Finite Group Databasebeta

← #28339163

Subgroups — #28339163

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×#16812080C211yesmaximal, characteristic
2C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
3C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
4#20292521C211yesmaximal, characteristic, complemented
5#20292515C211yesmaximal, characteristic, complemented
6#20292512C211yesmaximal, characteristic, complemented
7#20292510C211yesmaximal, characteristic, complemented
8#20292507C211yesmaximal, characteristic, complemented
9#20292503C211yesmaximal, characteristic, complemented
10#20292503C211yesmaximal, characteristic, complemented
11#20292500C211yesmaximal, characteristic, complemented
12#20292499C211yesmaximal, characteristic, complemented
13#20292486C211yesmaximal, characteristic, complemented
14#95×D107C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20C22×C4×D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22#16812080C418yescomplemented
23C4×C22⋊D107—24no—
24(C22⋊D107)⋊C4—24no—
25C2×C4⋊Dic107C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×C4×Dic107C2211yescharacteristic
38C2×C4×Dic107C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C22×D4×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50C2×C22⋊C4×C107C2211yescharacteristic
51C2×C22⋊C4×C107C2211yescharacteristic
52#95×C107C2211yescharacteristic
53#95×C107C2211yescharacteristic
54C23×D107C2311yescharacteristic
55C2×C22⋊D107—22no—
56C2×C22⋊D107—22no—
57C2×C22⋊D107—22no—
58C2×C22⋊D107—22no—
59C2×C22⋊D107C2×C414yescomplemented
60C2×C22⋊D107C2×C414yescomplemented
61C22×Dic107—22no—
62C22×Dic107—41no—
63C22×Dic107—41no—
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C2×C4×D107—22no—
72C2×C4×D107—41no—
73C2×C4×D107—41no—
74C2×C4×D107—41no—
75C2×C4×D107—41no—
76C2×C4×D107—41no—
77C2×C4×D107—41no—
78C2×C4×D107C2×C414yescomplemented
79C2×C4×D107D414yescomplemented
80C4⋊Dic107C2×C414yescomplemented
81Dic107⋊C4—22no—
82Dic107⋊C4—22no—
83Dic107⋊C4—41no—
84Dic107⋊C4—41no—
85C22⋊Dic107—22no—
86C22⋊Dic107—41no—
87C22⋊Dic107—41no—
88C22⋊Dic107C2×C414yescomplemented
89C22⋊Dic107C2×C414yescomplemented
90(C2×D107)⋊C4—22no—
91(C2×D107)⋊C4—22no—
92C4×Dic107—22no—
93C4×Dic107—22no—
94C24×C107C2311yescharacteristic
95C24×C107C2311yescharacteristic
96C2×D4×C107—41no—
97C2×D4×C107—41no—
98C2×D4×C107C2×C414yescomplemented
99C22×C4×C107—22no—
100C22×C4×C107—41no—
101C22×C4×C107—41no—
102C22×C4×C107C2311yescharacteristic
103C22×C4×C107C2311yescharacteristic
104C22×C4×C107C2311yescharacteristic
105C22×C4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22⋊C4×C107—22no—
108C22⋊C4×C107—41no—
109C22⋊C4×C107—41no—
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×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107C22×C412yes—
123C23×C107C22×C412yes—
124C23×C107C2411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107C22×C412yes—
130C22×D107C2×D412yes—
131C22×D107#4112yes—
132D4×C107—44no—
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—44no—
147C2×C4×C107C22×C412yes—
148C2×C4×C107C2×D412yes—
149C22⋊D107—28no—
150C22⋊D107—28no—
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—21no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—21no—
166C2×Dic107—21no—
167C2×Dic107—21no—
168C2×Dic107—44no—
169C2×Dic107C22×C412yes—
170C2×Dic107C22×C412yes—
171C2×Dic107C22×C412yes—
172C2×Dic107C2×D412yes—
173C4×D107—44no—
174C4×D107—44no—
175C4×D107—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—41no—
190C22×C107C23×C411yescharacteristic
191C22×C107C22×D411yescharacteristic
192C22×C107C2×#4111yescharacteristic
193C22×C107H(5,𝔽2)11yescharacteristic
194C22×C107H(5,𝔽2)11yescharacteristic
195C22×C107H(5,𝔽2)11yescharacteristic
196C22×C1072−1+411yescharacteristic
197C2×D107—22no—
198C2×D107—22no—
199C2×D107—22no—
200C2×D107—22no—
201C2×D107—22no—
202C2×D107—22no—
203C2×D107C4×D414yes—
204C4×C107—22no—
205C4×C107—22no—
206C4×C107—41no—
207C4×C107—41no—
208C4×C107—41no—
209Dic107—22no—
210Dic107—22no—
211Dic107—22no—
212Dic107—22no—
213Dic107—41no—
214Dic107—41no—
215Dic107—41no—
216C2×C107—42no—
217C2×C107—42no—
218C2×C107C2×C4×D411yescharacteristic
219C2×C107#51811yescharacteristic
220C2×C107#52011yescharacteristic
221C2×C107#53411yescharacteristic
222C2×C107#53711yescharacteristic
223C2×C107#53811yescharacteristic
224C2×C107#53911yescharacteristic
225D107—24no—
226#2325—1071nomaximal, Sylow
227C107#232511yescharacteristic, complemented, Sylow
228C2×#121—1071no—
229C2×C4×D4—1071no—
230C2×C4×D4—1071no—
231#388—1071no—
232#388—1071no—
233#387—1071no—
234#387—1071no—
235#386—1071no—
236#386—1071no—
237#386—1071no—
238#386D10711yescharacteristic, complemented
239#385—1071no—
240#385—1071no—
241#380—1071no—
242C2×#95—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4C2×D10711yescharacteristic
246C23×C4—1071no—
247C23×C4—1071no—
248C23×C4—1071no—
249C23×C4—1071no—
250C23×C4C2×D10711yescharacteristic
251#121—1078no—
252C4×D4—2144no—
253C4×D4—2144no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4—1071no—
257C2×C4⋊C4—1071no—
258C2×C4⋊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—
271#95—1071no—
272#95—1071no—
273#95—1071no—
274#95—1071no—
275#95—1071no—
276#95—1071no—
277#95—1071no—
278#95—1071no—
279#95C2×D10711yescharacteristic
280#95C2×D10711yescharacteristic
281C24—1071no—
282C24C22×D10711yescharacteristic
283C24C22×D10711yescharacteristic
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—1074no—
287C2×D4—1074no—
288C2×D4—2142no—
289C2×D4—2142no—
290C2×D4—2142no—
291C2×D4—2142no—
292C2×D4C4×D10714yescomplemented
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—1071no—
297C22×C4—1071no—
298C22×C4—1071no—
299C22×C4—1071no—
300C22×C4—1074no—
301C22×C4—1074no—
302C22×C4—1071no—
303C22×C4—1071no—
304C22×C4—1071no—
305C22×C4—4281no—
306C22×C4—4281no—
307C22×C4—4281no—
308C22×C4—4281no—
309C22×C4—4281no—
310C22×C4—4281no—
311C22×C4—4281no—
312C22×C4—4281no—
313C22×C4—2142no—
314C22×C4—2142no—
315C22×C4C22×D10711yescharacteristic
316C22×C4C22×D10711yescharacteristic
317C22×C4C22×D10711yescharacteristic
318C22×C4C22×D10711yescharacteristic
319C22×C4C22×D10711yescharacteristic
320C4⋊C4—1074no—
321C4⋊C4—4281no—
322C4⋊C4—4281no—
323C4⋊C4—2142no—
324C4⋊C4—2142no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1074no—
329C22⋊C4—1074no—
330C22⋊C4—4281no—
331C22⋊C4—4281no—
332C22⋊C4—2142no—
333C22⋊C4—2142no—
334C22⋊C4—2142no—
335C42—2142no—
336C42—2142no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—1072no—
350C23—1072no—
351C23—1072no—
352C23—2141no—
353C23—2141no—
354C23—2141no—
355C23—2141no—
356C23C2×C4×D10712yes—
357C23C2×C4×D10712yes—
358C23C23×D10711yescharacteristic
359D4—44no—
360D4—2148no—
361D4—2148no—
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—44no—
376C2×C4—1072no—
377C2×C4—1072no—
378C2×C4—1072no—
379C2×C4—1072no—
380C2×C4—4284no—
381C2×C4—4284no—
382C2×C4—4284no—
383C2×C4—4284no—
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×C4—2141no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4C2×C4×D10712yes—
402C2×C4D4×D10712yes—
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—41no—
417C22—1074no—
418C22—2142no—
419C22—2142no—
420C22—2142no—
421C22—2142no—
422C22—2142no—
423C22—2142no—
424C22C22×C4×D10711yescharacteristic
425C22C2×D4×D10711yescharacteristic
426C22#41×D10711yescharacteristic
427C22#1681215411yescharacteristic
428C22#1681215411yescharacteristic
429C22#1681215411yescharacteristic
430C22#1681215611yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—4281no—
437C4—4281no—
438C4—4281no—
439C4—2142no—
440C4—2142no—
441C4—2142no—
442C4—2142no—
443C2—42no—
444C2—42no—
445C2—2144no—
446C2C4×D4×D10711yescharacteristic
447C2#2029310011yescharacteristic
448C2#2029310611yescharacteristic
449C2#2029318711yescharacteristic
450C2#2029320611yescharacteristic
451C2#2029320911yescharacteristic
452C2#2029322011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#386×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812080C2
hypercenter#386D107
hypocenterC107#2325
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D107C2semidirect
3. C2×(C22⋊D107)⋊C4C2semidirect
4. #20292521C2semidirect
5. #20292515C2semidirect
6. #20292512C2semidirect
7. #20292510C2semidirect
8. #20292507C2semidirect
9. #20292503C2semidirect
10. #20292503C2semidirect
11. #20292500C2semidirect
12. #20292499C2semidirect
13. #20292486C2semidirect
14. #95×D107C2semidirect
15. #386×C107C2semidirect
22. #16812080C4semidirect
59. C2×C22⋊D107C2×C4semidirect
60. C2×C22⋊D107C2×C4semidirect
78. C2×C4×D107C2×C4semidirect
79. C2×C4×D107D4semidirect
80. C4⋊Dic107C2×C4semidirect
88. C22⋊Dic107C2×C4semidirect
89. C22⋊Dic107C2×C4semidirect
98. C2×D4×C107C2×C4semidirect
227. C107#2325semidirect
238. #386D107semidirect
292. C2×D4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120802
C2×C4×C22⋊D1072
C2×(C22⋊D107)⋊C42
#202925212
#202925152
#202925122
#202925102
#202925072
#202925032
#202925002
#202924992
#202924862
#95×D1072
#386×C1072
#2325107