Finite Group Databasebeta

← #13256825

Subgroups — #13256825

Order 1792.

416 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 125 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×#2571C211yesmaximal, characteristic
2#11727333C211yesmaximal, characteristic
3#11727243C212yesmaximal
4#11725440C211yesmaximal, characteristic
5#11725425C212yesmaximal
6C2×#6432C2211yescharacteristic
7C7×#400—22no—
8C7×#389—22no—
9C7×#384C2211yescharacteristic
10C7×#382—22no—
11C4×C7×C4⋊C4C2212yes—
12C2×#6384—22no—
13C2×C4⋊(C7⋊C8)C2212yes—
14C2×C4×C7⋊C8C2211yescharacteristic
15#89324—22no—
16#89324—22no—
17#10352—71nomaximal, Sylow
18C22×Dic14—21no—
19C2×C7×C4⋊C4—22no—
20C2×C7×C4⋊C4—22no—
21C2×C7×C4⋊C4D412yes—
22C2×C7×C4⋊C4D412yes—
23C2×C7×C4⋊C4D412yes—
24C2×C42×C7—22no—
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7—22no—
27C22×C7⋊C8—21no—
28C22×C7⋊C8—22no—
29#6432—22no—
30#6432—21no—
31#6432Q812yes—
32C7×#95—22no—
33C7×#95—24no—
34C7×#95—22no—
35#6384—44no—
36C4⋊(C7⋊C8)—22no—
37C4⋊(C7⋊C8)—22no—
38C4×C7⋊C8—22no—
39#2571D711yescharacteristic
40#1987—72no—
41#1956—71no—
42#1296—71no—
43#1279—72no—
44C22×C4×C7—22no—
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7C2×D411yescharacteristic
48C22×C4×C7C2×D412yes—
49C22×Dic7—41no—
50C2×Dic14—22no—
51C2×Dic14—42no—
52C2×Dic14—42no—
53C2×Dic14—41no—
54C7×C4⋊C4—22no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—42no—
60C7×C4⋊C4—42no—
61C7×C4⋊C4—82no—
62C42×C7—42no—
63C42×C7—42no—
64C42×C7C2×Q811yescharacteristic
65C42×C7#4112yes—
66C42×C7#4111yescharacteristic
67C4⋊Dic7—24no—
68C4⋊Dic7—42no—
69C2×C7⋊C8—22no—
70C2×C7⋊C8—41no—
71C2×C7⋊C8—41no—
72C2×C7⋊C8—42no—
73C2×C7⋊C8—42no—
74C2×C7⋊C8—42no—
75C2×C4⋊Q8—71no—
76C2×C4⋊C8—72no—
77C2×#103—142no—
78C2×C4×C8—71no—
79#400—22no—
80#389—22no—
81#384C2×D711yescharacteristic
82#382—22no—
83C4×C4⋊C4C2×D712yes—
84#340—142no—
85#340—142no—
86C23×C7#12011yescharacteristic
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—42no—
90C2×C4×C7—42no—
91C2×C4×C7—42no—
92C2×C4×C7—42no—
93C2×C4×C7—44no—
94C2×C4×C7—42no—
95C2×C4×C7—42no—
96C2×C4×C7—42no—
97C2×C4×C7C22⋊Q811yescharacteristic
98C2×C4×C7C22⋊Q812yes—
99C2×C4×C7#12411yescharacteristic
100C2×C4×C7#12412yes—
101C2×C4×C7#13512yes—
102C2×C4×C7#13512yes—
103C2×C4×C7#13712yes—
104C2×C4×C7#13712yes—
105C2×Dic7—42no—
106C2×Dic7—81no—
107C2×Dic7—81no—
108Dic14—44no—
109Dic14—82no—
110C7⋊C8—42no—
111C7⋊C8—82no—
112C22×Q8—141no—
113C22×C8—141no—
114C22×C8—142no—
115C4⋊Q8—72no—
116C4⋊Q8—142no—
117C4⋊Q8—141no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—142no—
121C2×C4⋊C4C22⋊D712yes—
122C2×C4⋊C4C22⋊D712yes—
123C2×C4⋊C4C22⋊D712yes—
124C2×C42—22no—
125C2×C42C22×D711yescharacteristic
126C4⋊C8—142no—
127C4⋊C8—142no—
128#103—284no—
129C4×C8—142no—
130#95—22no—
131#95—24no—
132#95—22no—
133C22×C7#39311yescharacteristic
134C22×C7#45212yes—
135C22×C7#45212yes—
136C22×C7#45311yescharacteristic
137C22×C7#45411yescharacteristic
138C4×C7—21no—
139C4×C7—21no—
140C4×C7—82no—
141C4×C7—82no—
142C4×C7—82no—
143C4×C7#47912yes—
144C4×C7#47912yes—
145C4×C7#48812yes—
146C4×C7#49112yes—
147Dic7—82no—
148C2×Q8—142no—
149C2×Q8—282no—
150C2×Q8—282no—
151C2×Q8—281no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—281no—
156C22×C4C2×C22⋊D711yescharacteristic
157C22×C4C2×C22⋊D712yes—
158C2×C8—142no—
159C2×C8—281no—
160C2×C8—281no—
161C2×C8—282no—
162C2×C8—282no—
163C2×C8—282no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—82no—
172C4⋊C4—144no—
173C4⋊C4—282no—
174C42—42no—
175C42—42no—
176C42Q8×D711yescharacteristic
177C42#116112yes—
178C42#116111yescharacteristic
179C2×C7#163011yescharacteristic
180C2×C7#163411yescharacteristic
181C2×C7#163812yes—
182C2×C7#202612yes—
183C2×C7#202711yescharacteristic
184C23#651611yescharacteristic
185Q8—284no—
186Q8—562no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—282no—
198C2×C4—561no—
199C2×C4—561no—
200C2×C4#650911yescharacteristic
201C2×C4#650912yes—
202C2×C4#651011yescharacteristic
203C2×C4#651012yes—
204C2×C4#651312yes—
205C2×C4#651312yes—
206C2×C4#651412yes—
207C2×C4#651412yes—
208C8—282no—
209C8—562no—
210C7#1035211yescharacteristic, complemented, Sylow
211C22#9000311yescharacteristic
212C22#9001112yes—
213C22#9001112yes—
214C22#9001311yescharacteristic
215C22#9001711yescharacteristic
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—562no—
222C4#8983512yes—
223C4#8983512yes—
224C4#8984012yes—
225C4#8984112yes—
226C2#1172949812yes—
227C2#1172965711yescharacteristic
228C2#1172968211yescharacteristic
229C2#1172968612yes—
230C2#1172969811yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2571C2
socle, prime radicalC23×C7#120
hypercenter#2571D7
hypocenterC7#10352
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
210. C7#10352semidirect

Maximal subgroups

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

subgroupindex
C7×#25712
#117273332
#117272432
#117254402
#117254252
#103527