Finite Group Databasebeta

← #14224141

Subgroups — #14224141

Order 1792.

1830 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743330C212yesmaximal
2C7×#3160C211yesmaximal, characteristic, complemented
3#11740782C212yesmaximal, complemented
4#11739379C214yesmaximal, complemented
5#11739355C218yesmaximal, complemented
6#11739334C214yesmaximal, complemented
7#11736629C214yesmaximal
8#11736614C214yesmaximal, complemented
9C2×#89471C211yesmaximal, characteristic, complemented
10C4×#6464C211yesmaximal, characteristic
11C7×#533C2212yes—
12C22×Dic28C2214yes—
13C2×#6464C2212yes—
14#90419C2218yes—
15#90388C2218yes—
16#90333—28no—
17#90275—24no—
18#90261—24no—
19#90217C2214yes—
20#90207C2214yes—
21C4×#1157C2211yescharacteristic
22C2×#6432C2212yes—
23C7×#500C2212yes—
24C7×#499C2214yes—
25C7×#498C2212yes—
26C7×#427C2214yes—
27C2×C4×C7×C8C2211yescharacteristic
28#89873C2212yes—
29#89865C2218yes—
30#89841C2218yes—
31#89776C2218yes—
32C4×#1136C2211yescharacteristic
33#89659C22116yes—
34#89645C22116yes—
35#89640C22116yes—
36#89625C22116yes—
37#89471C2212yes—
38#89471C2214yes—
39#89471C2212yes—
40#89461C2214yes—
41C4×#1131C2212yes—
42C4×C8×D7C2212yes—
43#60715—71nomaximal, Sylow
44C2×#1157C2311yescharacteristic
45C22×Dic14C2314yes—
46C22×C7×C8C2312yes—
47C7×C4⋊Q8C2314yes—
48C7×#125C2314yes—
49C7×C22⋊Q8—24no—
50C4×C7×Q8—24no—
51C7×#117C2314yes—
52C2×C42×C7C2311yescharacteristic
53#6509—28no—
54Q8×Dic7—28no—
55C4×C22⋊D7D414yes—
56#6489C2311yescharacteristic
57C2×C4⋊Dic7—24no—
58C2×#1136C2312yes—
59C2×Dic28—216no—
60C2×Dic28C2318yes—
61C2×Dic28C2318yes—
62#6464—28no—
63C2×#1131C2314yes—
64C2×C8×D7C2314yes—
65#6461—216no—
66#6460C23116yes—
67#6455—28no—
68#6453—216no—
69#6453—28no—
70#6450C23116yes—
71#6449—28no—
72#6440—28no—
73#6438—24no—
74C4×D28D412yes—
75#6435C2312yes—
76C42×D7C2312yes—
77#6432—24no—
78#6432—24no—
79#6432C2314yes—
80#6432C2314yes—
81C4×Dic14—28no—
82C4×Dic14D412yes—
83C7×#107C2318yes—
84C7×#106C2318yes—
85C4×C7×C8C2311yescharacteristic
86C4×C7×C8C2312yes—
87C4×C7×C8C2311yescharacteristic
88#6389C2314yes—
89C8×Dic7C2314yes—
90#6384C23132yes—
91#6377C2312yes—
92C4×C7⋊C8C2312yes—
93#3509—72no—
94#3185—74no—
95#3177—78no—
96#3170—74no—
97#3160D711yescharacteristic, complemented
98C2×#494—71no—
99#3139—72no—
100#3059—74no—
101#2950—74no—
102C4×#131—71no—
103C2×C7×Q8—24no—
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2411yescharacteristic
106C2×C22⋊D7C2×D412yes—
107C22×Dic7—24no—
108#1157—24no—
109C2×D28C2×D411yescharacteristic
110C2×C4×D7C2×D414yes—
111C2×C4×D7C2412yes—
112C2×Dic14—216no—
113C2×Dic14—24no—
114C2×Dic14C2×D411yescharacteristic
115C2×Dic14C2418yes—
116C2×C7×C8#4118yes—
117C2×C7×C8C2412yes—
118C2×C7×C8C2412yes—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—28no—
121C7×C4⋊C4C2418yes—
122C7×C22⋊C4—24no—
123C42×C7—24no—
124C42×C7C2×D412yes—
125C42×C7C2411yescharacteristic
126C42×C7C2411yescharacteristic
127C22⋊Dic7C2×D412yes—
128(C2×D7)⋊C4—216no—
129(C2×D7)⋊C4C2×D414yes—
130C4⋊Dic7—28no—
131C4⋊Dic7—28no—
132C4⋊Dic7—216no—
133C4⋊Dic7C2×D412yes—
134Dic7⋊C4—216no—
135Dic7⋊C4C2×D414yes—
136C4×Dic7—216no—
137C4×Dic7C2×D414yes—
138C4×Dic7C2412yes—
139#1136—24no—
140C2×C7⋊C8C2414yes—
141Dic28—216no—
142Dic28—216no—
143#1131—28no—
144C8×D7—28no—
145C22×Q16—74no—
146C2×#131—72no—
147#557—144no—
148#556—144no—
149#552—148no—
150#541—78no—
151#536—74no—
152#533—78no—
153#533C2×D712yes—
154C2×C4⋊Q8—72no—
155#519—74no—
156C4×#41—71no—
157#500C2×D712yes—
158#499C2×D714yes—
159#498C2×D712yes—
160#494—72no—
161#494—74no—
162#494—72no—
163#491—78no—
164#487—78no—
165#470—716no—
166#467—716no—
167#441—716no—
168C4×Q16—716no—
169#427C2×D714yes—
170#419—78no—
171#405—72no—
172#405—74no—
173C4×M16—71no—
174C4×M16—72no—
175C2×C4×C8—72no—
176C2×C4×C8C2×D711yescharacteristic
177C23×C7C22×D411yescharacteristic
178C22×D7C22×D412yes—
179C7×Q8—48no—
180C2×C4×C7—24no—
181C2×C4×C7—28no—
182C2×C4×C7C22×D412yes—
183C2×C4×C7C22×D411yescharacteristic
184C2×C4×C7C22×D412yes—
185C2×C4×C7C2×#4114yes—
186C2×C4×C7C2511yescharacteristic
187C22⋊D7—24no—
188C2×Dic7—28no—
189C2×Dic7—48no—
190C2×Dic7C22×D414yes—
191C2×Dic7C22×D412yes—
192D28—22no—
193C4×D7—28no—
194C4×D7—24no—
195Dic14—22no—
196Dic14—216no—
197Dic14—48no—
198C7×C8C2×#4114yes—
199C7×C8C2×#4114yes—
200C7⋊C8—24no—
201C2×#41—71no—
202C22×Q8—74no—
203C2×Q16—78no—
204C2×Q16—78no—
205C2×Q16—1416no—
206#131—148no—
207C2×M16—72no—
208C2×M16—74no—
209C22×C8—74no—
210C22×C8C22×D712yes—
211C4⋊Q8—74no—
212C4⋊Q8—74no—
213C4⋊Q8—144no—
214C4⋊Q8—144no—
215C4⋊Q8C22×D714yes—
216#126—1416no—
217#125—148no—
218#125—1416no—
219#125C22×D714yes—
220#124—144no—
221#123—148no—
222C22⋊Q8—24no—
223C22⋊Q8—716no—
224C22⋊Q8—148no—
225C4×Q8—24no—
226C4×Q8—72no—
227C4×Q8—716no—
228C4×Q8—148no—
229C4×Q8—148no—
230C4×D4—74no—
231C4×D4—72no—
232#117—71no—
233#117—72no—
234#117—148no—
235#117—148no—
236#117C22×D714yes—
237C2×C4⋊C4—144no—
238C2×C42—72no—
239C2×C42C22×D711yescharacteristic
240#107C22×D718yes—
241#106C22×D718yes—
242#103—732no—
243#97—74no—
244#97—72no—
245C4×C8—74no—
246C4×C8—72no—
247C4×C8C22×D711yescharacteristic
248C4×C8C22×D711yescharacteristic
249C4×C8C22×D712yes—
250C22×C7—22no—
251C22×C7#53212yes—
252C22×C7C23×D411yescharacteristic
253C2×D7—22no—
254C2×D7—24no—
255C4×C7—22no—
256C4×C7—48no—
257C4×C7#53212yes—
258C4×C7#57814yes—
259C4×C7C22×#4112yes—
260Dic7—22no—
261Dic7—24no—
262Dic7—48no—
263#41—144no—
264C2×Q8—24no—
265C2×Q8—71no—
266C2×Q8—78no—
267C2×Q8—144no—
268C2×Q8—1416no—
269C2×D4—72no—
270C2×D4—71no—
271C22×C4—74no—
272C22×C4—72no—
273C22×C4—144no—
274C22×C4D4×D712yes—
275C22×C4C23×D711yescharacteristic
276Q16—1416no—
277Q16—1416no—
278M16—144no—
279M16—148no—
280C2×C8—74no—
281C2×C8—148no—
282C2×C8#116118yes—
283C2×C8C23×D712yes—
284C2×C8C23×D712yes—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—74no—
288C4⋊C4—72no—
289C4⋊C4—1416no—
290C4⋊C4—1416no—
291C4⋊C4—148no—
292C4⋊C4—148no—
293C4⋊C4C23×D718yes—
294C22⋊C4—24no—
295C22⋊C4—72no—
296C22⋊C4—74no—
297C22⋊C4—1416no—
298C42—24no—
299C42—74no—
300C42—72no—
301C42—1416no—
302C42D4×D712yes—
303C42C23×D711yescharacteristic
304C42C23×D711yescharacteristic
305C2×C7—22no—
306C2×C7C2×#53211yescharacteristic
307C2×C7C2×#57812yes—
308D7—42no—
309C23—72no—
310C23C2×D4×D711yescharacteristic
311Q8—48no—
312Q8—142no—
313Q8—1416no—
314Q8—288no—
315D4—144no—
316D4—142no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—74no—
320C2×C4—72no—
321C2×C4—148no—
322C2×C4—144no—
323C2×C4—148no—
324C2×C4—288no—
325C2×C4C2×D4×D712yes—
326C2×C4C2×D4×D711yescharacteristic
327C2×C4C2×D4×D712yes—
328C2×C4C2×#116114yes—
329C2×C4C24×D711yescharacteristic
330C8—144no—
331C8C2×#116114yes—
332C8C2×#116114yes—
333C7#6071511yescharacteristic, complemented, Sylow
334C22—22no—
335C22—142no—
336C22—144no—
337C22#9041612yes—
338C22C22×D4×D711yescharacteristic
339C4—22no—
340C4—48no—
341C4—142no—
342C4—144no—
343C4—288no—
344C4#9041612yes—
345C4#9046214yes—
346C4C22×#116112yes—
347C2—22no—
348C2—282no—
349C2C2×#9041611yescharacteristic
350C2C2×#9046212yes—
subgroupG/H
centerC22C22×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3160C2
socleC22×C7C23×D4
prime radicalC2×#1157C23
hypercenter#3160D7
hypocenterC7#60715
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. C7×#3160C2semidirect
3. #11740782C2semidirect
4. #11739379C2semidirect
5. #11739355C2semidirect
6. #11739334C2semidirect
8. #11736614C2semidirect
9. C2×#89471C2semidirect
97. #3160D7semidirect
333. C7#60715semidirect

Maximal subgroups

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

subgroupindex
#117433302
C7×#31602
#117407822
#117393792
#117393552
#117393342
#117366292
#117366142
C2×#894712
C4×#64642
#607157