Finite Group Databasebeta

← #13251226

Subgroups — #13251226

Order 1792.

554 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2983C211yesmaximal, characteristic, complemented
2#11739921C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic, complemented
4#11728528C212yesmaximal
5#11725552C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6510—21no—
8C2×#6437C2211yescharacteristic
9C7×#446—24no—
10C7×#432—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#9406—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—44no—
32C2×(C2×D7)⋊C4—24no—
33C2×(C2×D7)⋊C4—24no—
34C2×C4⋊Dic7—21no—
35C2×C4×Dic7—21no—
36C2×#1136—22no—
37#6437—22no—
38#6437—22no—
39C7×C4⋊C8—24no—
40C7×C4⋊C8D414yes—
41C7×C22⋊C8—22no—
42C7×#97—24no—
43C4×C7×C8—22no—
44#6410—42no—
45C7⋊#95C2×C412yes—
46C4⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—22no—
48#2983D711yescharacteristic, complemented
49#2548—71no—
50#1801—71no—
51#1545—72no—
52#1303—72no—
53C23×D7—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—21no—
56C2×C7×Q8—22no—
57C2×C7×Q8—21no—
58C22×C4×C7C22⋊C412yes—
59C22×C4×C7C22⋊C412yes—
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×Dic7—22no—
64C2×D28—24no—
65C2×D28—44no—
66C2×D28—41no—
67C2×D28—42no—
68C2×D28—41no—
69C2×D28—42no—
70C2×D28—41no—
71C2×D28—41no—
72C2×D28—82no—
73C2×C7×C8—24no—
74C7×C4⋊C4—24no—
75C7×C4⋊C4—22no—
76C42×C7—22no—
77C42×C7—22no—
78C42×C7C2×D412yes—
79C42×C7#4112yes—
80(C2×D7)⋊C4—48no—
81(C2×D7)⋊C4—84no—
82C4⋊Dic7—42no—
83C4×Dic7—42no—
84#1136—42no—
85#1136—42no—
86C2×C7⋊C8—42no—
87C2×#127—71no—
88C2×#124—141no—
89C2×C4×Q8C2×D711yescharacteristic
90#446—24no—
91#432—22no—
92#431C2×D712yes—
93#423—72no—
94C2×#101—141no—
95#394—74no—
96#390—141no—
97#390—142no—
98C4×C4⋊C4—71no—
99#343—141no—
100#331—144no—
101C23×C7C2×C22⋊C411yescharacteristic
102C22×D7—44no—
103C22×D7—44no—
104C22×D7—42no—
105C22×D7—42no—
106C22×D7—42no—
107C7×Q8—42no—
108C7×Q8—42no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7C4×D412yes—
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7#12112yes—
119C2×C4×C7#12311yescharacteristic
120C2×C4×C7AGL(1,ℤ/8)12yes—
121C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C7#13711yescharacteristic
123C2×Dic7—42no—
124C2×Dic7—42no—
125C2×Dic7—42no—
126D28—84no—
127D28—82no—
128D28—82no—
129D28—82no—
130D28—82no—
131C7×C8—44no—
132C7⋊C8—82no—
133C22×Q8D2811yescharacteristic
134C22×D4—72no—
135C22×D4—141no—
136C22×D4—141no—
137C2×M16—142no—
138#127—142no—
139#127—142no—
140#124—284no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—141no—
144C2×C4⋊C4C4×D712yes—
145C2×C4⋊C4D2811yescharacteristic
146C2×C22⋊C4—144no—
147C2×C22⋊C4—144no—
148C2×C42—141no—
149C2×C42C22⋊D712yes—
150C2×C42C22×D711yescharacteristic
151C4⋊C8—24no—
152C4⋊C8—142no—
153C4⋊C8—142no—
154C4⋊C8C22⋊D714yes—
155#101—282no—
156C22⋊C8—22no—
157#97—24no—
158C4×C8—22no—
159#95—72no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7#38611yescharacteristic
163C22×C7#41711yescharacteristic
164C22×C7#41811yescharacteristic
165C2×D7—84no—
166C2×D7—84no—
167C2×D7—82no—
168C2×D7—82no—
169C2×D7—82no—
170C4×C7—21no—
171C4×C7—22no—
172C4×C7—42no—
173C4×C7—42no—
174C4×C7#45412yes—
175C4×C7#46812yes—
176C4×C7#46912yes—
177Dic7—82no—
178C24—142no—
179C2×Q8—21no—
180C2×Q8—22no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×D4—144no—
184C2×D4—284no—
185C2×D4—282no—
186C2×D4—281no—
187C2×D4—281no—
188C2×D4—281no—
189C2×D4—282no—
190C2×D4—281no—
191C2×D4—562no—
192C22×C4—142no—
193C22×C4(C2×D7)⋊C412yes—
194C22×C4(C2×D7)⋊C412yes—
195C22×C4C2×C4×D711yescharacteristic
196C22×C4C2×D2811yescharacteristic
197C22×C4C2×C22⋊D711yescharacteristic
198M16—282no—
199M16—282no—
200C2×C8—24no—
201C2×C8—282no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—282no—
205C22⋊C4—288no—
206C22⋊C4—564no—
207C42—22no—
208C42—22no—
209C42—282no—
210C42#115712yes—
211C42C2×C22⋊D712yes—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#188811yescharacteristic
215C2×C7#189511yescharacteristic
216C2×C7#191411yescharacteristic
217D7—162no—
218C23—284no—
219C23—284no—
220C23—282no—
221C23—282no—
222C23—282no—
223C23C2×(C2×D7)⋊C411yescharacteristic
224Q8—42no—
225Q8—42no—
226D4—564no—
227D4—562no—
228D4—562no—
229D4—562no—
230D4—562no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—282no—
239C2×C4—282no—
240C2×C4—282no—
241C2×C4#647112yes—
242C2×C4#647111yescharacteristic
243C2×C4#647211yescharacteristic
244C2×C4C4×C22⋊D712yes—
245C2×C4#649211yescharacteristic
246C2×C4#649312yes—
247C2×C4#651611yescharacteristic
248C8—44no—
249C8—562no—
250C7#940611yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—564no—
254C22—564no—
255C22—562no—
256C22—562no—
257C22—562no—
258C22#8990311yescharacteristic
259C22#8990511yescharacteristic
260C22#8998511yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—562no—
266C4#8990612yes—
267C4#8990712yes—
268C4#9001712yes—
269C2—21no—
270C2—21no—
271C2—1122no—
272C2#1172941511yescharacteristic
273C2#1172974411yescharacteristic
274C2#1172978411yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2983C2
socleC22×C7#386
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenter#2983D7
hypocenterC7#9406
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
1. C7×#2983C2semidirect
3. #11729391C2semidirect
5. #11725552C2semidirect
48. #2983D7semidirect
250. C7#9406semidirect

Maximal subgroups

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

subgroupindex
C7×#29832
#117399212
#117293912
#117285282
#117255522
#94067