Finite Group Databasebeta

← #28340160

Subgroups — #28340160

Order 13696.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812107C211yesmaximal, characteristic, complemented
2C2×#16812086C211yesmaximal, characteristic
3C22×Dic107⋊C4C211yesmaximal, characteristic
4#20292558C211yesmaximal, characteristic, complemented
5#20292557C211yesmaximal, characteristic, complemented
6#20292547C211yesmaximal, characteristic, complemented
7#20292546C211yesmaximal, characteristic, complemented
8#20292522C211yesmaximal, characteristic, complemented
9#20292522C211yesmaximal, characteristic, complemented
10#20292500C211yesmaximal, characteristic
11#20292499C211yesmaximal, characteristic
12#20292496C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic
14C22⋊C4×Dic107C211yesmaximal, characteristic
15C2×C22⋊Q8×C107C211yesmaximal, characteristic
16C22×Dic214C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19#16812107—24no—
20#16812086—24no—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4—24no—
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×Dic107⋊C4C2211yescharacteristic
30C2×Dic107⋊C4C2211yescharacteristic
31C2×Dic107⋊C4C2211yescharacteristic
32C2×Dic107⋊C4C2211yescharacteristic
33C2×Dic107⋊C4C2211yescharacteristic
34C2×Dic107⋊C4C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C4×Dic107C2211yescharacteristic
38C2×C4×Dic107C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C22×Q8×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49C22⋊Q8×C107—24no—
50C2×C4⋊C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic
52C2×C4⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×Dic214—22no—
56C2×Dic214—22no—
57C2×Dic214—41no—
58C22×Dic107—22no—
59C22×Dic107—22no—
60C22×Dic107—22no—
61C22×Dic107—22no—
62C22×Dic107—41no—
63C22×Dic107Q814yescomplemented
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C22×Dic107C2311yescharacteristic
69C22×Dic107C2311yescharacteristic
70C22×Dic107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C4⋊Dic107—22no—
73Dic107⋊C4—22no—
74Dic107⋊C4—22no—
75Dic107⋊C4—22no—
76Dic107⋊C4—22no—
77Dic107⋊C4—22no—
78Dic107⋊C4—22no—
79Dic107⋊C4—22no—
80Dic107⋊C4—22no—
81Dic107⋊C4—22no—
82Dic107⋊C4—41no—
83Dic107⋊C4—41no—
84Dic107⋊C4D414yes—
85C22⋊Dic107—22no—
86C22⋊Dic107Q814yescomplemented
87C4×Dic107—22no—
88C4×Dic107—22no—
89C24×C107C2311yescharacteristic
90C2×Q8×C107—22no—
91C2×Q8×C107—22no—
92C2×Q8×C107—41no—
93C22×C4×C107—22no—
94C22×C4×C107—22no—
95C22×C4×C107—41no—
96C22×C4×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C4⋊C4×C107—22no—
103C4⋊C4×C107—22no—
104C4⋊C4×C107—22no—
105C22⋊C4×C107—22no—
106C22⋊C4×C107D414yes—
107C23×C107—21no—
108C23×C107—21no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107C2×D412yes—
112C23×C107C2×Q812yes—
113C23×C107C2×Q812yes—
114C23×C107C2411yescharacteristic
115Q8×C107—44no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—44no—
132C2×C4×C107C2×D412yes—
133C2×C4×C107C2×D412yes—
134C2×C4×C107C2×D412yes—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—28no—
144C2×Dic107—21no—
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—44no—
154C2×Dic107C2×D412yes—
155C2×Dic107C2×D412yes—
156C2×Dic107C2×Q812yes—
157C2×Dic107C2×Q812yes—
158C2×Dic107C2×Q812yes—
159C2×Dic107C2×Q812yes—
160C2×Dic107#4112yes—
161Dic214—44no—
162C22×C107—22no—
163C22×C107—22no—
164C22×C107—22no—
165C22×C107—22no—
166C22×C107—22no—
167C22×C107—22no—
168C22×C107C4⋊Q814yescomplemented
169C22×C107C22×D411yescharacteristic
170C22×C107C22×D411yescharacteristic
171C22×C107C22×Q811yescharacteristic
172C22×C107C22×Q811yescharacteristic
173C22×C107C2×#4111yescharacteristic
174C22×C107H(5,𝔽2)11yescharacteristic
175C22×C1072−1+411yescharacteristic
176C4×C107—22no—
177C4×C107—22no—
178C4×C107—22no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107—41no—
182Dic107—22no—
183Dic107—22no—
184Dic107—22no—
185Dic107—22no—
186Dic107—41no—
187Dic107—41no—
188Dic107—41no—
189Dic107C22⋊Q814yescomplemented
190C2×C107—24no—
191C2×C107C2×C22⋊Q811yescharacteristic
192C2×C107C2×C4⋊Q811yescharacteristic
193C2×C107#54411yescharacteristic
194C2×C107D4211yescharacteristic
195C2×C107#54711yescharacteristic
196C2×C107D4×Q811yescharacteristic
197C2×C107#55411yescharacteristic
198#2440—1071nomaximal, Sylow
199C107#244011yescharacteristic, complemented, Sylow
200C2×C4⋊Q8—1071no—
201C2×C22⋊Q8—1071no—
202C2×C22⋊Q8D10711yescharacteristic
203C22×C4⋊C4—1071no—
204#398—1071no—
205#398—1071no—
206#395—1071no—
207#395—1071no—
208#389—1071no—
209#389—1071no—
210#385—1071no—
211#385—1071no—
212#384—1071no—
213#380—1071no—
214C4×C22⋊C4—1071no—
215C22×Q8—1071no—
216C22×Q8C2×D10711yescharacteristic
217C23×C4—1071no—
218C23×C4—1071no—
219C23×C4C2×D10711yescharacteristic
220C4⋊Q8—2144no—
221C22⋊Q8—24no—
222C22⋊Q8—2144no—
223C2×C4⋊C4—1071no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1071no—
226C2×C4⋊C4—1071no—
227C2×C4⋊C4—1071no—
228C2×C4⋊C4—1071no—
229C2×C4⋊C4—1071no—
230C2×C4⋊C4—1071no—
231C2×C4⋊C4—1071no—
232C2×C4⋊C4—1071no—
233C2×C4⋊C4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C4⋊C4—2144no—
237C2×C4⋊C4C2×D10711yescharacteristic
238C2×C4⋊C4C2×D10711yescharacteristic
239C2×C4⋊C4C2×D10711yescharacteristic
240C2×C22⋊C4—1071no—
241C2×C22⋊C4—1071no—
242C2×C22⋊C4C2×D10711yescharacteristic
243C2×C22⋊C4C2×D10711yescharacteristic
244C2×C42—1071no—
245C2×C42—1071no—
246#95—1071no—
247#95—1071no—
248#95—1071no—
249#95—1071no—
250#95—1071no—
251#95—1071no—
252#95—1071no—
253#95—1071no—
254C24C22×D10711yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—4281no—
259C2×Q8—2142no—
260C2×Q8—2142no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—1071no—
265C22×C4—1074no—
266C22×C4—1071no—
267C22×C4—1071no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—1071no—
273C22×C4—4281no—
274C22×C4—2142no—
275C22×C4—2142no—
276C22×C4—2142no—
277C22×C4—2142no—
278C22×C4C22×D10711yescharacteristic
279C22×C4C22×D10711yescharacteristic
280C22×C4C22×D10711yescharacteristic
281C22×C4C22×D10711yescharacteristic
282C22×C4C22×D10711yescharacteristic
283C22×C4C22×D10711yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—1074no—
288C4⋊C4—4281no—
289C4⋊C4—4281no—
290C4⋊C4—2142no—
291C4⋊C4—2142no—
292C4⋊C4—2142no—
293C4⋊C4—2142no—
294C4⋊C4—2142no—
295C4⋊C4—2142no—
296C4⋊C4—2142no—
297C4⋊C4—2142no—
298C4⋊C4—2142no—
299C4⋊C4—2142no—
300C22⋊C4—22no—
301C22⋊C4—1074no—
302C22⋊C4—2142no—
303C22⋊C4C22⋊D10714yes—
304C42—2142no—
305C42—2142no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×C22⋊D10712yes—
311C23Q8×D10712yes—
312C23Q8×D10712yes—
313C23C23×D10711yescharacteristic
314Q8—44no—
315Q8—4284no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—1072no—
333C2×C4—1072no—
334C2×C4—1072no—
335C2×C4—1072no—
336C2×C4—1072no—
337C2×C4—1072no—
338C2×C4—1072no—
339C2×C4—4284no—
340C2×C4—2148no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2141no—
347C2×C4—2141no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4—2141no—
353C2×C4—2141no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4—2141no—
358C2×C4D4×D10712yes—
359C2×C4C2×C22⋊D10712yes—
360C2×C4C2×C22⋊D10712yes—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22#1681210714yescomplemented
368C22C2×D4×D10711yescharacteristic
369C22C22×C22⋊D10711yescharacteristic
370C22C2×Q8×D10711yescharacteristic
371C22C2×Q8×D10711yescharacteristic
372C22#41×D10711yescharacteristic
373C22#1681215411yescharacteristic
374C22#1681215611yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1074no—
382C4—4281no—
383C4—4281no—
384C4—4281no—
385C4—2142no—
386C4—2142no—
387C4—2142no—
388C4—2142no—
389C2—24no—
390C2C22⋊Q8×D10711yescharacteristic
391C2C2×#1681210711yescharacteristic
392C2#2029325611yescharacteristic
393C2D4×C22⋊D10711yescharacteristic
394C2#2029327611yescharacteristic
395C2#2029329011yescharacteristic
396C2#2029333111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC24×C107C23
hypercenterC2×C22⋊Q8D107
hypocenterC107#2440
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812107C2semidirect
4. #20292558C2semidirect
5. #20292557C2semidirect
6. #20292547C2semidirect
7. #20292546C2semidirect
8. #20292522C2semidirect
9. #20292522C2semidirect
12. #20292496C2semidirect
63. C22×Dic107Q8semidirect
86. C22⋊Dic107Q8semidirect
168. C22×C107C4⋊Q8semidirect
189. Dic107C22⋊Q8semidirect
199. C107#2440semidirect
367. C22#16812107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121072
C2×#168120862
C22×Dic107⋊C42
#202925582
#202925572
#202925472
#202925462
#202925222
#202925002
#202924992
#202924962
#202924852
C22⋊C4×Dic1072
C2×C22⋊Q8×C1072
#2440107