Finite Group Databasebeta

← #13254457

Subgroups — #13254457

Order 1792.

502 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2817C211yesmaximal, characteristic
2#11740172C212yesmaximal
3#11729443C212yesmaximal
4#11727288C211yesmaximal, characteristic
5#11725435C211yesmaximal, characteristic
6C2×C7×C22⋊Q8C2212yes—
7C2×#6489C2211yescharacteristic
8C7×#400—22no—
9C7×#397—22no—
10C7×#395—22no—
11C7×#381C2211yescharacteristic
12#89997—22no—
13#89983C2211yescharacteristic
14#89979C2212yes—
15#89817—24no—
16#89816—22no—
17#89815—22no—
18#89736—22no—
19#89333—24no—
20#89322—22no—
21#9985—71nomaximal, Sylow
22C22×C7×Q8D412yes—
23C23×C4×C7C2311yescharacteristic
24C7×C22⋊Q8—22no—
25C7×C22⋊Q8—22no—
26C7×C22⋊Q8—22no—
27C7×C22⋊Q8—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4D412yes—
33C2×C22⋊Dic7—21no—
34C2×C22⋊Dic7—21no—
35#6489—22no—
36#6489—22no—
37C2×C4⋊Dic7D412yes—
38C2×C4×Dic7—21no—
39C2×#1136—22no—
40C7×#95—24no—
41C7×#95—22no—
42C7×#95—22no—
43#6409—24no—
44#6409—24no—
45C7⋊#95—22no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)D414yes—
48#6382—24no—
49#6381—24no—
50#3095—72no—
51#2817D711yescharacteristic
52#1820—71no—
53#1511—72no—
54#1306—71no—
55C24×C7C2×D411yescharacteristic
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—44no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—42no—
63C22×C4×C7C2×D412yes—
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D412yes—
66C22×C4×C7#4111yescharacteristic
67C22×Dic7—22no—
68C22×Dic7—41no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—82no—
74C7×C22⋊C4—42no—
75C7×C22⋊C4—42no—
76C7×C22⋊C4—42no—
77C22⋊Dic7—41no—
78C22⋊Dic7—41no—
79C22⋊Dic7—42no—
80C4⋊Dic7—22no—
81C4⋊Dic7—22no—
82C4×Dic7—42no—
83#1136—44no—
84C2×C7⋊C8—24no—
85C2×C22⋊Q8C2×D712yes—
86C2×#117—71no—
87#485—144no—
88#484—142no—
89#483—142no—
90#428—142no—
91#419—142no—
92#407—72no—
93#400—22no—
94#397—22no—
95#395—22no—
96#381—71no—
97#381C2×D711yescharacteristic
98#342—142no—
99#328—144no—
100C23×C7—22no—
101C23×C7—42no—
102C23×C7#12011yescharacteristic
103C23×C7#12111yescharacteristic
104C23×C7#12711yescharacteristic
105C7×Q8—44no—
106C7×Q8—82no—
107C2×C4×C7—24no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—44no—
112C2×C4×C7—44no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—44no—
117C2×C4×C7#12012yes—
118C2×C4×C7#12112yes—
119C2×Dic7—42no—
120C2×Dic7—82no—
121C2×Dic7—81no—
122C2×Dic7—82no—
123C7⋊C8—44no—
124C22×Q8C22⋊D712yes—
125C23×C4C22×D711yescharacteristic
126C2×M16—142no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131#117—142no—
132#117—142no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4C22⋊D712yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—141no—
139C2×C22⋊C4—141no—
140C2×C22⋊C4C22⋊D712yes—
141C2×C42—141no—
142#107—144no—
143#106—144no—
144#103—144no—
145#103—144no—
146C22⋊C8—74no—
147C22⋊C8—142no—
148#95—24no—
149#95—22no—
150#95—22no—
151#95—142no—
152C22×C7—22no—
153C22×C7—22no—
154C22×C7—22no—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7—82no—
158C22×C7#39211yescharacteristic
159C22×C7#45812yes—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—82no—
163C4×C7—82no—
164C4×C7—82no—
165Dic7—82no—
166Dic7—161no—
167C24C2×C22⋊D711yescharacteristic
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—44no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—142no—
176C22×C4—281no—
177C22×C4D4×D711yescharacteristic
178C22×C4D4×D712yes—
179C22×C4#115911yescharacteristic
180C22×C4C2×C22⋊D712yes—
181M16—284no—
182C2×C8—144no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—142no—
189C4⋊C4—142no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—281no—
194C22⋊C4—281no—
195C22⋊C4—282no—
196C42—282no—
197C2×C7—22no—
198C2×C7—41no—
199C2×C7—41no—
200C2×C7#200912yes—
201C2×C7#202211yescharacteristic
202C23—22no—
203C23—42no—
204C23#650111yescharacteristic
205C23#650311yescharacteristic
206C23#651611yescharacteristic
207Q8—44no—
208Q8—82no—
209C2×C4—24no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—282no—
220C2×C4—562no—
221C2×C4—561no—
222C2×C4—562no—
223C2×C4#650012yes—
224C2×C4#650212yes—
225C8—284no—
226C7#998511yescharacteristic, complemented, Sylow
227C22—22no—
228C22—21no—
229C22—21no—
230C22—22no—
231C22—22no—
232C22—82no—
233C22#8999511yescharacteristic
234C22#9001512yes—
235C4—22no—
236C4—22no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4—562no—
241C4—1121no—
242C2—22no—
243C2—41no—
244C2—41no—
245C2#1172960311yescharacteristic
246C2#1172962312yes—
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2817C2
socleC22×C7#392
prime radicalC24×C7C2×D4
hypercenter#2817D7
hypocenterC7#9985
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
226. C7#9985semidirect

Maximal subgroups

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

subgroupindex
C7×#28172
#117401722
#117294432
#117272882
#117254352
#99857