteristic
427C22#41×D10711yescharacteristic
428C22C2×#1313381811yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—1074no—
436C4—4281no—
437C4—4281no—
438C4—4281no—
439C4—2142no—
440C4—2142no—
441C2—81no—
442C2—8561no—
443C2—2144no—
444C2#121×D10711yescharacteristic
445C2#123×D10711yescharacteristic
446C2#2029313611yescharacteristic
447C2#2029315511yescharacteristic
448C2#2029316011yescharacteristic
449C2#2029317411yescharacteristic
450C2#2029317611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#388×C107C2
prime radicalC2×#16812082C2
hypercenter#388D107
hypocenterC107#2854
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
2. C2×#16812091C2semidirect
4. C2×C42×D107C2semidirect
5. #20292527C2semidirect
6. #20292527C2semidirect
7. #20292524C2semidirect
8. #20292522C2semidirect
9. #20292519C2semidirect
10. #20292518C2semidirect
11. #20292515C2semidirect
12. #20292505C2semidirect
13. #20292503C2semidirect
14. #20292486C2semidirect
15. #388×C107C2semidirect
80. C2×C4×D107D4semidirect
214. Dic107#123semidirect
226. C107#2854semidirect
238. #388D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120912
C2×#168120822
C2×C42×D1072
#202925272
#202925242
#202925222
#202925192
#202925182
#202925152
#202925052
#202925032
#202924862
#388×C1072
#2854107