Finite Group Databasebeta

← #28339613

Subgroups — #28339613

Order 13696.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812080C211yesmaximal, characteristic
2#20292527C211yesmaximal, characteristic, complemented
3#20292527C211yesmaximal, characteristic, complemented
4#20292521C211yesmaximal, characteristic, complemented
5#20292515C211yesmaximal, characteristic, complemented
6#20292512C211yesmaximal, characteristic, complemented
7#20292510C211yesmaximal, characteristic, complemented
8#20292507C211yesmaximal, characteristic, complemented
9#20292506C211yesmaximal, characteristic, complemented
10#20292504C211yesmaximal, characteristic, complemented
11#20292500C211yesmaximal, characteristic, complemented
12#20292499C211yesmaximal, characteristic, complemented
13#20292489C211yesmaximal, characteristic, complemented
14#20292486C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812080C418yescomplemented
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×C4×Dic107C2211yescharacteristic
36C2×C4×Dic107C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C22×D4×C107C2211yescharacteristic
46C23×C4×C107C2211yescharacteristic
47C2×C22⋊C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2211yescharacteristic
49C2×C22⋊C4×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52C23×D107C2311yescharacteristic
53C2×C22⋊D107—41no—
54C2×C22⋊D107—41no—
55C2×C22⋊D107—41no—
56C2×C22⋊D107—41no—
57C2×C22⋊D107C2×C414yescomplemented
58C2×C22⋊D107C2×C414yescomplemented
59C22×Dic107—41no—
60C22×Dic107—41no—
61C22×Dic107—41no—
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C2×C4×D107—41no—
70C2×C4×D107—41no—
71C2×C4×D107C2×C414yescomplemented
72C4⋊Dic107C2×C414yescomplemented
73Dic107⋊C4—41no—
74Dic107⋊C4—41no—
75Dic107⋊C4—41no—
76Dic107⋊C4—41no—
77C22⋊Dic107—41no—
78C22⋊Dic107—41no—
79C22⋊Dic107—41no—
80C22⋊Dic107C2×C414yescomplemented
81C22⋊Dic107C2×C414yescomplemented
82(C2×D107)⋊C4—41no—
83(C2×D107)⋊C4—41no—
84(C2×D107)⋊C4—41no—
85(C2×D107)⋊C4—41no—
86C4×Dic107—41no—
87C4×Dic107—41no—
88C24×C107C2311yescharacteristic
89C24×C107C2311yescharacteristic
90C2×D4×C107—41no—
91C2×D4×C107—41no—
92C2×D4×C107C2×C414yes—
93C22×C4×C107—41no—
94C22×C4×C107—41no—
95C22×C4×C107—41no—
96C22×C4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22⋊C4×C107—41no—
102C22⋊C4×C107—41no—
103C22⋊C4×C107—41no—
104C23×C107—21no—
105C23×C107—21no—
106C23×C107—21no—
107C23×C107—21no—
108C23×C107—21no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—21no—
114C23×C107—21no—
115C23×C107—21no—
116C23×C107C22×C412yes—
117C23×C107C22×C412yes—
118C23×C107C2411yescharacteristic
119C22×D107—21no—
120C22×D107—21no—
121C22×D107—21no—
122C22×D107—21no—
123C22×D107—21no—
124C22×D107—21no—
125C22×D107C22×C412yes—
126D4×C107—44no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
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—44no—
142C2×C4×C107C22×C412yes—
143C22⋊D107—44no—
144C22⋊D107—44no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—21no—
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—44no—
164C2×Dic107C22×C412yes—
165C2×Dic107C22×C412yes—
166C2×Dic107C22×C412yes—
167C4×D107—44no—
168C22×C107—22no—
169C22×C107—22no—
170C22×C107—22no—
171C22×C107—22no—
172C22×C107—22no—
173C22×C107—22no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107C23×C411yescharacteristic
183C22×C107H(5,𝔽2)11yescharacteristic
184C22×C107H(5,𝔽2)11yescharacteristic
185C22×C107H(5,𝔽2)11yescharacteristic
186C22×C107H(5,𝔽2)11yescharacteristic
187C22×C107H(5,𝔽2)11yescharacteristic
188C22×C1072−1+411yescharacteristic
189C2×D107—22no—
190C2×D107—22no—
191C2×D107—22no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C4×C107—22no—
197C4×C107—41no—
198C4×C107—41no—
199C4×C107—41no—
200C4×C107—41no—
201Dic107—22no—
202Dic107—22no—
203Dic107—22no—
204Dic107—41no—
205Dic107—41no—
206Dic107—41no—
207Dic107—41no—
208C2×C107—42no—
209C2×C107—42no—
210C2×C107#51811yescharacteristic
211C2×C107#51811yescharacteristic
212C2×C107#52011yescharacteristic
213C2×C107#56011yescharacteristic
214C2×C107#56011yescharacteristic
215C2×C107#56211yescharacteristic
216C2×C107#56211yescharacteristic
217D107—42no—
218#2377—1071nomaximal, Sylow
219C107#237711yescharacteristic, complemented, Sylow
220C2×#121—1071no—
221#390—1071no—
222#390—1071no—
223#388—1071no—
224#388—1071no—
225#387—1071no—
226#387—1071no—
227#386—1071no—
228#386—1071no—
229#386—1071no—
230#386D10711yescharacteristic, complemented
231#385—1071no—
232#385—1071no—
233#381—1071no—
234#380—1071no—
235C22×D4—1071no—
236C22×D4—1071no—
237C22×D4C2×D10711yescharacteristic
238C23×C4—1071no—
239C23×C4—1071no—
240C23×C4C2×D10711yescharacteristic
241#121—1078no—
242C2×C4⋊C4—1071no—
243C2×C4⋊C4—1071no—
244C2×C4⋊C4—1071no—
245C2×C4⋊C4—1071no—
246C2×C4⋊C4—1071no—
247C2×C22⋊C4—1071no—
248C2×C22⋊C4—1071no—
249C2×C22⋊C4—1071no—
250C2×C22⋊C4—1071no—
251C2×C22⋊C4—1071no—
252C2×C22⋊C4—1071no—
253C2×C22⋊C4—1071no—
254C2×C22⋊C4—1071no—
255C2×C22⋊C4—1071no—
256C2×C22⋊C4C2×D10711yescharacteristic
257C2×C22⋊C4C2×D10711yescharacteristic
258C2×C22⋊C4C2×D10711yescharacteristic
259C2×C42—1071no—
260C2×C42—1071no—
261#95—1071no—
262#95—1071no—
263#95—1071no—
264#95—1071no—
265#95—1071no—
266#95—1071no—
267#95—1071no—
268#95—1071no—
269#95C2×D10711yescharacteristic
270#95C2×D10711yescharacteristic
271C24—1071no—
272C24C22×D10711yescharacteristic
273C24C22×D10711yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—1074no—
277C2×D4—1074no—
278C2×D4—4281no—
279C2×D4—4281no—
280C2×D4—4281no—
281C2×D4—4281no—
282C2×D4C4×D10714yes—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—1071no—
287C22×C4—1071no—
288C22×C4—1071no—
289C22×C4—1074no—
290C22×C4—1071no—
291C22×C4—1071no—
292C22×C4—1071no—
293C22×C4—1071no—
294C22×C4—4281no—
295C22×C4—4281no—
296C22×C4—4281no—
297C22×C4—4281no—
298C22×C4—4281no—
299C22×C4C22×D10711yescharacteristic
300C22×C4C22×D10711yescharacteristic
301C22×C4C22×D10711yescharacteristic
302C22×C4C22×D10711yescharacteristic
303C22×C4C22×D10711yescharacteristic
304C4⋊C4—1074no—
305C4⋊C4—4281no—
306C4⋊C4—4281no—
307C4⋊C4—4281no—
308C4⋊C4—4281no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—1074no—
313C22⋊C4—1074no—
314C22⋊C4—4281no—
315C22⋊C4—4281no—
316C22⋊C4—4281no—
317C22⋊C4—4281no—
318C22⋊C4—4281no—
319C22⋊C4—4281no—
320C22⋊C4—4281no—
321C42—4281no—
322C42—4281no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—1072no—
336C23—2141no—
337C23—2141no—
338C23—2141no—
339C23—2141no—
340C23—2141no—
341C23—2141no—
342C23C2×C4×D10712yes—
343C23C2×C4×D10712yes—
344C23C23×D10711yescharacteristic
345D4—44no—
346D4—4284no—
347D4—4284no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—44no—
363C2×C4—1072no—
364C2×C4—1072no—
365C2×C4—1072no—
366C2×C4—4284no—
367C2×C4—4284no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4—2141no—
376C2×C4—2141no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4—2141no—
386C2×C4C2×C4×D10712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—2142no—
406C22—2142no—
407C22—2142no—
408C22C22×C4×D10711yescharacteristic
409C22#1681215411yescharacteristic
410C22#1681215411yescharacteristic
411C22#1681215411yescharacteristic
412C22#1681215411yescharacteristic
413C22#1681215511yescharacteristic
414C22#1681215611yescharacteristic
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—4281no—
421C4—4281no—
422C4—4281no—
423C4—4281no—
424C4—2142no—
425C4—2142no—
426C4—2142no—
427C2—42no—
428C2—42no—
429C2—4282no—
430C2#2029309911yescharacteristic
431C2#2029310011yescharacteristic
432C2#2029310611yescharacteristic
433C2#2029336511yescharacteristic
434C2#2029336611yescharacteristic
435C2#2029337211yescharacteristic
436C2#2029337411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#386×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812080C2
hypercenter#386D107
hypocenterC107#2377
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20292527C2semidirect
3. #20292527C2semidirect
4. #20292521C2semidirect
5. #20292515C2semidirect
6. #20292512C2semidirect
7. #20292510C2semidirect
8. #20292507C2semidirect
9. #20292506C2semidirect
10. #20292504C2semidirect
11. #20292500C2semidirect
12. #20292499C2semidirect
13. #20292489C2semidirect
14. #20292486C2semidirect
15. #386×C107C2semidirect
20. #16812080C4semidirect
57. C2×C22⋊D107C2×C4semidirect
58. C2×C22⋊D107C2×C4semidirect
71. C2×C4×D107C2×C4semidirect
72. C4⋊Dic107C2×C4semidirect
80. C22⋊Dic107C2×C4semidirect
81. C22⋊Dic107C2×C4semidirect
219. C107#2377semidirect
230. #386D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120802
#202925272
#202925212
#202925152
#202925122
#202925102
#202925072
#202925062
#202925042
#202925002
#202924992
#202924892
#202924862
#386×C1072
#2377107