Finite Group Databasebeta

← #13336870

Subgroups — #13336870

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#494C211yesmaximal, characteristic, complemented
2#11740660C212yesmaximal
3#11735903C211yesmaximal, characteristic
4#11729527C212yesmaximal, complemented
5#11729060C214yesmaximal, complemented
6#11729059C214yesmaximal, complemented
7#11727483C211yesmaximal, characteristic, complemented
8C22×C7×Q16—21no—
9C22×C7×Q16—21no—
10C2×C7×C4⋊Q8C2212yes—
11C2×#6464—21no—
12#90168C2211yescharacteristic
13C7×#494—21no—
14C7×#494—21no—
15C7×#494C2212yes—
16C7×#494C2212yes—
17C2×C4×C7×C8C2211yescharacteristic
18#89882—21no—
19#89860—22no—
20#89859C2218yes—
21#89850—22no—
22#89739—24no—
23#89697C2212yes—
24#89696C2211yescharacteristic
25#89460—21no—
26#89460C2212yes—
27C8×D28—21no—
28C8×D28C2212yes—
29#89342C2218yes—
30#89253C2212yes—
31#89252C2212yes—
32#19188—71nomaximal, Sylow
33C22×C7×Q8—22no—
34C2×#1157—21no—
35C2×C7×Q16—22no—
36C2×C7×Q16—22no—
37C2×C7×Q16—22no—
38C2×C7×Q16—22no—
39C2×C7×Q16—44no—
40C2×C7×Q16—44no—
41C22×C7×C8D411yescharacteristic
42C22×C7×C8D411yescharacteristic
43C7×C4⋊Q8—22no—
44C7×C4⋊Q8—22no—
45C7×C4⋊Q8—22no—
46C7×C4⋊Q8C2314yes—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—22no—
49C2×C42×C7C2311yescharacteristic
50#6493—21no—
51#6493D412yes—
52C2×#1136—21no—
53#6464—44no—
54C2×#1131—22no—
55C2×C8×D7—22no—
56#6438—21no—
57#6437D411yescharacteristic
58C4×D28—21no—
59C4×D28C2312yes—
60#6432D411yescharacteristic
61C4×C7×C8D411yescharacteristic
62C4×C7×C8D411yescharacteristic
63C4×C7×C8C2311yescharacteristic
64C4×C7×C8C2311yescharacteristic
65C22⋊(C7⋊C8)—22no—
66(C2×D7)⋊C8—22no—
67(C2×D7)⋊C8—22no—
68#6383—28no—
69#6381—28no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)C2314yes—
72C4⋊(C7⋊C8)C2312yes—
73C4×C7⋊C8—22no—
74#3226—72no—
75C2×#494D711yescharacteristic, complemented
76#2931—71no—
77#1689—74no—
78#1669—74no—
79#1564—71no—
80#1546—72no—
81C2×C7×Q8—24no—
82C2×C7×Q8—42no—
83C2×C7×Q8—44no—
84C2×C7×Q8—44no—
85C22×C4×C7—42no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C2×C22⋊D7—22no—
90#1157—42no—
91#1157—42no—
92C2×D28—21no—
93C2×D28—21no—
94C2×D28C2×D412yes—
95C2×C4×D7—22no—
96C2×Dic14—21no—
97C7×Q16—44no—
98C7×Q16—44no—
99C7×Q16—44no—
100C7×Q16—44no—
101C2×C7×C8—22no—
102C2×C7×C8D814yes—
103C2×C7×C8C2×D411yescharacteristic
104C2×C7×C8C2×D411yescharacteristic
105C2×C7×C8C2×D411yescharacteristic
106C2×C7×C8C2×D411yescharacteristic
107C7×C4⋊C4—24no—
108C7×C4⋊C4—24no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—42no—
111C42×C7C2×D411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7C2411yescharacteristic
115(C2×D7)⋊C4—22no—
116(C2×D7)⋊C4—22no—
117C4⋊Dic7—21no—
118C4⋊Dic7C2×D412yes—
119#1136—42no—
120C2×C7⋊C8—24no—
121C2×C7⋊C8—22no—
122#1131—44no—
123C8×D7—44no—
124C22×Q16—21no—
125C22×Q16—21no—
126C2×#131—141no—
127#532—71no—
128C2×C4⋊Q8C2×D712yes—
129#500—142no—
130#494—21no—
131#494—21no—
132#494C2×D712yes—
133#494C2×D712yes—
134#486—142no—
135#480—144no—
136#474—78no—
137#436—72no—
138#436—141no—
139C8×D4—72no—
140C8×D4—141no—
141#431—72no—
142#423—71no—
143#408—141no—
144C2×C4×C8C2×D711yescharacteristic
145#335—72no—
146#334—72no—
147#332—78no—
148C23×C7#12111yescharacteristic
149C22×D7—22no—
150C7×Q8—44no—
151C7×Q8—44no—
152C7×Q8—84no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—44no—
156C2×C4×C7—84no—
157C2×C4×C7#12111yescharacteristic
158C2×C4×C7#12111yescharacteristic
159C2×C4×C7#12111yescharacteristic
160C2×C4×C7C2×D812yes—
161C2×C4×C7AGL(1,ℤ/8)12yes—
162C2×C4×C7C22×D411yescharacteristic
163C2×C4×C7C22×D411yescharacteristic
164C2×C4×C7C2×#4111yescharacteristic
165C22⋊D7—44no—
166C2×Dic7—22no—
167D28—24no—
168D28—42no—
169D28—42no—
170C4×D7—42no—
171C4×D7—42no—
172Dic14—42no—
173C7×C8—21no—
174C7×C8—21no—
175C7×C8C2×D812yes—
176C7×C8C2×D812yes—
177C7⋊C8—42no—
178C7⋊C8—44no—
179C2×#41—141no—
180C22×Q8—22no—
181C2×Q16—22no—
182C2×Q16—22no—
183C2×Q16—22no—
184C2×Q16—22no—
185C2×Q16—44no—
186C2×Q16—44no—
187#131—284no—
188C2×M16—141no—
189C2×M16—142no—
190C22×C8—142no—
191C22×C8C22⋊D711yescharacteristic
192C22×C8C22⋊D711yescharacteristic
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—71no—
197C4⋊Q8C22×D714yes—
198#127—71no—
199#124—141no—
200#121—72no—
201#121—141no—
202C4×D4—72no—
203C4×D4—141no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C42C22×D711yescharacteristic
207#107—148no—
208C4⋊C8—72no—
209C4⋊C8—74no—
210C4⋊C8—141no—
211#102—148no—
212C22⋊C8—142no—
213C22⋊C8—142no—
214C22⋊C8—142no—
215C4×C8—142no—
216C4×C8C22⋊D711yescharacteristic
217C4×C8C22⋊D711yescharacteristic
218C4×C8C22×D711yescharacteristic
219C4×C8C22×D711yescharacteristic
220C22×C7—21no—
221C22×C7—21no—
222C22×C7#45912yes—
223C22×C7C2×#12111yescharacteristic
224C2×D7—42no—
225C2×D7—42no—
226C2×D7—42no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—84no—
230C4×C7#45912yes—
231C4×C7#47312yes—
232C4×C7C22×D811yescharacteristic
233C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
234C4×C7#57812yes—
235Dic7—42no—
236#41—282no—
237#41—282no—
238C2×Q8—24no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—141no—
243C2×D4—72no—
244C2×D4—142no—
245C2×D4—141no—
246C2×D4—141no—
247C22×C4—42no—
248C22×C4—142no—
249C22×C4D4×D711yescharacteristic
250C22×C4#115911yescharacteristic
251C22×C4C2×C22⋊D711yescharacteristic
252Q16—44no—
253Q16—44no—
254Q16—44no—
255Q16—44no—
256M16—282no—
257M16—284no—
258C2×C8—22no—
259C2×C8—144no—
260C2×C8—142no—
261C2×C8—284no—
262C2×C8#114114yes—
263C2×C8C2×C22⋊D711yescharacteristic
264C2×C8C2×C22⋊D711yescharacteristic
265C2×C8C2×C22⋊D711yescharacteristic
266C2×C8C2×C22⋊D711yescharacteristic
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—72no—
272C4⋊C4—141no—
273C22⋊C4—142no—
274C22⋊C4—142no—
275C42D4×D711yescharacteristic
276C42#115911yescharacteristic
277C42C2×C22⋊D711yescharacteristic
278C42C23×D711yescharacteristic
279C2×C7—22no—
280C2×C7C2×#45911yescharacteristic
281C2×C7#305911yescharacteristic
282C2×C7C2×#47311yescharacteristic
283D7—82no—
284C23—142no—
285C23#650111yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—84no—
289Q8—282no—
290D4—144no—
291D4—284no—
292D4—282no—
293D4—282no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—84no—
298C2×C4—142no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4C2×#114112yes—
302C2×C4#649512yes—
303C2×C4#650111yescharacteristic
304C2×C4#650111yescharacteristic
305C2×C4#650111yescharacteristic
306C2×C4C2×D4×D711yescharacteristic
307C2×C4C2×#115911yescharacteristic
308C2×C4C22×C22⋊D711yescharacteristic
309C8—21no—
310C8—21no—
311C8—284no—
312C8—282no—
313C8C2×#114112yes—
314C8C2×#114112yes—
315C7#1918811yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—282no—
319C22—282no—
320C22—282no—
321C22#8984412yes—
322C22C2×#650111yescharacteristic
323C4—21no—
324C4—21no—
325C4—84no—
326C4—282no—
327C4#8984412yes—
328C4#8996412yes—
329C4#9045712yes—
330C4C22×#114111yescharacteristic
331C4C2×#649511yescharacteristic
332C2—22no—
333C2—562no—
334C2C2×#8984411yescharacteristic
335C2#1174133211yescharacteristic
336C2C2×#8996411yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#494C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenterC2×#494D7
hypocenterC7#19188
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C2×C7×#494C2semidirect
4. #11729527C2semidirect
5. #11729060C2semidirect
6. #11729059C2semidirect
7. #11727483C2semidirect
75. C2×#494D7semidirect
315. C7#19188semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4942
#117406602
#117359032
#117295272
#117290602
#117290592
#117274832
#191887