Finite Group Databasebeta

← #13247432

Subgroups — #13247432

Order 1792.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2357C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11725434C212yesmaximal, complemented
5#11725424C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C7×#390—21no—
8C7×#388—21no—
9C7×#388—21no—
10C7×#382C411yescharacteristic, complemented
11C7×#381C411yescharacteristic, complemented
12C7×#381C2211yescharacteristic
13C4×C7×C4⋊C4—21no—
14C4×C7×C4⋊C4C411yescharacteristic, complemented
15C4×C7×C22⋊C4C411yescharacteristic, complemented
16C4×C7×C22⋊C4C2211yescharacteristic
17C2×#6408—21no—
18C4×C22⋊Dic7C2212yes—
19#89979C2212yes—
20#89323—21no—
21#89321—21no—
22#89320—21no—
23#8778—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C4×C7×D4—22no—
28C4×C7×D4—22no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C22⋊C4—21no—
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C42×C7—21no—
39C2×C42×C7—21no—
40C2×C42×C7—21no—
41C2×C42×C7C2×C411yescharacteristic
42C2×C22⋊Dic7—22no—
43C2×C4×Dic7—22no—
44C2×#1136—22no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95C2×C411yescharacteristic
49C7×#95C2×C411yescharacteristic
50C7×#95C2×C411yescharacteristic
51C7×#95C2×C411yescharacteristic
52#6408—24no—
53C7⋊#95—22no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56#2357D711yescharacteristic
57C4×#99—71no—
58#1465—71no—
59#1297—72no—
60#1284—72no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—81no—
73C22×C4×C7—81no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7#4112yes—
80C22×C4×C7#4112yes—
81C22×Dic7—42no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—42no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—81no—
90C7×C22⋊C4—81no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—42no—
94C42×C7—81no—
95C42×C7—81no—
96C22⋊Dic7—24no—
97C22⋊Dic7—42no—
98C4×Dic7—82no—
99#1136—82no—
100C2×C7⋊C8—42no—
101C2×C4×D4C2×D711yescharacteristic
102C2×#99—141no—
103#407—72no—
104#390—21no—
105#388—21no—
106#388—21no—
107#382Dic711yescharacteristic, complemented
108#381Dic711yescharacteristic, complemented
109#381C2×D711yescharacteristic
110C4×C4⋊C4—21no—
111C4×C4⋊C4Dic711yescharacteristic, complemented
112C4×C22⋊C4—72no—
113C4×C22⋊C4Dic711yescharacteristic, complemented
114C4×C22⋊C4C2×D711yescharacteristic
115#342—141no—
116#324—141no—
117#323—141no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7C2×C22⋊C411yescharacteristic
128C23×C7#12312yes—
129C7×D4—44no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—81no—
153C2×C4×C7—81no—
154C2×C4×C7#10412yes—
155C2×C4×C7#11711yescharacteristic
156C2×C4×C7#11711yescharacteristic
157C2×C4×C7#12312yes—
158C2×Dic7—44no—
159C2×Dic7—82no—
160C2×Dic7—82no—
161C7⋊C8—82no—
162C22×D4C22⋊D711yescharacteristic
163C23×C4C2×Dic711yescharacteristic
164C23×C4C22×D711yescharacteristic
165C2×M16—142no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C22⋊D711yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—142no—
176C2×C22⋊C4C22⋊D711yescharacteristic
177C2×C22⋊C4C22⋊D711yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—142no—
182C2×C42C2×Dic711yescharacteristic
183#99—144no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—142no—
190#95C2×Dic711yescharacteristic
191#95C2×Dic711yescharacteristic
192#95C2×Dic711yescharacteristic
193#95C2×Dic711yescharacteristic
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—22no—
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—81no—
203C22×C7—81no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7#38111yescharacteristic
207C22×C7C2×#10411yescharacteristic
208C22×C7#42111yescharacteristic
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—42no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—81no—
217C4×C7—81no—
218C4×C7—81no—
219Dic7—84no—
220C24C22⋊Dic711yescharacteristic
221C24C2×C22⋊D711yescharacteristic
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—282no—
234C22×C4C22⋊Dic711yescharacteristic
235C22×C4C22⋊Dic711yescharacteristic
236C22×C4C22⋊Dic711yescharacteristic
237C22×C4#115712yes—
238C22×C4#115712yes—
239C22×C4C22×Dic711yescharacteristic
240C22×C4C2×C22⋊D711yescharacteristic
241M16—562no—
242C2×C8—282no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—144no—
253C22⋊C4—282no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42—81no—
258C42—81no—
259C42—562no—
260C2×C7—21no—
261C2×C7—21no—
262C2×C7—42no—
263C2×C7—81no—
264C2×C7#182611yescharacteristic
265C2×C7#182711yescharacteristic
266C2×C7#182811yescharacteristic
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#649212yes—
277C23C2×C22⋊Dic711yescharacteristic
278D4—44no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—284no—
304C2×C4—562no—
305C2×C4—562no—
306C2×C4#641112yes—
307C2×C4#648911yescharacteristic
308C2×C4#648911yescharacteristic
309C2×C4#649212yes—
310C8—562no—
311C7#877811yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#8998311yescharacteristic
325C22C2×#641111yescharacteristic
326C22#9000811yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—564no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#1172934411yescharacteristic
343C2#1172939811yescharacteristic
344C2#1172940711yescharacteristic
subgroupG/H
centerC22#89983
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2357C2
socleC22×C7#381
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2357D7
hypocenterC7#8778
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
4. #11725434C2semidirect
5. #11725424C2semidirect
10. C7×#382C4semidirect
11. C7×#381C4semidirect
14. C4×C7×C4⋊C4C4semidirect
15. C4×C7×C22⋊C4C4semidirect
107. #382Dic7semidirect
108. #381Dic7semidirect
111. C4×C4⋊C4Dic7semidirect
113. C4×C22⋊C4Dic7semidirect
311. C7#8778semidirect

Maximal subgroups

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

subgroupindex
C7×#23572
C4×#64082
#117293562
#117254342
#117254242
#87787