Finite Group Databasebeta

← #28340022

Subgroups — #28340022

Order 13696.

632 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812095C211yesmaximal, characteristic, complemented
2C2×#16812087C211yesmaximal, characteristic, complemented
3C2×#16812082C211yesmaximal, characteristic
4C2×#16812074C211yesmaximal, characteristic, complemented
5#20292551C211yesmaximal, characteristic, complemented
6#20292544C211yesmaximal, characteristic, complemented
7#20292542C211yesmaximal, characteristic, complemented
8#20292528C211yesmaximal, characteristic, complemented
9#20292527C211yesmaximal, characteristic, complemented
10#20292527C211yesmaximal, characteristic, complemented
11#20292522C211yesmaximal, characteristic, complemented
12#20292517C211yesmaximal, characteristic, complemented
13#20292515C211yesmaximal, characteristic, complemented
14#20292474C211yesmaximal, characteristic, complemented
15#396×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21#16812095—24no—
22#16812087—24no—
23#16812082—24no—
24#16812074—24no—
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
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×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×C4×Dic107C2211yescharacteristic
42C2×C4×Dic107C2211yescharacteristic
43C2×C4×Dic107C2211yescharacteristic
44C2×C4×Dic107C2211yescharacteristic
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
56C23×D107C2311yescharacteristic
57C2×Dic214—22no—
58C2×Dic214—22no—
59C2×Dic214D414yescomplemented
60C2×D428—22no—
61C2×D428—22no—
62C2×D428—41no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—22no—
68C2×C22⋊D107—41no—
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4—41no—
82Dic107⋊C4D414yes—
83C22⋊Dic107—22no—
84C22⋊Dic107—22no—
85(C2×D107)⋊C4—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—
91(C2×D107)⋊C4—22no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—41no—
94C4×Dic107—22no—
95C4×Dic107—22no—
96C4×Dic107—22no—
97C4×Dic107—22no—
98C24×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C4⋊C4×C107—22no—
106C4⋊C4×C107—22no—
107C4⋊C4×C107—41no—
108C22⋊C4×C107—22no—
109C22⋊C4×C107—22no—
110C22⋊C4×C107—41no—
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×D107—21no—
131C22×D107—21no—
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×C107—21no—
149C2×C4×C107C2×D412yes—
150C2×C4×C107C2×D412yes—
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×Dic107C2×D412yes—
166C2×Dic107C2×D412yes—
167C2×Dic107C2×D412yes—
168C2×Dic107C2×D412yes—
169C2×Dic107#4112yes—
170C2×Dic107#4112yes—
171D428—44no—
172C4×D107—44no—
173Dic214—28no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C22×C107C22×D411yescharacteristic
183C22×C107C2×#4111yescharacteristic
184C22×C107C2×#4111yescharacteristic
185C22×C107C2×#4111yescharacteristic
186C22×C107H(5,𝔽2)11yescharacteristic
187C22×C1072−1+411yescharacteristic
188C2×D107—22no—
189C2×D107—22no—
190C2×D107—22no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C4×C107—22no—
203C4×C107—22no—
204C4×C107—41no—
205C4×C107—41no—
206C4×C107—41no—
207C4×C107—41no—
208Dic107—22no—
209Dic107—22no—
210Dic107—22no—
211Dic107—41no—
212Dic107—41no—
213Dic107#12114yes—
214C2×C107—81no—
215C2×C107C2×#12111yescharacteristic
216C2×C107#52911yescharacteristic
217C2×C107#54211yescharacteristic
218C2×C107#54611yescharacteristic
219C2×C107#54811yescharacteristic
220C2×C107#54811yescharacteristic
221C2×C107#55011yescharacteristic
222D107—42no—
223D107—81no—
224#2426—1071nomaximal, Sylow
225C107#242611yescharacteristic, complemented, Sylow
226C2×#124—1071no—
227C2×C22⋊Q8—1071no—
228C2×#121—1071no—
229C2×C4×Q8—1071no—
230#396—1071no—
231#396D10711yescharacteristic, complemented
232#394—1071no—
233#394—1071no—
234#390—1071no—
235#390—1071no—
236#390—1071no—
237#389—1071no—
238#388—1071no—
239#388—1071no—
240C4×C22⋊C4—1071no—
241C22×Q8—1071no—
242C22×D4—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C23×C4—1071no—
246#124—2144no—
247C22⋊Q8—2144no—
248#121—2144no—
249C4×Q8—2144no—
250C2×C4⋊C4—1071no—
251C2×C4⋊C4—1071no—
252C2×C4⋊C4—1071no—
253C2×C4⋊C4—1071no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4C2×D10711yescharacteristic
256C2×C4⋊C4C2×D10711yescharacteristic
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⋊C4—1071no—
268C2×C22⋊C4—1071no—
269C2×C22⋊C4C2×D10711yescharacteristic
270C2×C22⋊C4C2×D10711yescharacteristic
271C2×C22⋊C4C2×D10711yescharacteristic
272C2×C42—1071no—
273C2×C42—1071no—
274C2×C42—1071no—
275C2×C42—1071no—
276#95—1071no—
277#95—1071no—
278#95—1071no—
279#95C2×D10711yescharacteristic
280C24—1071no—
281C24—1071no—
282C24C22×D10711yescharacteristic
283C2×Q8—1074no—
284C2×Q8—2142no—
285C2×Q8—2142no—
286C2×D4—4281no—
287C2×D4—4281no—
288C2×D4—2142no—
289C2×D4—2142no—
290C2×D4—2142no—
291C2×D4—2142no—
292C2×D4—2142no—
293C2×D4—2142no—
294C2×D4—2142no—
295C22×C4—1071no—
296C22×C4—1071no—
297C22×C4—1071no—
298C22×C4—1071no—
299C22×C4—1071no—
300C22×C4—1071no—
301C22×C4—2142no—
302C22×C4—2142no—
303C22×C4—2142no—
304C22×C4C22×D10711yescharacteristic
305C22×C4C22×D10711yescharacteristic
306C22×C4C22×D10711yescharacteristic
307C22×C4C22×D10711yescharacteristic
308C22×C4C22×D10711yescharacteristic
309C22×C4C22×D10711yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—1074no—
314C4⋊C4—4281no—
315C4⋊C4—2142no—
316C4⋊C4—2142no—
317C4⋊C4—2142no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—4281no—
322C22⋊C4—2142no—
323C22⋊C4—2142no—
324C22⋊C4—2142no—
325C22⋊C4—2142no—
326C22⋊C4—2142no—
327C22⋊C4—2142no—
328C22⋊C4—2142no—
329C22⋊C4—2142no—
330C22⋊C4—2142no—
331C22⋊C4—2142no—
332C42—2142no—
333C42—2142no—
334C42—2142no—
335C42—2142no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—1072no—
344C23—2141no—
345C23—2141no—
346C23—2141no—
347C23—2141no—
348C23—2141no—
349C23—2141no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355C23—2141no—
356C23—2141no—
357C23C23×D10711yescharacteristic
358Q8—2148no—
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—21no—
378C2×C4—1072no—
379C2×C4—1072no—
380C2×C4—1072no—
381C2×C4—1072no—
382C2×C4—1072no—
383C2×C4—1072no—
384C2×C4—4284no—
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×C4D4×D10712yes—
398C2×C4D4×D10712yes—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—4281no—
415C22—4281no—
416C22—4281no—
417C22—2142no—
418C22—2142no—
419C22—2142no—
420C22C2×D4×D10711yescharacteristic
421C22C2×D4×D10711yescharacteristic
422C22#41×D10711yescharacteristic
423C22#41×D10711yescharacteristic
424C22#41×D10711yescharacteristic
425C22#1681215411yescharacteristic
426C22#1681215711yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—1074no—
434C4—4281no—
435C4—4281no—
436C4—2142no—
437C4—2142no—
438C4—2142no—
439C2—81no—
440C2—8561no—
441C2—4282no—
442C2#121×D10711yescharacteristic
443C2#2029316011yescharacteristic
444C2#2029324711yescharacteristic
445C2#2029326611yescharacteristic
446C2#2029328711yescharacteristic
447C2#2029328711yescharacteristic
448C2#2029329511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#396×C107C2
prime radicalC2×#16812082C2
hypercenter#396D107
hypocenterC107#2426
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#16812095C2semidirect
2. C2×#16812087C2semidirect
4. C2×#16812074C2semidirect
5. #20292551C2semidirect
6. #20292544C2semidirect
7. #20292542C2semidirect
8. #20292528C2semidirect
9. #20292527C2semidirect
10. #20292527C2semidirect
11. #20292522C2semidirect
12. #20292517C2semidirect
13. #20292515C2semidirect
14. #20292474C2semidirect
15. #396×C107C2semidirect
59. C2×Dic214D4semidirect
225. C107#2426semidirect
231. #396D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120952
C2×#168120872
C2×#168120822
C2×#168120742
#202925512
#202925442
#202925422
#202925282
#202925272
#202925222
#202925172
#202925152
#202924742
#396×C1072
#2426107