Finite Group Databasebeta

← #28339543

Subgroups — #28339543

Order 13696.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812089C211yesmaximal, characteristic
2#20292531C211yesmaximal, characteristic, complemented
3#20292531C211yesmaximal, characteristic, complemented
4#20292513C211yesmaximal, characteristic, complemented
5#20292512C211yesmaximal, characteristic, complemented
6#20292510C211yesmaximal, characteristic, complemented
7#20292501C211yesmaximal, characteristic
8#20292498C211yesmaximal, characteristic
9C4⋊C4×Dic107C211yesmaximal, characteristic, complemented
10C4×C4⋊Dic107C211yesmaximal, characteristic, complemented
11C4×(C2×D107)⋊C4C211yesmaximal, characteristic
12#20292474C211yesmaximal, characteristic
13#20292474C211yesmaximal, characteristic
14#95×D107C211yesmaximal, characteristic
15#387×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812089C418yescomplemented
21C2×C4⋊Dic107C2211yescharacteristic
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×C4×Dic107C2211yescharacteristic
31C2×C4×Dic107C2211yescharacteristic
32C2×C4×Dic107C2211yescharacteristic
33C2×C4×Dic107C2211yescharacteristic
34C2×C4×Dic107C2211yescharacteristic
35C107⋊#95C2211yescharacteristic
36C107⋊#95C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C2×C4⋊C4×C107C2211yescharacteristic
46C2×C4⋊C4×C107C2211yescharacteristic
47C2×C42×C107C2211yescharacteristic
48#95×C107C2211yescharacteristic
49#95×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52C23×D107C2311yescharacteristic
53C2×Dic214—41no—
54C2×Dic214—41no—
55C2×Dic214C2×C414yescomplemented
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C2×C4×D107—22no—
64C2×C4×D107—22no—
65C2×C4×D107—22no—
66C2×C4×D107—22no—
67C2×C4×D107—41no—
68C2×C4×D107—41no—
69C2×C4×D107—41no—
70C2×C4×D107—41no—
71C2×C4×D107C2×C414yes—
72C4⋊Dic107—22no—
73C4⋊Dic107C2×C414yescomplemented
74C4⋊Dic107C2×C414yescomplemented
75Dic107⋊C4—41no—
76Dic107⋊C4—41no—
77(C2×D107)⋊C4—22no—
78(C2×D107)⋊C4C2×C414yes—
79(C2×D107)⋊C4C2×C414yes—
80(C2×D107)⋊C4Q814yescomplemented
81C4×Dic107—22no—
82C4×Dic107—41no—
83C4×Dic107—41no—
84C4×Dic107—41no—
85C4×Dic107—41no—
86C22×C4×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C4⋊C4×C107—22no—
94C4⋊C4×C107C2×C414yescomplemented
95C42×C107—22no—
96C23×C107C2411yescharacteristic
97C22×D107—21no—
98C22×D107—21no—
99C22×D107—21no—
100C22×D107—21no—
101C22×D107C22×C412yes—
102C22×D107C2×Q812yes—
103C22×D107#4112yes—
104C2×C4×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107C22×C412yes—
121C2×C4×C107C22×C412yes—
122C2×C4×C107C22×C412yes—
123C2×C4×C107C2×Q812yes—
124C2×C4×C107#4112yes—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107C22×C412yes—
142C2×Dic107C22×C412yes—
143C2×Dic107C22×C412yes—
144C2×Dic107C2×Q812yes—
145C2×Dic107#4112yes—
146C4×D107—44no—
147C4×D107—44no—
148C4×D107—44no—
149Dic214—44no—
150C22×C107C23×C411yescharacteristic
151C22×C107C22×Q811yescharacteristic
152C22×C107C2×#4111yescharacteristic
153C22×C107C2×#4111yescharacteristic
154C22×C107C2×#4111yescharacteristic
155C22×C107H(5,𝔽2)11yescharacteristic
156C22×C107H(5,𝔽2)11yescharacteristic
157C2×D107—22no—
158C2×D107—22no—
159C2×D107—22no—
160C2×D107—22no—
161C2×D107—22no—
162C2×D107—22no—
163C2×D107C4×Q814yes—
164C4×C107—22no—
165C4×C107—22no—
166C4×C107—22no—
167C4×C107—22no—
168C4×C107—22no—
169C4×C107—41no—
170C4×C107—41no—
171Dic107—22no—
172Dic107—22no—
173Dic107—22no—
174Dic107—22no—
175Dic107—22no—
176Dic107—41no—
177Dic107—41no—
178C2×C107C2×C4×Q811yescharacteristic
179C2×C107C4×#4111yescharacteristic
180C2×C107#51811yescharacteristic
181C2×C107#55111yescharacteristic
182C2×C107#55111yescharacteristic
183C2×C107#55411yescharacteristic
184C2×C107#55411yescharacteristic
185D107—24no—
186#2368—1071nomaximal, Sylow
187C107#236811yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1071no—
189#391—1071no—
190#391—1071no—
191#387—1071no—
192#387—1071no—
193#387—1071no—
194#387D10711yescharacteristic, complemented
195#385—1071no—
196#385—1071no—
197C4×C4⋊C4—1071no—
198C4×C4⋊C4—1071no—
199C4×C22⋊C4—1071no—
200C4×C22⋊C4—1071no—
201C4×C22⋊C4—1071no—
202C2×#95—1071no—
203C22×Q8—1071no—
204C23×C4—1071no—
205C23×C4—1071no—
206C23×C4—1071no—
207C22⋊Q8—1078no—
208C2×C4⋊C4—1071no—
209C2×C4⋊C4—1071no—
210C2×C4⋊C4—1071no—
211C2×C4⋊C4—1071no—
212C2×C4⋊C4—1071no—
213C2×C4⋊C4C2×D10711yescharacteristic
214C2×C4⋊C4C2×D10711yescharacteristic
215C2×C22⋊C4—1071no—
216C2×C22⋊C4—1071no—
217C2×C22⋊C4—1071no—
218C2×C22⋊C4—1071no—
219C2×C42—1071no—
220C2×C42—1071no—
221C2×C42—1071no—
222C2×C42—1071no—
223C2×C42—1071no—
224C2×C42C2×D10711yescharacteristic
225#95—1071no—
226#95—1071no—
227#95—1071no—
228#95—1071no—
229#95—1071no—
230#95—1071no—
231#95—1071no—
232#95—1071no—
233#95—1071no—
234#95—1071no—
235#95C2×D10711yescharacteristic
236#95C2×D10711yescharacteristic
237#95C2×D10711yescharacteristic
238#95C2×D10711yescharacteristic
239C24—1071no—
240C2×Q8—1074no—
241C2×Q8—4281no—
242C2×Q8—4281no—
243C22×C4—1071no—
244C22×C4—1071no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1074no—
249C22×C4—1071no—
250C22×C4—1071no—
251C22×C4—4281no—
252C22×C4—4281no—
253C22×C4—4281no—
254C22×C4—4281no—
255C22×C4—2142no—
256C22×C4—2142no—
257C22×C4—2142no—
258C22×C4—2142no—
259C22×C4C22×D10711yescharacteristic
260C22×C4C22×D10711yescharacteristic
261C22×C4C22×D10711yescharacteristic
262C22×C4C22×D10711yescharacteristic
263C22×C4C22×D10711yescharacteristic
264C22×C4C22×D10711yescharacteristic
265C22×C4C22×D10711yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—1074no—
268C4⋊C4—1074no—
269C4⋊C4—4281no—
270C4⋊C4—4281no—
271C4⋊C4—2142no—
272C4⋊C4C4×D10714yescomplemented
273C22⋊C4—1074no—
274C22⋊C4—1074no—
275C22⋊C4—1074no—
276C22⋊C4—2142no—
277C42—22no—
278C42—4281no—
279C42—4281no—
280C42—4281no—
281C42—4281no—
282C42—2142no—
283C23—1072no—
284C23—1072no—
285C23—1072no—
286C23—2141no—
287C23—2141no—
288C23—2141no—
289C23—2141no—
290C23C23×D10711yescharacteristic
291Q8—4284no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—1072no—
309C2×C4—1072no—
310C2×C4—1072no—
311C2×C4—1072no—
312C2×C4—1072no—
313C2×C4—4284no—
314C2×C4—4284no—
315C2×C4—4284no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4C2×C4×D10712yes—
333C2×C4C2×C4×D10712yes—
334C2×C4C2×C4×D10712yes—
335C2×C4Q8×D10712yes—
336C2×C4#1313381712yes—
337C22—1074no—
338C22—2142no—
339C22—2142no—
340C22—2142no—
341C22—2142no—
342C22—2142no—
343C22—2142no—
344C22C22×C4×D10711yescharacteristic
345C22C2×Q8×D10711yescharacteristic
346C22#41×D10711yescharacteristic
347C22#41×D10711yescharacteristic
348C22C2×#1313381711yescharacteristic
349C22#1681215511yescharacteristic
350C22#1681215511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—4281no—
359C4—4281no—
360C4—2142no—
361C4—2142no—
362C4—2142no—
363C4—2142no—
364C4—2142no—
365C2—2144no—
366C2C4×Q8×D10711yescharacteristic
367C2C4×#1313381711yescharacteristic
368C2#2029310111yescharacteristic
369C2#2029330011yescharacteristic
370C2#2029330211yescharacteristic
371C2#2029333311yescharacteristic
372C2#2029333511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#387×C107C2
socle, cosocleC23×C107C24
prime radicalC23×D107C23
hypercenter#387D107
hypocenterC107#2368
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #20292531C2semidirect
3. #20292531C2semidirect
4. #20292513C2semidirect
5. #20292512C2semidirect
6. #20292510C2semidirect
9. C4⋊C4×Dic107C2semidirect
10. C4×C4⋊Dic107C2semidirect
15. #387×C107C2semidirect
20. #16812089C4semidirect
55. C2×Dic214C2×C4semidirect
73. C4⋊Dic107C2×C4semidirect
74. C4⋊Dic107C2×C4semidirect
80. (C2×D107)⋊C4Q8semidirect
94. C4⋊C4×C107C2×C4semidirect
187. C107#2368semidirect
194. #387D107semidirect
272. C4⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
#202925312
#202925132
#202925122
#202925102
#202925012
#202924982
C4⋊C4×Dic1072
C4×C4⋊Dic1072
C4×(C2×D107)⋊C42
#202924742
#95×D1072
#387×C1072
#2368107