Finite Group Databasebeta

← #13251511

Subgroups — #13251511

Order 1792.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2548C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic, complemented
3#11729514C212yesmaximal
4#11729379C211yesmaximal, characteristic, complemented
5#11725427C212yesmaximal, complemented
6C2×C7×#127C2211yescharacteristic
7C2×C7×#124C412yes—
8C2×C4×C7×Q8C2211yescharacteristic
9C7×#394—21no—
10C7×#394—21no—
11C7×#390—21no—
12C7×#390—21no—
13C7×#390C412yes—
14C4×C7×C4⋊C4C2211yescharacteristic
15#89796—24no—
16#89698—21no—
17#89698C2212yes—
18#89698C2212yes—
19#89697—21no—
20#89339—24no—
21#89323—21no—
22#89323—21no—
23#9458—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4C2×C412yes—
28C7×#127—22no—
29C7×#127—22no—
30C7×#124—44no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C42×C7C2×C412yes—
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C42×C7C2311yescharacteristic
45C7×#95—21no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)D414yes—
52#6377—22no—
53#6377—22no—
54#6377—22no—
55C4×C7⋊C8—22no—
56#2983—71no—
57#2548D711yescharacteristic
58#1555—72no—
59#1467—71no—
60#1285—72no—
61C24×C7—22no—
62C2×C7×Q8—22no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×D4—24no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—42no—
70C2×C7×D4—41no—
71C2×C7×D4—42no—
72C2×C7×D4—41no—
73C2×C7×D4—82no—
74C22×C4×C7—22no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×C4⋊C4—22no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—42no—
86C7×C22⋊C4—44no—
87C7×C22⋊C4—44no—
88C7×C22⋊C4—82no—
89C7×C22⋊C4—82no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—42no—
95C42×C7C2×D412yes—
96C42×C7#4112yes—
97C2×C7⋊C8—24no—
98C2×C7⋊C8—42no—
99C2×#127C2×D711yescharacteristic
100C2×#124Dic712yes—
101C2×C4×Q8C2×D711yescharacteristic
102#446—144no—
103#432—72no—
104#432—72no—
105#432—141no—
106#431—141no—
107#394—21no—
108#394—21no—
109#390—21no—
110#390—21no—
111#390Dic712yes—
112C4×C4⋊C4C2×D711yescharacteristic
113#331—144no—
114#324—141no—
115#324—141no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—44no—
125C7×D4—84no—
126C7×D4—82no—
127C7×D4—84no—
128C7×D4—82no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7#10012yes—
146C2×C4×C7C4×D412yes—
147C2×C4×C7#12011yescharacteristic
148C2×C4×C7#12112yes—
149C2×C4×C7#12311yescharacteristic
150C7⋊C8—44no—
151C7⋊C8—82no—
152C22×Q8C2×Dic711yescharacteristic
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C2×Dic712yes—
156#127—22no—
157#127—22no—
158#124—44no—
159C4×Q8—22no—
160C4×Q8—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic711yescharacteristic
163C2×C4⋊C4C22⋊D711yescharacteristic
164C2×C4⋊C4C22⋊D711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42C2×Dic712yes—
170C2×C42C22⋊D711yescharacteristic
171C2×C42C22⋊D711yescharacteristic
172C2×C42C22×D711yescharacteristic
173C4⋊C8—74no—
174C4⋊C8—142no—
175C4⋊C8—142no—
176C4⋊C8—142no—
177C22⋊C8—142no—
178C22⋊C8—142no—
179#97—142no—
180#97—142no—
181#97—142no—
182C4×C8—142no—
183#95—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—82no—
187C22×C7—82no—
188C22×C7—82no—
189C22×C7—82no—
190C22×C7—82no—
191C22×C7—82no—
192C22×C7—82no—
193C22×C7#38611yescharacteristic
194C22×C7C2×#10011yescharacteristic
195C22×C7#41311yescharacteristic
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—22no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—82no—
204C4×C7#45412yes—
205C24—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—82no—
218C22×C4—22no—
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22⋊Dic711yescharacteristic
223C22×C4C22×Dic711yescharacteristic
224C22×C4C2×C22⋊D711yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C2×C8—144no—
227C2×C8—282no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42D4×D712yes—
242C42#115912yes—
243C2×C7—21no—
244C2×C7—21no—
245C2×C7—162no—
246C2×C7#188811yescharacteristic
247C2×C7#190011yescharacteristic
248C2×C7#191011yescharacteristic
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23C2×C22⋊Dic711yescharacteristic
257Q8—44no—
258D4—84no—
259D4—82no—
260D4—84no—
261D4—82no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4#640712yes—
279C2×C4D4×Dic712yes—
280C2×C4#649811yescharacteristic
281C2×C4#650011yescharacteristic
282C2×C4#650212yes—
283C8—284no—
284C8—562no—
285C7#945811yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22C2×#640711yescharacteristic
296C22#8999211yescharacteristic
297C22#9000911yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4#8983412yes—
307C2—21no—
308C2—21no—
309C2—162no—
310C2#1172943711yescharacteristic
311C2#1172947811yescharacteristic
312C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2548C2
socleC22×C7#386
prime radicalC2×C7×#127C22
cosocleC2×C42×C7C23
hypercenter#2548D7
hypocenterC7#9458
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #11739881C2semidirect
4. #11729379C2semidirect
5. #11725427C2semidirect
285. C7#9458semidirect

Maximal subgroups

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

subgroupindex
C7×#25482
#117398812
#117295142
#117293792
#117254272
#94587