Finite Group Databasebeta

← #13251161

Subgroups — #13251161

Order 1792.

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

normal subgroups only

This group has 558 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×#2972C211yesmaximal, characteristic
2#11739827C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11728532C212yesmaximal, complemented
5#11725552C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6499—21no—
8C2×#6432C2211yescharacteristic
9C7×C8×D4—24no—
10C7×#431C2212yes—
11C2×C7×C22⋊C8—22no—
12C2×#6407—21no—
13#89984—22no—
14#89746—21no—
15#89725C414yes—
16C4×C4⋊Dic7C2211yescharacteristic
17#89696C2212yes—
18#89325—21no—
19#89281—24no—
20#9393—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7D412yes—
23C22×Dic14—21no—
24C22×C7×C8—24no—
25C4×C7×D4—22no—
26C4×C7×D4—22no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C22⋊C4C2×C412yes—
29C2×C42×C7C2311yescharacteristic
30C2×C22⋊Dic7—24no—
31#6499—42no—
32#6499—42no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7C2×C412yes—
36C2×C4×Dic7—21no—
37C2×#1136—22no—
38#6432—22no—
39#6432—22no—
40C7×C4⋊C8D414yes—
41C7×C22⋊C8—24no—
42C7×C22⋊C8—22no—
43C4×C7×C8—22no—
44#6407—42no—
45C7⋊#95—24no—
46C7⋊#95C2×C412yes—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49#2972D711yescharacteristic
50#2547—71no—
51#1800—71no—
52#1546—72no—
53#1303—72no—
54C24×C7C22⋊C412yes—
55C2×C7×D4—22no—
56C2×C7×D4—22no—
57C2×C7×D4D812yes—
58C2×C7×D4SD1612yes—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—44no—
62C22×C4×C7C22⋊C412yes—
63C22×C4×C7C22×C411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×Dic7—22no—
67C22×Dic7—22no—
68C2×Dic14—42no—
69C2×Dic14—41no—
70C2×Dic14—41no—
71C2×Dic14—42no—
72C2×C7×C8—24no—
73C2×C7×C8—84no—
74C7×C4⋊C4—22no—
75C7×C22⋊C4—24no—
76C42×C7C2×D412yes—
77C42×C7#4112yes—
78C22⋊Dic7—48no—
79C4⋊Dic7—24no—
80C4⋊Dic7—42no—
81C4⋊Dic7—41no—
82C4⋊Dic7—41no—
83C4×Dic7—41no—
84C4×Dic7—41no—
85#1136—42no—
86#1136—42no—
87C2×C7⋊C8—42no—
88C2×C4⋊Q8—71no—
89C2×#124—141no—
90C2×C4×D4C2×D711yescharacteristic
91C8×D4—24no—
92#431C2×D712yes—
93#423—72no—
94C2×#100—141no—
95C2×C22⋊C8—22no—
96#397—74no—
97#391—141no—
98#380—142no—
99C4×C4⋊C4—71no—
100#343—141no—
101#332—144no—
102C23×C7—22no—
103C23×C7—22no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7#10214yes—
107C23×C7C2×C22⋊C411yescharacteristic
108C7×D4—24no—
109C7×D4—24no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—41no—
113C2×C4×C7—44no—
114C2×C4×C7—42no—
115C2×C4×C7—82no—
116C2×C4×C7C4×D412yes—
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7#12112yes—
119C2×C4×C7#12311yescharacteristic
120C2×C4×C7C2×D811yescharacteristic
121C2×C4×C7C2×SD1611yescharacteristic
122C2×C4×C7#13512yes—
123C2×Dic7—42no—
124C2×Dic7—42no—
125C2×Dic7—42no—
126C2×Dic7—42no—
127C2×Dic7—42no—
128C2×Dic7—42no—
129Dic14—82no—
130Dic14—82no—
131C7×C8—44no—
132C7⋊C8—82no—
133C22×Q8—141no—
134C22×D4D2811yescharacteristic
135C23×C4C22⋊D712yes—
136C2×M16—142no—
137C22×C8—24no—
138C4⋊Q8—142no—
139C4⋊Q8—142no—
140#124—282no—
141#124—282no—
142C4×D4—22no—
143C4×D4—22no—
144C2×C4⋊C4—72no—
145C2×C4⋊C4—141no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4D2811yescharacteristic
148C2×C22⋊C4—144no—
149C2×C22⋊C4C4×D712yes—
150C2×C42—141no—
151C2×C42C22×D711yescharacteristic
152C4⋊C8—142no—
153C4⋊C8—142no—
154C4⋊C8C22⋊D714yes—
155#100—282no—
156C22⋊C8—24no—
157C22⋊C8—22no—
158C4×C8—22no—
159#95—72no—
160#95—144no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7—24no—
164C22×C7—44no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7—82no—
168C22×C7#38611yescharacteristic
169C22×C7C2×#10211yescharacteristic
170C22×C7#41611yescharacteristic
171C4×C7—21no—
172C4×C7—22no—
173C4×C7—42no—
174C4×C7#45512yes—
175C4×C7#46512yes—
176C4×C7#46612yes—
177Dic7—82no—
178Dic7—82no—
179C24(C2×D7)⋊C412yes—
180C2×Q8—281no—
181C2×Q8—281no—
182C2×Q8—282no—
183C2×Q8—282no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4#113212yes—
187C2×D4D5612yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—142no—
192C22×C4—142no—
193C22×C4(C2×D7)⋊C412yes—
194C22×C4C2×C4×D711yescharacteristic
195C22×C4C2×D2811yescharacteristic
196C22×C4C2×C22⋊D711yescharacteristic
197M16—282no—
198M16—282no—
199C2×C8—24no—
200C2×C8—84no—
201C2×C8—282no—
202C4⋊C4—22no—
203C4⋊C4—144no—
204C4⋊C4—282no—
205C4⋊C4—281no—
206C4⋊C4—281no—
207C22⋊C4—24no—
208C22⋊C4—288no—
209C42—281no—
210C42—281no—
211C42#115712yes—
212C42C2×C22⋊D712yes—
213C2×C7—21no—
214C2×C7—21no—
215C2×C7—44no—
216C2×C7#188511yescharacteristic
217C2×C7#189311yescharacteristic
218C2×C7#191511yescharacteristic
219C23—22no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23#639514yes—
224C23C2×(C2×D7)⋊C411yescharacteristic
225Q8—562no—
226Q8—562no—
227D4—24no—
228D4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—282no—
236C2×C4—282no—
237C2×C4—282no—
238C2×C4—282no—
239C2×C4—282no—
240C2×C4—282no—
241C2×C4C2×#113211yescharacteristic
242C2×C4C2×D5611yescharacteristic
243C2×C4#646712yes—
244C2×C4C4×C22⋊D712yes—
245C2×C4#649211yescharacteristic
246C2×C4#649312yes—
247C2×C4#651611yescharacteristic
248C8—44no—
249C8—562no—
250C7#939311yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—24no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22C2×#639511yescharacteristic
259C22#8988511yescharacteristic
260C22#8998511yescharacteristic
261C4—22no—
262C4—21no—
263C4—42no—
264C4—562no—
265C4—562no—
266C4#8988612yes—
267C4#8988712yes—
268C4#9001412yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#1172940311yescharacteristic
273C2#1172972711yescharacteristic
274C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2972C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2972D7
hypocenterC7#9393
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #11728532C2semidirect
5. #11725552C2semidirect
250. C7#9393semidirect

Maximal subgroups

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

subgroupindex
C7×#29722
#117398272
#117293712
#117285322
#117255522
#93937