Finite Group Databasebeta

← #13251186

Subgroups — #13251186

Order 1792.

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

normal subgroups only

This group has 574 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
1C7×#2982C211yesmaximal, characteristic, complemented
2#11739921C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic, complemented
4#11728528C212yesmaximal
5#11725552C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6510—21no—
8C2×#6437C2211yescharacteristic
9C7×C8×Q8—24no—
10C7×#431—22no—
11C7×#431C2212yes—
12C2×#6410—21no—
13#89763C414yes—
14#89760—21no—
15#89711—22no—
16C4×C4⋊Dic7C2211yescharacteristic
17#89696C2212yes—
18#89325—21no—
19#89280—24no—
20#9398—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×D28—21no—
23C22×D28—21no—
24C22×D28C2×C412yes—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C2×C7×C4⋊C4C2×C412yes—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C42×C7D412yes—
30C2×C42×C7C2311yescharacteristic
31#6510—42no—
32#6510—42no—
33C2×(C2×D7)⋊C4—24no—
34C2×(C2×D7)⋊C4—24no—
35C2×C4⋊Dic7—21no—
36C2×C4×Dic7—21no—
37C2×#1136—22no—
38#6437—22no—
39#6437—22no—
40C7×C4⋊C8—24no—
41C7×C4⋊C8D414yes—
42C7×C22⋊C8—22no—
43C4×C7×C8—24no—
44C4×C7×C8—22no—
45#6410—42no—
46C7⋊#95C2×C412yes—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49#2982D711yescharacteristic, complemented
50#2548—71no—
51#1801—71no—
52#1545—72no—
53#1303—72no—
54C23×D7—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8D812yes—
58C2×C7×Q8SD1612yes—
59C22×C4×C7C22⋊C412yes—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—22no—
65C2×D28—24no—
66C2×D28—44no—
67C2×D28—41no—
68C2×D28—42no—
69C2×D28—41no—
70C2×D28—41no—
71C2×D28—41no—
72C2×D28—42no—
73C2×D28—82no—
74C2×C7×C8—24no—
75C7×C4⋊C4—24no—
76C7×C4⋊C4—22no—
77C42×C7—22no—
78C42×C7—22no—
79C42×C7C2×D412yes—
80C42×C7#4112yes—
81(C2×D7)⋊C4—48no—
82(C2×D7)⋊C4—84no—
83C4⋊Dic7—42no—
84C4×Dic7—41no—
85C4×Dic7—41no—
86#1136—42no—
87#1136—42no—
88C2×C7⋊C8—42no—
89C2×#127—71no—
90C2×#124—141no—
91C2×C4×Q8C2×D711yescharacteristic
92C8×Q8—24no—
93#431—22no—
94#431C2×D712yes—
95#423—72no—
96C2×#101—141no—
97#394—74no—
98#390—141no—
99#390—142no—
100C4×C4⋊C4—71no—
101#343—141no—
102#331—144no—
103C23×C7C2×C22⋊C411yescharacteristic
104C22×D7—44no—
105C22×D7—44no—
106C22×D7—42no—
107C22×D7—42no—
108C22×D7—42no—
109C7×Q8—24no—
110C7×Q8—24no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—41no—
116C2×C4×C7—42no—
117C2×C4×C7#10214yes—
118C2×C4×C7C4×D412yes—
119C2×C4×C7#12011yescharacteristic
120C2×C4×C7#12112yes—
121C2×C4×C7#12311yescharacteristic
122C2×C4×C7C2×D811yescharacteristic
123C2×C4×C7C2×SD1611yescharacteristic
124C2×C4×C7#13512yes—
125C2×Dic7—42no—
126C2×Dic7—42no—
127C2×Dic7—42no—
128D28—84no—
129D28—82no—
130D28—82no—
131D28—82no—
132D28—82no—
133C7×C8—44no—
134C7⋊C8—82no—
135C22×Q8D2811yescharacteristic
136C22×D4—72no—
137C22×D4—141no—
138C22×D4—141no—
139C2×M16—142no—
140#127—142no—
141#127—142no—
142#124—282no—
143#124—282no—
144C4×Q8—22no—
145C4×Q8—22no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4C4×D712yes—
148C2×C4⋊C4D2811yescharacteristic
149C2×C22⋊C4—144no—
150C2×C22⋊C4—144no—
151C2×C42—141no—
152C2×C42C22⋊D712yes—
153C2×C42C22×D711yescharacteristic
154C4⋊C8—24no—
155C4⋊C8—142no—
156C4⋊C8—142no—
157C4⋊C8C22⋊D714yes—
158#101—282no—
159C22⋊C8—22no—
160C4×C8—24no—
161C4×C8—22no—
162#95—72no—
163C22×C7—21no—
164C22×C7—21no—
165C22×C7#38611yescharacteristic
166C22×C7C2×#10211yescharacteristic
167C22×C7#41611yescharacteristic
168C2×D7—84no—
169C2×D7—84no—
170C2×D7—82no—
171C2×D7—82no—
172C2×D7—82no—
173C4×C7—21no—
174C4×C7—24no—
175C4×C7—22no—
176C4×C7—42no—
177C4×C7#45412yes—
178C4×C7#46512yes—
179C4×C7#46612yes—
180Dic7—82no—
181C24—142no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8#113212yes—
185C2×Q8D5612yes—
186C2×D4—144no—
187C2×D4—284no—
188C2×D4—281no—
189C2×D4—281no—
190C2×D4—282no—
191C2×D4—281no—
192C2×D4—282no—
193C2×D4—281no—
194C2×D4—562no—
195C22×C4—142no—
196C22×C4(C2×D7)⋊C412yes—
197C22×C4(C2×D7)⋊C412yes—
198C22×C4C2×C4×D711yescharacteristic
199C22×C4C2×D2811yescharacteristic
200C22×C4C2×C22⋊D711yescharacteristic
201M16—282no—
202M16—282no—
203C2×C8—24no—
204C2×C8—282no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—282no—
208C22⋊C4—288no—
209C22⋊C4—564no—
210C42—22no—
211C42—22no—
212C42—281no—
213C42—281no—
214C42#115712yes—
215C42C2×C22⋊D712yes—
216C2×C7—21no—
217C2×C7—21no—
218C2×C7#188811yescharacteristic
219C2×C7#189311yescharacteristic
220C2×C7#191411yescharacteristic
221D7—162no—
222C23—284no—
223C23—284no—
224C23—282no—
225C23—282no—
226C23—282no—
227C23C2×(C2×D7)⋊C411yescharacteristic
228Q8—24no—
229Q8—24no—
230D4—564no—
231D4—562no—
232D4—562no—
233D4—562no—
234D4—562no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—282no—
242C2×C4—282no—
243C2×C4—282no—
244C2×C4#639514yes—
245C2×C4C2×#113211yescharacteristic
246C2×C4C2×D5611yescharacteristic
247C2×C4#646712yes—
248C2×C4C4×C22⋊D712yes—
249C2×C4#649211yescharacteristic
250C2×C4#649312yes—
251C2×C4#651611yescharacteristic
252C8—44no—
253C8—562no—
254C7#939811yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—564no—
258C22—564no—
259C22—562no—
260C22—562no—
261C22—562no—
262C22C2×#639511yescharacteristic
263C22#8988511yescharacteristic
264C22#8998511yescharacteristic
265C4—21no—
266C4—24no—
267C4—22no—
268C4—42no—
269C4—562no—
270C4#8988612yes—
271C4#8988712yes—
272C4#9001712yes—
273C2—21no—
274C2—21no—
275C2—1122no—
276C2#1172941511yescharacteristic
277C2#1172972711yescharacteristic
278C2#1172978411yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2982C2
socleC22×C7#386
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenter#2982D7
hypocenterC7#9398
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
3. #11729391C2semidirect
5. #11725552C2semidirect
49. #2982D7semidirect
254. C7#9398semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117399212
#117293912
#117285282
#117255522
#93987