Finite Group Databasebeta

← #28339149

Subgroups — #28339149

Order 13696.

668 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×#16812082C211yesmaximal, characteristic
2C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
3C2×#16812067C211yesmaximal, characteristic, complemented
4#20292529C211yesmaximal, characteristic, complemented
5#20292527C211yesmaximal, characteristic, complemented
6#20292527C211yesmaximal, characteristic, complemented
7#20292524C211yesmaximal, characteristic, complemented
8#20292517C211yesmaximal, characteristic, complemented
9#20292505C211yesmaximal, characteristic, complemented
10#20292504C211yesmaximal, characteristic, complemented
11#20292502C211yesmaximal, characteristic, complemented
12#20292501C211yesmaximal, characteristic, complemented
13#20292486C211yesmaximal, characteristic, complemented
14C4×Dic107⋊C4C211yesmaximal, characteristic, complemented
15#388×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C23×Dic107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812082C418yescomplemented
23(C22⋊D107)⋊C4—24no—
24#16812067—24no—
25C2×C4⋊Dic107C2211yescharacteristic
26C2×C4⋊Dic107C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
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×C4×Dic107C2211yescharacteristic
41C2×C4×Dic107C2211yescharacteristic
42C2×C4×Dic107C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C2×C4⋊C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C42×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54C23×D107C2311yescharacteristic
55C23×D107C2311yescharacteristic
56C2×D428—41no—
57C2×D428—41no—
58C2×D428C2×C414yescomplemented
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—41no—
62C2×C22⋊D107—41no—
63C2×C22⋊D107C2×C414yes—
64C2×C22⋊D107C2×C414yescomplemented
65C22×Dic107—22no—
66C22×Dic107—41no—
67C22×Dic107—41no—
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C2×C4×D107—22no—
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—41no—
78C2×C4×D107—41no—
79C2×C4×D107C2×C414yescomplemented
80C4⋊Dic107—22no—
81C4⋊Dic107—41no—
82Dic107⋊C4—22no—
83Dic107⋊C4—22no—
84Dic107⋊C4—41no—
85Dic107⋊C4C2×C414yescomplemented
86C22⋊Dic107—41no—
87(C2×D107)⋊C4—22no—
88(C2×D107)⋊C4—22no—
89(C2×D107)⋊C4—22no—
90(C2×D107)⋊C4—41no—
91(C2×D107)⋊C4—41no—
92(C2×D107)⋊C4—41no—
93(C2×D107)⋊C4—41no—
94(C2×D107)⋊C4C2×C414yescomplemented
95C4×Dic107—22no—
96C4×Dic107—41no—
97C4×Dic107D414yescomplemented
98C24×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C4⋊C4×C107—22no—
106C22⋊C4×C107—22no—
107C22⋊C4×C107—41no—
108C22⋊C4×C107C2×C414yescomplemented
109C42×C107—41no—
110C23×C107—21no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—21no—
115C23×C107—21no—
116C23×C107C22×C412yes—
117C23×C107C2411yescharacteristic
118C22×D107—21no—
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×D107C22×C412yes—
131C22×D107C22×C412yes—
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×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×C107C22×C412yes—
148C2×C4×C107C22×C412yes—
149C2×C4×C107C2×D412yes—
150C22⋊D107—28no—
151C22⋊D107—44no—
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—21no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—44no—
166C2×Dic107C22×C412yes—
167C2×Dic107C22×C412yes—
168C2×Dic107C2×D412yes—
169C2×Dic107C2×D412yes—
170C2×Dic107#4112yes—
171D428—44no—
172C4×D107—28no—
173C4×D107—44no—
174C22×C107—22no—
175C22×C107—22no—
176C22×C107—22no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C23×C411yescharacteristic
182C22×C107C22×D411yescharacteristic
183C22×C107C2×#4111yescharacteristic
184C22×C107H(5,𝔽2)11yescharacteristic
185C22×C107H(5,𝔽2)11yescharacteristic
186C22×C107H(5,𝔽2)11yescharacteristic
187C22×C1072−1+411yescharacteristic
188C2×D107—22no—
189C2×D107—22no—
190C2×D107—22no—
191C2×D107—22no—
192C2×D107—22no—
193C2×D107—22no—
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—22no—
205C4×C107—41no—
206C4×C107—41no—
207C4×C107—41no—
208Dic107—22no—
209Dic107—22no—
210Dic107—41no—
211Dic107—41no—
212Dic107—41no—
213Dic107C4×D414yes—
214C2×C107—42no—
215C2×C107C2×C4×D411yescharacteristic
216C2×C107#51811yescharacteristic
217C2×C107#52011yescharacteristic
218C2×C107#53511yescharacteristic
219C2×C107#53711yescharacteristic
220C2×C107#54011yescharacteristic
221C2×C107#54211yescharacteristic
222D107—42no—
223D107—42no—
224#2324—1071nomaximal, Sylow
225C107#232411yescharacteristic, complemented, Sylow
226C2×#121—1071no—
227C2×C4×D4—1071no—
228C2×#117—1071no—
229#390—1071no—
230#390—1071no—
231#390—1071no—
232#389—1071no—
233#388—1071no—
234#388D10711yescharacteristic, complemented
235#386—1071no—
236#386—1071no—
237#385—1071no—
238#385—1071no—
239#380—1071no—
240C4×C4⋊C4—1071no—
241C22×D4—1071no—
242C22×D4—1071no—
243C22×D4—1071no—
244C23×C4—1071no—
245C23×C4—1071no—
246C23×C4—1071no—
247#121—1078no—
248C4×D4—2144no—
249#117—2144no—
250C2×C4⋊C4—1071no—
251C2×C4⋊C4—1071no—
252C2×C4⋊C4—1071no—
253C2×C4⋊C4—1071no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4C2×D10711yescharacteristic
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×C42C2×D10711yescharacteristic
273#95—1071no—
274#95—1071no—
275#95—1071no—
276#95—1071no—
277#95C2×D10711yescharacteristic
278#95C2×D10711yescharacteristic
279C24—1071no—
280C24—1071no—
281C24C22×D10711yescharacteristic
282C2×D4—1074no—
283C2×D4—1074no—
284C2×D4—1074no—
285C2×D4—4281no—
286C2×D4—4281no—
287C2×D4—4281no—
288C2×D4—4281no—
289C2×D4—2142no—
290C2×D4—2142no—
291C22×C4—1071no—
292C22×C4—1071no—
293C22×C4—1071no—
294C22×C4—1074no—
295C22×C4—1071no—
296C22×C4—1071no—
297C22×C4—1071no—
298C22×C4—4281no—
299C22×C4—4281no—
300C22×C4—4281no—
301C22×C4—4281no—
302C22×C4—2142no—
303C22×C4—2142no—
304C22×C4—2142no—
305C22×C4—2142no—
306C22×C4C22×D10711yescharacteristic
307C22×C4C22×D10711yescharacteristic
308C22×C4C22×D10711yescharacteristic
309C22×C4C22×D10711yescharacteristic
310C22×C4C22×D10711yescharacteristic
311C22×C4C22×D10711yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—1074no—
314C4⋊C4—4281no—
315C4⋊C4—4281no—
316C4⋊C4—2142no—
317C4⋊C4—2142no—
318C4⋊C4—2142no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—1074no—
322C22⋊C4—4281no—
323C22⋊C4—4281no—
324C22⋊C4—4281no—
325C22⋊C4—4281no—
326C22⋊C4—4281no—
327C22⋊C4—2142no—
328C22⋊C4—2142no—
329C22⋊C4—2142no—
330C22⋊C4C4×D10714yescomplemented
331C42—41no—
332C42—1074no—
333C42—4281no—
334C42—2142no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—1072no—
342C23—1072no—
343C23—2141no—
344C23—2141no—
345C23—2141no—
346C23—2141no—
347C23—2141no—
348C23—2141no—
349C23—2141no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355C23C2×C4×D10712yes—
356C23C23×D10711yescharacteristic
357D4—4284no—
358D4—4284no—
359D4—2148no—
360C2×C4—21no—
361C2×C4—21no—
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—1072no—
376C2×C4—1072no—
377C2×C4—1072no—
378C2×C4—1072no—
379C2×C4—1072no—
380C2×C4—4284no—
381C2×C4—4284no—
382C2×C4—2148no—
383C2×C4—2141no—
384C2×C4—2141no—
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×C4C2×C4×D10712yes—
397C2×C4C2×C4×D10712yes—
398C2×C4D4×D10712yes—
399C22—22no—
400C22—22no—
401C22—22no—
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—2142no—
415C22—2142no—
416C22—2142no—
417C22—2142no—
418C22—2142no—
419C22—2142no—
420C22C22×C4×D10711yescharacteristic
421C22C2×D4×D10711yescharacteristic
422C22#41×D10711yescharacteristic
423C22#1681215411yescharacteristic
424C22#1681215511yescharacteristic
425C22#1681215511yescharacteristic
426C22#1681215611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—1074no—
434C4—4281no—
435C4—4281no—
436C4—4281no—
437C4—2142no—
438C4—2142no—
439C2—42no—
440C2—4282no—
441C2—4282no—
442C2C4×D4×D10711yescharacteristic
443C2#2029309911yescharacteristic
444C2#2029310711yescharacteristic
445C2#2029319511yescharacteristic
446C2#2029320411yescharacteristic
447C2#2029322911yescharacteristic
448C2#2029324311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#388×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812082C2
hypercenter#388D107
hypocenterC107#2324
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D107)⋊C4C2semidirect
3. C2×#16812067C2semidirect
4. #20292529C2semidirect
5. #20292527C2semidirect
6. #20292527C2semidirect
7. #20292524C2semidirect
8. #20292517C2semidirect
9. #20292505C2semidirect
10. #20292504C2semidirect
11. #20292502C2semidirect
12. #20292501C2semidirect
13. #20292486C2semidirect
14. C4×Dic107⋊C4C2semidirect
15. #388×C107C2semidirect
22. #16812082C4semidirect
58. C2×D428C2×C4semidirect
64. C2×C22⋊D107C2×C4semidirect
79. C2×C4×D107C2×C4semidirect
85. Dic107⋊C4C2×C4semidirect
94. (C2×D107)⋊C4C2×C4semidirect
97. C4×Dic107D4semidirect
108. C22⋊C4×C107C2×C4semidirect
225. C107#2324semidirect
234. #388D107semidirect
330. C22⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120822
C2×(C22⋊D107)⋊C42
C2×#168120672
#202925292
#202925272
#202925242
#202925172
#202925052
#202925042
#202925022
#202925012
#202924862
C4×Dic107⋊C42
#388×C1072
#2324107