Finite Group Databasebeta

← #13243065

Subgroups — #13243065

Order 1792.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2724C211yesmaximal, characteristic
2#11727410C211yesmaximal, characteristic
3#11727366C212yesmaximal, complemented
4#11725445C211yesmaximal, characteristic
5#11725444C212yesmaximal, complemented
6C2×C7×#123—21no—
7C7×#398—21no—
8C7×#397—22no—
9C7×#390C2211yescharacteristic
10C7×#389C2212yes—
11C7×#385—22no—
12C2×#6406—21no—
13C2×#6382—21no—
14#89707C2211yescharacteristic
15#89324—22no—
16#89324—22no—
17#89319C2212yes—
18#89319C2211yescharacteristic
19#7824—71nomaximal, Sylow
20C22×C7×D4D411yescharacteristic
21C23×C4×C7—21no—
22C7×#123—42no—
23C7×#123—42no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4D412yes—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C4⋊C4D412yes—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—21no—
31C2×C7×C22⋊C4—22no—
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—22no—
35C22×C7⋊C8—21no—
36C22×C7⋊C8—22no—
37C7×#95—22no—
38C7×#95—21no—
39C7×#95—22no—
40#6406—42no—
41#6382—42no—
42#2724D711yescharacteristic
43#2100—72no—
44#2094—71no—
45#1271—72no—
46#1270—71no—
47C24×C7—21no—
48C2×C7×D4—21no—
49C2×C7×D4—41no—
50C2×C7×D4—41no—
51C2×C7×D4SD1612yes—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7—42no—
56C22×C4×C7—81no—
57C22×C4×C7—81no—
58C22×C4×C7C2×D411yescharacteristic
59C22×C4×C7C2×D412yes—
60C22×Dic7—41no—
61C7×C4⋊C4—22no—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—42no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—42no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—81no—
68C7×C22⋊C4—42no—
69C7×C22⋊C4—44no—
70C7×C22⋊C4—42no—
71C7×C22⋊C4—42no—
72C42×C7—41no—
73C4⋊Dic7—41no—
74C4⋊Dic7—44no—
75C4⋊Dic7—41no—
76C2×C7⋊C8—22no—
77C2×C7⋊C8—41no—
78C2×C7⋊C8—41no—
79C2×C7⋊C8—42no—
80C2×C7⋊C8—42no—
81C2×C7⋊C8—42no—
82C2×#123—21no—
83C2×#106—141no—
84C2×#102—141no—
85#398—21no—
86#397—22no—
87#390C2×D711yescharacteristic
88#389C2×D712yes—
89#385—22no—
90#384—71no—
91#340—142no—
92#340—142no—
93#336—71no—
94#336—72no—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—41no—
98C23×C7—42no—
99C23×C7—41no—
100C23×C7#12011yescharacteristic
101C7×D4—42no—
102C7×D4—81no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—84no—
116C2×C4×C7C2×SD1611yescharacteristic
117C2×C4×C7#13512yes—
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C7#13712yes—
120C2×Dic7—81no—
121C2×Dic7—82no—
122C7⋊C8—42no—
123C7⋊C8—82no—
124C22×D4C22⋊D711yescharacteristic
125C23×C4—21no—
126C22×C8—141no—
127C22×C8—142no—
128#123—42no—
129#123—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—142no—
133C2×C4⋊C4—141no—
134C2×C4⋊C4C22⋊D712yes—
135C2×C4⋊C4C22⋊D712yes—
136C2×C4⋊C4C22⋊D711yescharacteristic
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C42C22×D711yescharacteristic
141#106—282no—
142#102—282no—
143#95—22no—
144#95—21no—
145#95—22no—
146C22×C7—42no—
147C22×C7—44no—
148C22×C7—82no—
149C22×C7—81no—
150C22×C7—81no—
151C22×C7#45011yescharacteristic
152C22×C7#45212yes—
153C22×C7#45311yescharacteristic
154C22×C7#45412yes—
155C22×C7#45811yescharacteristic
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—22no—
159C4×C7—22no—
160C4×C7—82no—
161C4×C7—81no—
162C4×C7—82no—
163Dic7—161no—
164C24—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4#114212yes—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—281no—
176C22×C4C2×C22⋊D711yescharacteristic
177C22×C4C2×C22⋊D712yes—
178C2×C8—142no—
179C2×C8—281no—
180C2×C8—281no—
181C2×C8—282no—
182C2×C8—282no—
183C2×C8—282no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—281no—
192C4⋊C4—281no—
193C4⋊C4—284no—
194C22⋊C4—42no—
195C22⋊C4—44no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C42—41no—
199C2×C7—82no—
200C2×C7#161311yescharacteristic
201C2×C7#161512yes—
202C2×C7#163011yescharacteristic
203C2×C7#163512yes—
204C2×C7#165711yescharacteristic
205C23—22no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23#651611yescharacteristic
211D4—42no—
212D4—81no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—84no—
226C2×C4—561no—
227C2×C4—562no—
228C2×C4#649511yescharacteristic
229C2×C4C2×#114211yescharacteristic
230C2×C4#651312yes—
231C2×C4#651412yes—
232C8—282no—
233C8—562no—
234C7#782411yescharacteristic, complemented, Sylow
235C22—44no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—82no—
240C22#8999011yescharacteristic
241C22#9001112yes—
242C22#9001311yescharacteristic
243C22#9001511yescharacteristic
244C22#9001712yes—
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—1121no—
253C2—82no—
254C2#1172959811yescharacteristic
255C2#1172964112yes—
256C2#1172966512yes—
257C2#1172969811yescharacteristic
258C2#1172970911yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2724C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenter#2724D7
hypocenterC7#7824
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
3. #11727366C2semidirect
5. #11725444C2semidirect
234. C7#7824semidirect

Maximal subgroups

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

subgroupindex
C7×#27242
#117274102
#117273662
#117254452
#117254442
#78247