Finite Group Databasebeta

← #13264413

Subgroups — #13264413

Order 1792.

460 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2563C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11725452C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C7×#393C412yescomplemented
8C7×#393C2212yes—
9C7×#390—21no—
10C7×#382—21no—
11C7×#380—21no—
12C7×#380—21no—
13C4×C7×C4⋊C4C411yescharacteristic, complemented
14C2×#6408—21no—
15C4×C22⋊Dic7—21no—
16#89979—21no—
17#89323—21no—
18#89321C2212yes—
19#89320C2212yes—
20#11787—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7C2×C411yescharacteristic
39C2×C22⋊Dic7—22no—
40C2×C4×Dic7—41no—
41C2×#1136—41no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95C2×C412yes—
45#6408—24no—
46C7⋊#95—41no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—41no—
49#2563D711yescharacteristic
50C4×#99—71no—
51#1465—71no—
52#1351—74no—
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×Dic7—42no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C22⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—82no—
82C7×C22⋊C4—82no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—42no—
86C22⋊Dic7—24no—
87C22⋊Dic7—42no—
88C4×Dic7—82no—
89#1136—82no—
90C2×C7⋊C8—42no—
91C2×C4⋊Q8Dic711yescharacteristic, complemented
92C2×C4×D4C2×D711yescharacteristic
93C2×#99—141no—
94#407—141no—
95#393Dic712yescomplemented
96#393C2×D712yes—
97#390—21no—
98#382—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4Dic711yescharacteristic, complemented
102C4×C22⋊C4—141no—
103#342—72no—
104#324—141no—
105#323—72no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#13712yes—
117C7×Q8—84no—
118C7×D4—44no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—81no—
137C2×C4×C7—81no—
138C2×C4×C7#9911yescharacteristic
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#10412yes—
141C2×Dic7—44no—
142C2×Dic7—82no—
143C2×Dic7—82no—
144C7⋊C8—82no—
145C22×Q8C2×Dic712yes—
146C22×D4C22⋊D711yescharacteristic
147C23×C4C22⋊D711yescharacteristic
148C23×C4C22⋊D711yescharacteristic
149C2×M16—281no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4×D4—41no—
153C4×D4—41no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D711yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—142no—
161C2×C22⋊C4C2×Dic711yescharacteristic
162C2×C22⋊C4C22×D711yescharacteristic
163C2×C42—21no—
164C2×C42—281no—
165C2×C42C2×Dic711yescharacteristic
166#99—144no—
167C22⋊C8—142no—
168C22⋊C8—281no—
169#95—22no—
170#95—22no—
171#95—281no—
172#95C2×Dic712yes—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—81no—
184C22×C7—81no—
185C22×C7C2×#9911yescharacteristic
186C22×C7#41911yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—21no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7—82no—
194C4×C7—82no—
195C4×C7#35412yes—
196Dic7—84no—
197C24C22⋊Dic711yescharacteristic
198C24C2×C22⋊D711yescharacteristic
199C2×Q8—42no—
200C2×Q8—82no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—282no—
213C22×C4C22⋊Dic711yescharacteristic
214C22×C4C22⋊Dic711yescharacteristic
215C22×C4C22⋊Dic711yescharacteristic
216C22×C4C22×Dic711yescharacteristic
217C22×C4C2×C22⋊D711yescharacteristic
218M16—562no—
219C2×C8—282no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—144no—
230C22⋊C4—282no—
231C42—21no—
232C42—21no—
233C42—42no—
234C42—562no—
235C2×C7—21no—
236C2×C7—21no—
237C2×C7—42no—
238C2×C7—81no—
239C2×C7#150811yescharacteristic
240C2×C7#212211yescharacteristic
241C2×C7C2×#35411yescharacteristic
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#650512yes—
252C23C2×C22⋊Dic711yescharacteristic
253Q8—84no—
254D4—44no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—284no—
275C2×C4—562no—
276C2×C4—562no—
277C2×C4#640811yescharacteristic
278C2×C4#640811yescharacteristic
279C2×C4#641112yes—
280C8—562no—
281C7#1178711yescharacteristic, complemented, Sylow
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#640811yescharacteristic
295C22#8999711yescharacteristic
296C22C2×#641111yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—564no—
305C4#8933612yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#1172944211yescharacteristic
311C2C2×#8933611yescharacteristic
312C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2563C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2563D7
hypocenterC7#11787
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
4. #11725452C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
7. C7×#393C4semidirect
13. C4×C7×C4⋊C4C4semidirect
91. C2×C4⋊Q8Dic7semidirect
95. #393Dic7semidirect
101. C4×C4⋊C4Dic7semidirect
281. C7#11787semidirect

Maximal subgroups

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

subgroupindex
C7×#25632
C4×#64082
#117293562
#117254522
#117877