Finite Group Databasebeta

← #13254789

Subgroups — #13254789

Order 1792.

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

normal subgroups only

This group has 476 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
2#11740813C212yesmaximal
3#11729578C212yesmaximal
4#11727592C211yesmaximal, characteristic
5#11725466C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C4×Dic14C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#427—22no—
11C2×C7×#97C2211yescharacteristic
12#89863—22no—
13#89862—22no—
14#89822—22no—
15#89816—22no—
16C4×#1144—24no—
17#89697C2212yes—
18C2×C4⋊(C7⋊C8)C2211yescharacteristic
19#89345—22no—
20Q8⋊(C7⋊C8)—24no—
21#10037—71nomaximal, Sylow
22C22×Dic14D411yescharacteristic
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×#1144—24no—
34C2×C4⋊Dic7D411yescharacteristic
35C2×Dic7⋊C4—21no—
36C2×C4×Dic7—21no—
37C22×C7⋊C8—21no—
38C4×Dic14—22no—
39C4×Dic14—22no—
40C7×#107—22no—
41C7×#107—22no—
42C7×#106—24no—
43C7×#97—22no—
44#6409—24no—
45C22⋊(C7⋊C8)—22no—
46#6384—24no—
47#6381—24no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4×C7⋊C8D414yes—
51#3162D711yescharacteristic
52#3104—72no—
53#1985—71no—
54#1481—72no—
55#1275—71no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7#4111yescharacteristic
61C22×Dic7—21no—
62C22×Dic7—21no—
63C2×Dic14—22no—
64C2×Dic14—42no—
65C2×C7×C8—22no—
66C2×C7×C8—22no—
67C2×C7×C8Q1614yescomplemented
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C22⋊C4—42no—
75C42×C7—42no—
76C42×C7C2×D412yes—
77C42×C7C2×D412yes—
78#1144—48no—
79C4⋊Dic7—22no—
80Dic7⋊C4—42no—
81C4×Dic7—41no—
82C4×Dic7—41no—
83C2×C7⋊C8—24no—
84C2×C7⋊C8—41no—
85C2×C7⋊C8—42no—
86#533C2×D712yes—
87C2×C4×Q8—71no—
88#501—22no—
89#501—22no—
90#484—142no—
91#477—142no—
92#462—142no—
93#451—142no—
94C4×Q16—144no—
95#431—72no—
96#427—22no—
97C2×C4⋊C8—71no—
98C2×#97C2×D711yescharacteristic
99#336—142no—
100Q8⋊C8—144no—
101C23×C7#12111yescharacteristic
102C7×Q8—44no—
103C7×Q8—44no—
104C2×C4×C7—41no—
105C2×C4×C7—41no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7#12012yes—
109C2×C4×C7#12011yescharacteristic
110C2×C4×C7#12112yes—
111C2×C4×C7#12711yescharacteristic
112C2×C4×C7C2×Q1612yes—
113C2×C4×C7#13712yes—
114C2×Dic7—22no—
115C2×Dic7—41no—
116C2×Dic7—41no—
117C2×Dic7—42no—
118C2×Dic7—41no—
119Dic14—24no—
120Dic14—42no—
121C7×C8—24no—
122C7⋊C8—44no—
123C7⋊C8—81no—
124C22×Q8—71no—
125C2×Q16—144no—
126C22×C8—141no—
127C22×C8C22⋊D712yes—
128C4⋊Q8C22⋊D712yes—
129C4⋊Q8C22⋊D712yes—
130#125—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—142no—
136C4×Q8—142no—
137#117—22no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—141no—
140C2×C42—141no—
141C2×C42C22×D711yescharacteristic
142#107—22no—
143#107—22no—
144#107—144no—
145#106—24no—
146C4⋊C8—142no—
147C4⋊C8—142no—
148#103—144no—
149#103—144no—
150C22⋊C8—142no—
151#97—22no—
152C4×C8—74no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39211yescharacteristic
156C22×C7#46212yes—
157C4×C7—22no—
158C4×C7—22no—
159C4×C7—82no—
160C4×C7—82no—
161C4×C7#45112yes—
162C4×C7#49411yescharacteristic
163C4×C7#49711yescharacteristic
164Dic7—42no—
165Dic7—81no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—142no—
169C2×Q8—282no—
170C22×C4—141no—
171C22×C4—141no—
172C22×C4D4×D711yescharacteristic
173C22×C4#115911yescharacteristic
174C22×C4C2×C22⋊D711yescharacteristic
175Q16—288no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—144no—
179C2×C8—281no—
180C2×C8—282no—
181C2×C8#114414yescomplemented
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—142no—
189C4⋊C4—282no—
190C22⋊C4—42no—
191C42—42no—
192C42—281no—
193C42—281no—
194C42D4×D712yes—
195C42C2×C22⋊D712yes—
196C2×C7—22no—
197C2×C7#201011yescharacteristic
198C2×C7#201611yescharacteristic
199C2×C7#201711yescharacteristic
200C23#650111yescharacteristic
201Q8—44no—
202Q8—44no—
203Q8—144no—
204Q8—282no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—142no—
210C2×C4—281no—
211C2×C4—281no—
212C2×C4—282no—
213C2×C4—281no—
214C2×C4#650012yes—
215C2×C4#650212yes—
216C2×C4#650311yescharacteristic
217C2×C4#650512yes—
218C2×C4C2×#114412yes—
219C2×C4#651611yescharacteristic
220C8—24no—
221C8—284no—
222C8—561no—
223C7#1003711yescharacteristic, complemented, Sylow
224C22—21no—
225C22—21no—
226C22#8986212yes—
227C22#8999511yescharacteristic
228C4—22no—
229C4—22no—
230C4—82no—
231C4—82no—
232C4—282no—
233C4—561no—
234C4#8985811yescharacteristic
235C4#8986411yescharacteristic
236C4#9000012yes—
237C2—22no—
238C2#1172954711yescharacteristic
239C2#1172987211yescharacteristic
240C2#1172987511yescharacteristic
subgroupG/H
centerC2×C4#6503
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3162C2
socleC22×C7#392
prime radicalC23×C7#121
hypercenter#3162D7
hypocenterC7#10037
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
67. C2×C7×C8Q16semidirect
181. C2×C8#1144semidirect
223. C7#10037semidirect

Maximal subgroups

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

subgroupindex
C7×#31622
#117408132
#117295782
#117275922
#117254662
#100377