Finite Group Databasebeta

← #13526837

Subgroups — #13526837

Order 1792.

2650 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C212yesmaximal
2C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
3#11739539C214yesmaximal
4#11739521C214yesmaximal, complemented
5#11730992C218yesmaximal, complemented
6#11730962C214yesmaximal, complemented
7#11730935C214yesmaximal, complemented
8#11730806C214yesmaximal, complemented
9C24×C4×C7C2211yescharacteristic
10C23×Dic14C2212yes—
11C22×C7×C22⋊C4C2212yes—
12C22×C42×C7C2212yes—
13C22×C22⋊Dic7C2214yes—
14C22×C4⋊Dic7C2212yes—
15C2×#6487—216no—
16C2×#6487C2214yes—
17C2×#6487C2214yes—
18C2×#6487C2214yes—
19C2×#6487C2214yes—
20C22×Dic7⋊C4C2214yes—
21C2×#6432C2214yescomplemented
22C4×C7×C22⋊C4C2214yes—
23C4×C7×C22⋊C4C2214yes—
24C2×C7×#95C2212yes—
25#89984C22116yes—
26#89722C22116yes—
27#89722C22116yes—
28#89707C2214yescomplemented
29#89701C2218yescomplemented
30#89700C22116yescomplemented
31#89433C22116yescomplemented
32#89428C22116yescomplemented
33#35011—71nomaximal, Sylow
34C25×C7C2311yescharacteristic
35C23×C4×C7—44no—
36C23×C4×C7D418yes—
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2314yes—
42C23×Dic7C2314yes—
43C23×Dic7C2314yes—
44C22×Dic14—28no—
45C22×Dic14—48no—
46C22×Dic14C2314yes—
47C22×Dic14C2314yes—
48C2×C7×C22⋊C4Q8116yescomplemented
49C2×C7×C22⋊C4C2318yes—
50C2×C42×C7—44no—
51C2×C42×C7C2314yes—
52C2×C42×C7C2314yes—
53C2×C22⋊Dic7—216no—
54C2×C22⋊Dic7C2318yes—
55C2×C22⋊Dic7C2318yes—
56C2×C4⋊Dic7—28no—
57C2×C4⋊Dic7C2314yes—
58C2×C4⋊Dic7C2314yes—
59C2×C4⋊Dic7C23116yes—
60C2×C4⋊Dic7C23116yes—
61#6487—216no—
62#6487—216no—
63#6487—216no—
64#6487—216no—
65C2×Dic7⋊C4—216no—
66C2×Dic7⋊C4C2318yes—
67C2×Dic7⋊C4C2318yes—
68#6432—48no—
69C7×#95C2318yes—
70C7⋊#95C23132yes—
71C22×C22⋊Q8—72no—
72#2857—74no—
73#2659—74no—
74#2583—78no—
75#2502—74no—
76#2434—74no—
77#2315—74no—
78C2×C4×C22⋊C4D711yescharacteristic
79C24×C7—42no—
80C24×C7—42no—
81C24×C7C2×D414yes—
82C24×C7C2×Q818yes—
83C24×C7C2411yescharacteristic
84C24×C7C2412yes—
85C22×C4×C7—24no—
86C22×C4×C7—28no—
87C22×C4×C7—416no—
88C22×C4×C7—44no—
89C22×C4×C7—44no—
90C22×C4×C7—42no—
91C22×C4×C7—44no—
92C22×C4×C7C2×D414yes—
93C22×C4×C7C2×D418yes—
94C22×C4×C7C2×D418yes—
95C22×C4×C7C2×D414yes—
96C22×C4×C7C2×Q8116yescomplemented
97C22×C4×C7C2412yes—
98C22×C4×C7C2411yescharacteristic
99C22×C4×C7C2418yes—
100C22×C4×C7C2411yescharacteristic
101C22×Dic7—28no—
102C22×Dic7—28no—
103C22×Dic7—48no—
104C22×Dic7—48no—
105C22×Dic7C2418yes—
106C22×Dic7C2418yes—
107C2×Dic14—28no—
108C2×Dic14—28no—
109C2×Dic14—216no—
110C2×Dic14—48no—
111C2×Dic14—432no—
112C7×C22⋊C4C2×Q8132yescomplemented
113C42×C7—28no—
114C42×C7—44no—
115C22⋊Dic7—216no—
116C22⋊Dic7—216no—
117C4⋊Dic7—28no—
118C4⋊Dic7—28no—
119C4⋊Dic7—416no—
120C4⋊Dic7—416no—
121Dic7⋊C4—216no—
122Dic7⋊C4—216no—
123C23×Q8—72no—
124C24×C4C2×D711yescharacteristic
125C2×C4⋊Q8—74no—
126C2×C22⋊Q8—74no—
127C2×C22⋊Q8—74no—
128C2×C22⋊Q8—74no—
129C2×C22⋊Q8—74no—
130C2×C22⋊Q8—1416no—
131C22×C4⋊C4—74no—
132C22×C4⋊C4—72no—
133C22×C22⋊C4—74no—
134C22×C22⋊C4C2×D712yes—
135C22×C42C2×D712yes—
136#398—716no—
137#395—716no—
138#391—78no—
139#389—716no—
140#385—716no—
141#385—716no—
142#384—74no—
143#380—716no—
144C4×C22⋊C4C2×D714yes—
145C4×C22⋊C4C2×D714yes—
146C2×#95C2×D712yes—
147C23×C7—24no—
148C23×C7—22no—
149C23×C7—24no—
150C23×C7—22no—
151C23×C7—44no—
152C23×C7—44no—
153C23×C7—416no—
154C23×C7C4⋊Q8116yescomplemented
155C23×C7C22×D412yes—
156C23×C7C22×D414yes—
157C23×C7C22×Q818yes—
158C23×C7C22×Q814yes—
159C23×C7C2511yescharacteristic
160C2×C4×C7—28no—
161C2×C4×C7—22no—
162C2×C4×C7—24no—
163C2×C4×C7—28no—
164C2×C4×C7—216no—
165C2×C4×C7—28no—
166C2×C4×C7—44no—
167C2×C4×C7—48no—
168C2×C4×C7—416no—
169C2×C4×C7C22×D412yes—
170C2×C4×C7C22×D414yes—
171C2×C4×C7C22×D412yes—
172C2×C4×C7C22×Q8116yes—
173C2×C4×C7H(5,𝔽2)14yes—
174C2×Dic7—28no—
175C2×Dic7—28no—
176C2×Dic7—216no—
177C2×Dic7—216no—
178C2×Dic7—416no—
179C2×Dic7—416no—
180Dic14—416no—
181Dic14—416no—
182C25C22×D711yescharacteristic
183C22×Q8—74no—
184C22×Q8—74no—
185C22×Q8—148no—
186C22×Q8—288no—
187C23×C4—44no—
188C23×C4—74no—
189C23×C4—74no—
190C23×C4D2818yes—
191C23×C4C22×D712yes—
192C23×C4C22×D711yescharacteristic
193C23×C4C22×D714yes—
194C23×C4C22×D712yes—
195C23×C4C22×D711yescharacteristic
196C4⋊Q8—288no—
197C22⋊Q8—1416no—
198C22⋊Q8—1416no—
199C22⋊Q8—1416no—
200C22⋊Q8—1416no—
201C2×C4⋊C4—78no—
202C2×C4⋊C4—78no—
203C2×C4⋊C4—74no—
204C2×C4⋊C4—74no—
205C2×C4⋊C4—716no—
206C2×C4⋊C4—716no—
207C2×C4⋊C4—1416no—
208C2×C4⋊C4—148no—
209C2×C22⋊C4—78no—
210C2×C22⋊C4—78no—
211C2×C22⋊C4—1416no—
212C2×C22⋊C4Dic14116yescomplemented
213C2×C22⋊C4C22×D718yes—
214C2×C42—44no—
215C2×C42C22×D714yes—
216C2×C42C22×D714yes—
217#95—732no—
218#95C22×D718yes—
219C22×C7—24no—
220C22×C7—22no—
221C22×C7—24no—
222C22×C7—22no—
223C22×C7—28no—
224C22×C7—216no—
225C22×C7—416no—
226C22×C7C2×C4⋊Q814yes—
227C22×C7C2×C4⋊Q818yes—
228C22×C7C23×D411yescharacteristic
229C22×C7C23×Q812yes—
230C22×C7C2×H(5,𝔽2)12yes—
231C22×C7#58512yes—
232C4×C7—24no—
233C4×C7—22no—
234C4×C7—216no—
235C4×C7#53414yes—
236Dic7—48no—
237Dic7—48no—
238C24—42no—
239C24—42no—
240C24C2×Dic1418yes—
241C24C2×D2814yes—
242C24C23×D711yescharacteristic
243C24C23×D712yes—
244C2×Q8—148no—
245C2×Q8—148no—
246C2×Q8—1416no—
247C2×Q8—288no—
248C2×Q8—2832no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—416no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—78no—
257C22×C4—78no—
258C22×C4—148no—
259C22×C4—148no—
260C22×C4—288no—
261C22×C4—288no—
262C22×C4C2×Dic14116yescomplemented
263C22×C4C2×D2818yes—
264C22×C4C2×D2814yes—
265C22×C4C2×D2818yes—
266C22×C4C2×D2814yes—
267C22×C4C23×D718yes—
268C22×C4C23×D712yes—
269C22×C4C23×D711yescharacteristic
270C22×C4C23×D711yescharacteristic
271C4⋊C4—1416no—
272C4⋊C4—1416no—
273C4⋊C4—148no—
274C4⋊C4—148no—
275C4⋊C4—2816no—
276C4⋊C4—2816no—
277C22⋊C4—1416no—
278C22⋊C4—1416no—
279C22⋊C4C2×Dic14132yescomplemented
280C42—28no—
281C42—44no—
282C2×C7—28no—
283C2×C7—24no—
284C2×C7C22×C4⋊Q811yescharacteristic
285C2×C7C2×#53411yescharacteristic
286C2×C7#344911yescharacteristic
287C2×C7#350114yes—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—44no—
293C23—416no—
294C23—44no—
295C23#6432116yescomplemented
296C23C22×Dic1418yes—
297C23C22×Dic1414yes—
298C23C22×D2812yes—
299C23C22×D2814yes—
300C23C24×D711yescharacteristic
301Q8—2816no—
302Q8—2816no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—216no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—416no—
312C2×C4—148no—
313C2×C4—148no—
314C2×C4—1416no—
315C2×C4—1416no—
316C2×C4—2816no—
317C2×C4—2816no—
318C2×C4C22×Dic14116yes—
319C2×C4C22×D2812yes—
320C2×C4C22×D2814yes—
321C2×C4C22×D2812yes—
322C2×C4#654814yes—
323C7#3501111yescharacteristic, complemented, Sylow
324C22—24no—
325C22—22no—
326C22—24no—
327C22—22no—
328C22—28no—
329C22—216no—
330C22—416no—
331C22C2×#643218yes—
332C22C2×#643214yes—
333C22C23×Dic1412yes—
334C22C23×D2811yescharacteristic
335C22C2×#654812yes—
336C22#9061612yes—
337C4—24no—
338C4—22no—
339C4—216no—
340C4—288no—
341C4—288no—
342C4#9018414yes—
343C2—28no—
344C2—24no—
345C2C22×#643211yescharacteristic
346C2C2×#9018411yescharacteristic
347C2#1174280311yescharacteristic
348C2#1174283414yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC2×C4×C7×C22⋊C4C2
prime radicalC25×C7C23
hypercenterC2×C4×C22⋊C4D7
hypocenterC7#35011
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11739521C2semidirect
5. #11730992C2semidirect
6. #11730962C2semidirect
7. #11730935C2semidirect
8. #11730806C2semidirect
21. C2×#6432C22semidirect
28. #89707C22semidirect
29. #89701C22semidirect
30. #89700C22semidirect
31. #89433C22semidirect
32. #89428C22semidirect
48. C2×C7×C22⋊C4Q8semidirect
96. C22×C4×C7C2×Q8semidirect
112. C7×C22⋊C4C2×Q8semidirect
154. C23×C7C4⋊Q8semidirect
212. C2×C22⋊C4Dic14semidirect
262. C22×C4C2×Dic14semidirect
279. C22⋊C4C2×Dic14semidirect
295. C23#6432semidirect
323. C7#35011semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C2×C4×C7×C22⋊C42
#117395392
#117395212
#117309922
#117309622
#117309352
#117308062
#350117