Finite Group Databasebeta

← #13248800

Subgroups — #13248800

Order 1792.

836 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2312C211yesmaximal, characteristic
2C2×#89719C211yesmaximal, characteristic
3C2×#89320C211yesmaximal, characteristic
4#11727279C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C23×Dic14C2211yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic7—21no—
9C7×#388—24no—
10C7×#380C2212yes—
11C7×#379—21no—
12C4×C7×C22⋊C4—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2212yes—
16C2×#6384—24no—
17#89719—22no—
18#89719C414yescomplemented
19#89320—22no—
20#9016—71nomaximal, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C22×Dic14—22no—
27C22×Dic14—22no—
28C22×Dic14—21no—
29C22×Dic14—44no—
30C22×Dic14C2×C412yes—
31C2×C7×C4⋊C4D414yes—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C412yes—
38C2×C7×C22⋊C4D412yes—
39C2×C42×C7—22no—
40C2×C22⋊Dic7—24no—
41C2×C22⋊Dic7—44no—
42C22×C7⋊C8—22no—
43C7×#95—22no—
44C7×#95—22no—
45C7⋊#95—24no—
46C7⋊#95—21no—
47C7⋊#95C2×C412yes—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50#6384—48no—
51C2×#393—71no—
52#2312D711yescharacteristic
53#1787—72no—
54C2×#323—71no—
55#1294—72no—
56C24×C7—21no—
57C24×C7—22no—
58C24×C7—21no—
59C24×C7—22no—
60C24×C7—42no—
61C24×C7—42no—
62C24×C7C22⋊C412yes—
63C24×C7C2×D411yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—24no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7C22×C411yescharacteristic
71C22×C4×C7C2×D412yes—
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7#4112yes—
74C22×Dic7—24no—
75C22×Dic7—22no—
76C22×Dic7—22no—
77C22×Dic7—42no—
78C22×Dic7—41no—
79C22×Dic7—41no—
80C22×Dic7—42no—
81C22×Dic7—42no—
82C22×Dic7—42no—
83C2×Dic14—28no—
84C2×Dic14—24no—
85C2×Dic14—44no—
86C2×Dic14—48no—
87C2×Dic14—44no—
88C2×Dic14—44no—
89C2×Dic14—42no—
90C7×C4⋊C4—44no—
91C7×C22⋊C4—24no—
92C7×C22⋊C4—24no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—82no—
96C7×C22⋊C4—82no—
97C42×C7—82no—
98C22⋊Dic7—48no—
99C2×C7⋊C8—44no—
100C2×C7⋊C8—42no—
101C23×Q8—71no—
102C22×C22⋊C4—141no—
103C22×C22⋊C4C2×D711yescharacteristic
104C2×#103—144no—
105C2×C22⋊C8—72no—
106#393—74no—
107#393—142no—
108#388—24no—
109#380C2×D712yes—
110#379—21no—
111C4×C22⋊C4—21no—
112C2×#95—71no—
113C2×#95—141no—
114#323—142no—
115C23×C7—22no—
116C23×C7—22no—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—48no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—42no—
126C23×C7—42no—
127C23×C7—81no—
128C23×C7—81no—
129C23×C7—84no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#12011yescharacteristic
132C23×C7#12311yescharacteristic
133C23×C7#13512yes—
134C23×C7#13712yes—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—44no—
145C2×C4×C7—81no—
146C2×C4×C7C4×D412yes—
147C2×C4×C7#12112yes—
148C2×Dic7—24no—
149C2×Dic7—42no—
150C2×Dic7—44no—
151C2×Dic7—42no—
152C2×Dic7—44no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—84no—
157Dic14—28no—
158Dic14—48no—
159Dic14—44no—
160C7⋊C8—82no—
161C25C22⋊D711yescharacteristic
162C22×Q8—72no—
163C22×Q8—141no—
164C22×Q8—142no—
165C22×Q8—142no—
166C22×Q8—284no—
167C23×C4—141no—
168C23×C4—141no—
169C23×C4C22⋊D711yescharacteristic
170C23×C4C22×D711yescharacteristic
171C22×C8—142no—
172C2×C4⋊C4C22⋊D714yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—144no—
179C2×C22⋊C4—284no—
180C2×C22⋊C4C4×D712yes—
181C2×C22⋊C4D2812yes—
182C2×C42—22no—
183#103—288no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#95—22no—
187#95—22no—
188#95—72no—
189#95—144no—
190#95—141no—
191C22×C7—24no—
192C22×C7—24no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—44no—
198C22×C7—42no—
199C22×C7—42no—
200C22×C7—42no—
201C22×C7—82no—
202C22×C7—84no—
203C22×C7—84no—
204C22×C7#38611yescharacteristic
205C22×C7#41611yescharacteristic
206C22×C7#41911yescharacteristic
207C22×C7#45212yes—
208C22×C7#45512yes—
209C22×C7B(4,𝔽2)12yes—
210C22×C7#48512yes—
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—42no—
214C4×C7—84no—
215Dic7—44no—
216Dic7—82no—
217C24—22no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24(C2×D7)⋊C412yes—
224C24C2×C22⋊D711yescharacteristic
225C2×Q8—148no—
226C2×Q8—144no—
227C2×Q8—284no—
228C2×Q8—284no—
229C2×Q8—284no—
230C2×Q8—288no—
231C2×Q8—282no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—144no—
238C22×C4—142no—
239C22×C4—142no—
240C22×C4—282no—
241C22×C4—281no—
242C22×C4—281no—
243C22×C4—282no—
244C22×C4—282no—
245C22×C4—282no—
246C22×C4(C2×D7)⋊C412yes—
247C22×C4C2×C4×D711yescharacteristic
248C22×C4C2×D2811yescharacteristic
249C22×C4#115712yes—
250C22×C4C2×C22⋊D712yes—
251C2×C8—284no—
252C2×C8—282no—
253C4⋊C4—44no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—288no—
261C42—82no—
262C2×C7—21no—
263C2×C7—21no—
264C2×C7—22no—
265C2×C7—82no—
266C2×C7#160614yes—
267C2×C7#187311yescharacteristic
268C2×C7#188511yescharacteristic
269C2×C7#188911yescharacteristic
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—48no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—81no—
283C23—81no—
284C23—84no—
285C23C2×(C2×D7)⋊C411yescharacteristic
286C23#649211yescharacteristic
287C23#651312yes—
288C23#651412yes—
289C23#651611yescharacteristic
290Q8—148no—
291Q8—288no—
292Q8—284no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—81no—
304C2×C4—144no—
305C2×C4—282no—
306C2×C4—284no—
307C2×C4—282no—
308C2×C4—284no—
309C2×C4—282no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—564no—
313C2×C4C4×C22⋊D712yes—
314C2×C4#649312yes—
315C8—562no—
316C7#901611yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22—24no—
320C22—24no—
321C22—21no—
322C22—21no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#8976812yes—
331C22#8977411yescharacteristic
332C22#8977611yescharacteristic
333C22#8998511yescharacteristic
334C22#9001112yes—
335C22#9001412yes—
336C22#9001612yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—84no—
341C4—284no—
342C4—562no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#1172933911yescharacteristic
348C2#1172940311yescharacteristic
349C2#1172940811yescharacteristic
350C2#1172963914yes—
subgroupG/H
centerC23#6516
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2312C2
socleC23×C7#120
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2312D7
hypocenterC7#9016
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
4. #11727279C2semidirect
5. #11725430C2semidirect
18. #89719C4semidirect
316. C7#9016semidirect

Maximal subgroups

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

subgroupindex
C7×#23122
C2×#897192
C2×#893202
#117272792
#117254302
#90167