Finite Group Databasebeta

← #13247232

Subgroups — #13247232

Order 1792.

514 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#11731763C211yesmaximal, characteristic
2#11728229C212yesmaximal
3C7×#1296C211yesmaximal, characteristic, complemented
4#11725508C212yesmaximal, complemented
5#11725440C211yesmaximal, characteristic, complemented
6C2×#6455C412yes—
7C2×C42×D7C2211yescharacteristic
8C2×C7×C4⋊C8C2212yes—
9C7×#384C2211yescharacteristic
10C2×(C2×D7)⋊C8—22no—
11#89759—22no—
12#89749C412yes—
13C4⋊C4×Dic7—22no—
14#89707C2211yescharacteristic
15C2×C4⋊(C7⋊C8)C2212yes—
16#89345—22no—
17#8728—71nomaximal, Sylow
18C22×C4×D7C2×C411yescharacteristic
19C22×C4×D7D412yes—
20C22×C7×C8—22no—
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4C2×C412yes—
23C2×C42×C7C2311yescharacteristic
24C2×(C2×D7)⋊C4—22no—
25C2×C4⋊Dic7—22no—
26C2×C4⋊Dic7C2×C412yes—
27C2×C4×Dic7—22no—
28C2×C4×Dic7C2×C411yescharacteristic
29C2×C4×Dic7D412yes—
30C22×C7⋊C8—22no—
31#6455—44no—
32C42×D7—22no—
33C42×D7—21no—
34C42×D7—21no—
35C7×C4⋊C8—22no—
36C7×C4⋊C8—22no—
37C7⋊#95—24no—
38(C2×D7)⋊C8—44no—
39C4⋊(C7⋊C8)—22no—
40C4⋊(C7⋊C8)—22no—
41#2285—71no—
42#1797—72no—
43#1296—72no—
44#1296—71no—
45#1296D711yescharacteristic, complemented
46C23×D7C22⋊C411yescharacteristic
47C22×C4×C7—22no—
48C22×C4×C7C22×C411yescharacteristic
49C22×C4×C7C2×D412yes—
50C22×Dic7—22no—
51C22×Dic7C22⋊C412yes—
52C22×Dic7C22⋊C411yescharacteristic
53C2×C4×D7—22no—
54C2×C4×D7—21no—
55C2×C4×D7—21no—
56C2×C4×D7—22no—
57C2×C4×D7—22no—
58C2×C4×D7—22no—
59C2×C4×D7—22no—
60C2×C4×D7—22no—
61C2×C4×D7—22no—
62C2×C4×D7—22no—
63C2×C7×C8—42no—
64C2×C7×C8—42no—
65C2×C7×C8—42no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C42×C7#4112yes—
70C42×C7#4111yescharacteristic
71C42×C7#4111yescharacteristic
72(C2×D7)⋊C4—44no—
73C4⋊Dic7—42no—
74C4⋊Dic7—42no—
75C4⋊Dic7—42no—
76C4×Dic7—21no—
77C4×Dic7—21no—
78C4×Dic7—22no—
79C4×Dic7—22no—
80C4×Dic7—44no—
81C2×C7⋊C8—42no—
82C2×C7⋊C8—42no—
83C2×C7⋊C8—42no—
84C2×#117—72no—
85C22×C42—71no—
86C2×C4⋊C8—72no—
87C2×C4⋊C8C2×D712yes—
88C2×C22⋊C8—142no—
89#384—71no—
90#384C2×D711yescharacteristic
91#382—72no—
92#380—142no—
93C4×C4⋊C4—142no—
94#336—142no—
95C23×C7C2×C22⋊C411yescharacteristic
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7#10112yes—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—42no—
104C2×C4×C7—44no—
105C2×C4×C7#11711yescharacteristic
106C2×C4×C7#11711yescharacteristic
107C2×C4×C7#12312yes—
108C2×C4×C7#12312yes—
109C2×C4×C7#13512yes—
110C2×C4×C7#13512yes—
111C2×C4×C7AGL(1,ℤ/8)12yes—
112C2×C4×C7#13712yes—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—42no—
119C2×Dic7—44no—
120C2×Dic7#10012yes—
121C4×D7—24no—
122C4×D7—24no—
123C4×D7—42no—
124C4×D7—42no—
125C4×D7—42no—
126C4×D7—42no—
127C7×C8—82no—
128C7⋊C8—82no—
129C23×C4—71no—
130C23×C4—72no—
131C22×C8—22no—
132C22×C8—142no—
133#117—284no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—142no—
137C2×C4⋊C4C4×D712yes—
138C2×C22⋊C4—142no—
139C2×C42—71no—
140C2×C42—72no—
141C2×C42—142no—
142C2×C42—141no—
143C2×C42—141no—
144C2×C42—142no—
145C2×C42C22×D711yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—142no—
149C4⋊C8—142no—
150C22⋊C8—284no—
151#95—144no—
152C22×C7#38111yescharacteristic
153C22×C7C2×#10011yescharacteristic
154C22×C7C2×#10111yescharacteristic
155C22×C7#41612yes—
156C22×C7#41712yes—
157C2×D7—24no—
158C2×D7—22no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—42no—
162C4×C7—21no—
163C4×C7—21no—
164C4×C7—82no—
165C4×C7#48212yes—
166C4×C7#48212yes—
167C4×C7#48512yes—
168C4×C7#48512yes—
169Dic7—22no—
170Dic7—42no—
171Dic7—82no—
172C24—71no—
173C22×C4—22no—
174C22×C4—71no—
175C22×C4—72no—
176C22×C4—142no—
177C22×C4—142no—
178C22×C4—141no—
179C22×C4—141no—
180C22×C4—142no—
181C22×C4—142no—
182C22×C4—142no—
183C22×C4—142no—
184C22×C4—142no—
185C22×C4—142no—
186C22×C4—142no—
187C22×C4C2×C4×D711yescharacteristic
188C22×C4D4×D712yes—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—282no—
193C2×C8—282no—
194C2×C8—282no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—282no—
199C4⋊C4—282no—
200C4⋊C4—282no—
201C22⋊C4—284no—
202C42—141no—
203C42—142no—
204C42—141no—
205C42—142no—
206C42—284no—
207C42#115712yes—
208C42#115911yescharacteristic
209C42#116111yescharacteristic
210C2×C7#154311yescharacteristic
211C2×C7#155212yes—
212C2×C7#156011yescharacteristic
213C2×C7#182111yescharacteristic
214C2×C7#182512yes—
215D7—42no—
216C23—72no—
217C23—141no—
218C23—141no—
219C23—142no—
220C23—142no—
221C23D7×C22⋊C411yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—72no—
227C2×C4—144no—
228C2×C4—144no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—282no—
235C2×C4—282no—
236C2×C4—282no—
237C2×C4—282no—
238C2×C4—282no—
239C2×C4—284no—
240C2×C4#643511yescharacteristic
241C2×C4#644612yes—
242C2×C4#645511yescharacteristic
243C2×C4#645712yes—
244C2×C4#647412yes—
245C2×C4#647812yes—
246C2×C4#647912yes—
247C2×C4#648212yes—
248C8—82no—
249C8—562no—
250C7#872811yescharacteristic, complemented, Sylow
251C22—144no—
252C22—142no—
253C22—281no—
254C22—281no—
255C22—282no—
256C22#8943611yescharacteristic
257C22D7×#10011yescharacteristic
258C22D7×#10111yescharacteristic
259C22#8954312yes—
260C22#8957612yes—
261C4—21no—
262C4—21no—
263C4—82no—
264C4—142no—
265C4—282no—
266C4—562no—
267C4#8955112yes—
268C4#8958612yes—
269C4#8963912yes—
270C4#8965712yes—
271C2—282no—
272C2#1172614311yescharacteristic
273C2#1172617912yes—
274C2#1172623611yescharacteristic
275C2#1172656712yes—
276C2#1172674411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1296C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1296D7
hypocenterC7#8728
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. C7×#1296C2semidirect
4. #11725508C2semidirect
5. #11725440C2semidirect
45. #1296D7semidirect
250. C7#8728semidirect

Maximal subgroups

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

subgroupindex
#117317632
#117282292
C7×#12962
#117255082
#117254402
#87287