Finite Group Databasebeta

← #13264474

Subgroups — #13264474

Order 1792.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2621C211yesmaximal, characteristic
2#11729348C211yesmaximal, characteristic
3#11729326C211yesmaximal, characteristic
4#11725455C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C7×#117C2211yescharacteristic
8C7×#398—21no—
9C7×#398C412yescomplemented
10C7×#397—21no—
11C7×#397C2212yes—
12C7×#391—21no—
13C7×#382C411yescharacteristic, complemented
14#89983—21no—
15#89698—21no—
16#89323—21no—
17#89321—21no—
18#89321C2212yes—
19#89320C2212yes—
20#11801—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C23×C4×C7D411yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C7×C22⋊Q8—44no—
26C7×#117—41no—
27C7×#117—41no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4C2×C412yes—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C22⋊Dic7—22no—
40C7×#95—21no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95C2×C412yes—
44C7⋊#95—41no—
45C7⋊#95—41no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48#6377—41no—
49#2621D711yescharacteristic
50#1820—71no—
51#1464—71no—
52#1344—74no—
53C24×C7C2×D411yescharacteristic
54C2×C7×Q8—42no—
55C2×C7×Q8—42no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C22×C4×C7—22no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—42no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—44no—
76C7×C4⋊C4—42no—
77C7×C22⋊C4—41no—
78C7×C22⋊C4—41no—
79C7×C22⋊C4—44no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7—21no—
84C22⋊Dic7—42no—
85C22⋊Dic7—42no—
86C2×C7⋊C8—42no—
87C2×C4⋊Q8Dic711yescharacteristic, complemented
88C2×C22⋊Q8—21no—
89C2×#117C2×D711yescharacteristic
90#432—141no—
91#398—21no—
92#398Dic712yescomplemented
93#397—21no—
94#397C2×D712yes—
95#391—21no—
96#382Dic711yescharacteristic, complemented
97#381—141no—
98#342—72no—
99#342—141no—
100#324—141no—
101#323—72no—
102C23×C7—21no—
103C23×C7—21no—
104C23×C7—41no—
105C23×C7—41no—
106C23×C7C2×C22⋊C411yescharacteristic
107C23×C7AGL(1,ℤ/8)12yes—
108C7×Q8—82no—
109C7×Q8—82no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7#9911yescharacteristic
132C2×C4×C7#9911yescharacteristic
133C2×Dic7—82no—
134C2×Dic7—82no—
135C2×Dic7—82no—
136C7⋊C8—82no—
137C22×Q8—21no—
138C23×C4C22⋊D711yescharacteristic
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C22⋊Q8—44no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic712yes—
148C2×C4⋊C4C2×Dic711yescharacteristic
149C2×C4⋊C4C22⋊D711yescharacteristic
150C2×C22⋊C4—22no—
151C2×C22⋊C4—142no—
152C2×C22⋊C4C22⋊D711yescharacteristic
153C2×C22⋊C4C22×D711yescharacteristic
154C2×C42C2×Dic711yescharacteristic
155C2×C42C22⋊D711yescharacteristic
156C4⋊C8—281no—
157C22⋊C8—142no—
158#97—281no—
159#95—21no—
160#95—22no—
161#95—22no—
162#95—281no—
163#95—281no—
164#95C2×Dic712yes—
165C22×C7—21no—
166C22×C7—21no—
167C22×C7—22no—
168C22×C7—42no—
169C22×C7—41no—
170C22×C7—41no—
171C22×C7—81no—
172C22×C7C2×#9911yescharacteristic
173C22×C7#41811yescharacteristic
174C22×C7#42111yescharacteristic
175C4×C7—21no—
176C4×C7—21no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7—41no—
180C4×C7—41no—
181C4×C7—41no—
182C4×C7—82no—
183C4×C7—82no—
184Dic7—162no—
185C24C2×C22⋊D711yescharacteristic
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—41no—
189C2×Q8—41no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—282no—
197C22×C4C22⋊Dic711yescharacteristic
198C22×C4C22⋊Dic711yescharacteristic
199C22×C4C22⋊Dic711yescharacteristic
200C22×C4C22⋊Dic711yescharacteristic
201C22×C4C22×Dic711yescharacteristic
202C22×C4C2×C22⋊D711yescharacteristic
203C2×C8—282no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—44no—
213C22⋊C4—282no—
214C22⋊C4—282no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C2×C7—21no—
220C2×C7—21no—
221C2×C7—42no—
222C2×C7#151011yescharacteristic
223C2×C7#212311yescharacteristic
224C2×C7#212911yescharacteristic
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23#649512yes—
230C23C2×C22⋊Dic711yescharacteristic
231Q8—82no—
232Q8—82no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—562no—
255C2×C4—562no—
256C2×C4—562no—
257C2×C4#640811yescharacteristic
258C2×C4#640811yescharacteristic
259C8—562no—
260C7#1180111yescharacteristic, complemented, Sylow
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22#8998711yescharacteristic
269C22C2×#640811yescharacteristic
270C22#9000811yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—1122no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#1172942811yescharacteristic
285C2#1172962411yescharacteristic
286C2#1172962511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2621C2
socleC22×C7C2×#99
prime radicalC24×C7C2×D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2621D7
hypocenterC7#11801
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
5. C2×C7×C4⋊Q8C4semidirect
9. C7×#398C4semidirect
13. C7×#382C4semidirect
87. C2×C4⋊Q8Dic7semidirect
92. #398Dic7semidirect
96. #382Dic7semidirect
260. C7#11801semidirect

Maximal subgroups

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

subgroupindex
C7×#26212
#117293482
#117293262
#117254552
#118017