Finite Group Databasebeta

← #13256043

Subgroups — #13256043

Order 1792.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic, complemented
2#11740800C212yesmaximal
3#11729592C212yesmaximal, complemented
4#11727577C211yesmaximal, characteristic
5#11725467C211yesmaximal, characteristic, complemented
6C7×#533C2212yes—
7C2×#6437C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#427—22no—
11C2×C7×#97C2211yescharacteristic
12C2×#6395—22no—
13#89860—22no—
14#89857—22no—
15#89818—24no—
16#89799—24no—
17#89708C2211yescharacteristic
18#89697C2212yes—
19#89333—24no—
20#10220—71nomaximal, Sylow
21C22×D28—21no—
22C22×D28—22no—
23C22×C7×C8D412yes—
24C7×C4⋊Q8D412yes—
25C7×C4⋊Q8D412yes—
26C7×#125—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C4×C7×Q8—22no—
30C4×C7×Q8—22no—
31C7×#117—22no—
32C2×C42×C7C2311yescharacteristic
33C2×#1143—28no—
34C2×C4⋊Dic7—21no—
35#6437—22no—
36#6437—21no—
37#6437D412yes—
38C7×#107—24no—
39C7×#106—22no—
40C7×#106—22no—
41C7×#97—22no—
42C7⋊#95—22no—
43C22⋊(C7⋊C8)D414yes—
44#6395—44no—
45#6383—28no—
46C4⋊(C7⋊C8)—22no—
47C4×C7⋊C8—22no—
48#3208—72no—
49#3162D711yescharacteristic, complemented
50#1961—71no—
51#1518—72no—
52#1281—71no—
53C23×D7—41no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C22×C4×C7C2×D412yes—
57C22×C4×C7C2×D411yescharacteristic
58C22×Dic7—41no—
59C2×D28—24no—
60C2×D28—22no—
61C2×D28—42no—
62C2×D28—42no—
63C2×D28—41no—
64C2×D28—42no—
65C2×D28—84no—
66C2×C7×C8—22no—
67C2×C7×C8—22no—
68C2×C7×C8SD1614yescomplemented
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—22no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C22⋊C4—42no—
76C42×C7—42no—
77C42×C7C2×D411yescharacteristic
78C42×C7C2×D412yes—
79C42×C7#4111yescharacteristic
80#1143—88no—
81C4⋊Dic7—42no—
82C2×C7⋊C8—24no—
83#533C2×D712yes—
84C2×#127—71no—
85#501—22no—
86#501—22no—
87#492—142no—
88#486—142no—
89#460—144no—
90#450—144no—
91#431—72no—
92#427—22no—
93C2×#102—142no—
94C2×#97C2×D711yescharacteristic
95#382—71no—
96#328—144no—
97C23×C7#12711yescharacteristic
98C22×D7—42no—
99C22×D7—84no—
100C22×D7—81no—
101C22×D7—81no—
102C7×Q8—44no—
103C7×Q8—82no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7#12012yes—
108C2×C4×C7#12011yescharacteristic
109C2×C4×C7#12111yescharacteristic
110C2×C4×C7#12112yes—
111C2×C4×C7C2×SD1612yes—
112C2×C4×C7#13712yes—
113C2×Dic7—82no—
114C2×Dic7—81no—
115D28—48no—
116D28—44no—
117D28—84no—
118D28—82no—
119C7×C8—24no—
120C7⋊C8—44no—
121C22×D4—142no—
122C22×D4—141no—
123C2×SD16—148no—
124C22×C8C22⋊D712yes—
125C4⋊Q8C22⋊D712yes—
126C4⋊Q8C22⋊D712yes—
127#127—72no—
128#127—141no—
129#127—142no—
130#125—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135#117—22no—
136C2×C4⋊C4—141no—
137C2×C42C22×D711yescharacteristic
138#107—24no—
139#106—22no—
140#106—22no—
141C4⋊C8—142no—
142#102—148no—
143#102—284no—
144C22⋊C8—74no—
145#97—22no—
146C4×C8—142no—
147#95—142no—
148C22×C7—22no—
149C22×C7#39211yescharacteristic
150C22×C7#49211yescharacteristic
151C22×C7#49711yescharacteristic
152C2×D7—84no—
153C2×D7—82no—
154C2×D7—162no—
155C2×D7—162no—
156C4×C7—22no—
157C4×C7—22no—
158C4×C7—82no—
159C4×C7—82no—
160C4×C7#44812yes—
161C4×C7#46112yes—
162Dic7—161no—
163C24—281no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×D4—144no—
167C2×D4—142no—
168C2×D4—282no—
169C2×D4—282no—
170C2×D4—282no—
171C2×D4—281no—
172C2×D4—564no—
173C22×C4—281no—
174C22×C4D4×D712yes—
175C22×C4C2×C22⋊D711yescharacteristic
176SD16—568no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—144no—
180C2×C8#114314yescomplemented
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—282no—
188C22⋊C4—42no—
189C42—42no—
190C42D4×D711yescharacteristic
191C42#115911yescharacteristic
192C42C2×C22⋊D712yes—
193C2×C7—21no—
194C2×C7—21no—
195C2×C7#200812yes—
196C2×C7#201311yescharacteristic
197D7—162no—
198C23—282no—
199C23—564no—
200C23—561no—
201C23—561no—
202C23#650311yescharacteristic
203Q8—44no—
204Q8—82no—
205D4—288no—
206D4—284no—
207D4—564no—
208D4—562no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—562no—
213C2×C4—561no—
214C2×C4#650012yes—
215C2×C4#650111yescharacteristic
216C2×C4#650212yes—
217C2×C4C2×#114312yes—
218C2×C4#650512yes—
219C2×C4#651611yescharacteristic
220C8—24no—
221C8—284no—
222C7#1022011yescharacteristic, complemented, Sylow
223C22—22no—
224C22—562no—
225C22—564no—
226C22—1122no—
227C22—1122no—
228C22#8985711yescharacteristic
229C22#8985811yescharacteristic
230C22#8999511yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—1121no—
236C4#8985512yes—
237C4#8999912yes—
238C2—21no—
239C2—21no—
240C2—1122no—
241C2#1172954511yescharacteristic
242C2#1172987312yes—
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3162C2
socleC22×C7#392
prime radicalC2×#6437C22
hypercenter#3162D7
hypocenterC7#10220
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
1. C7×#3162C2semidirect
3. #11729592C2semidirect
5. #11725467C2semidirect
49. #3162D7semidirect
68. C2×C7×C8SD16semidirect
180. C2×C8#1143semidirect
222. C7#10220semidirect

Maximal subgroups

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

subgroupindex
C7×#31622
#117408002
#117295922
#117275772
#117254672
#102207