Finite Group Databasebeta

← #13266818

Subgroups — #13266818

Order 1792.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2822C211yesmaximal, characteristic
2#11727467C212yesmaximal
3#11727297C211yesmaximal, characteristic
4#11725456C212yesmaximal
5#11725438C211yesmaximal, characteristic
6C2×C7×#125—21no—
7#90191C2211yescharacteristic
8C7×#400—22no—
9C7×#398—21no—
10C7×#398—22no—
11C7×#398C2212yes—
12C7×#382C2211yescharacteristic
13#89743—22no—
14#89736—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12251—71nomaximal, Sylow
20C7×#125—42no—
21C7×#125—42no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C4⋊C4D412yes—
29C2×C7×C4⋊C4C2311yescharacteristic
30C2×C42×C7D411yescharacteristic
31#6489—21no—
32#6487—22no—
33C2×#1136—21no—
34#6453—41no—
35#6451—41no—
36C7×#95—22no—
37C7×#95—21no—
38C7×#95—22no—
39C7×#95—22no—
40C7×#95D412yes—
41C22⋊(C7⋊C8)—22no—
42#6384—42no—
43#6382—41no—
44#6382—42no—
45#6381—41no—
46#2822D711yescharacteristic
47#2061—71no—
48#1611—72no—
49#1346—72no—
50#1324—71no—
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7C2×D411yescharacteristic
56C22×C4×C7C2×D412yes—
57C2×Dic14—22no—
58C7×C4⋊C4—41no—
59C7×C4⋊C4—41no—
60C7×C4⋊C4—42no—
61C7×C4⋊C4—42no—
62C7×C4⋊C4—44no—
63C7×C4⋊C4—42no—
64C7×C4⋊C4—42no—
65C7×C4⋊C4—42no—
66C7×C4⋊C4—81no—
67C42×C7—21no—
68C42×C7SD1612yes—
69C22⋊Dic7—41no—
70C4⋊Dic7—21no—
71C4⋊Dic7—22no—
72C4⋊Dic7—21no—
73Dic7⋊C4—42no—
74C4×Dic7—41no—
75#1136—22no—
76#1136—41no—
77C2×C7⋊C8—41no—
78C2×C7⋊C8—42no—
79#544—71no—
80C2×#125—21no—
81#483—142no—
82#428—141no—
83#400—22no—
84#398—21no—
85#398—22no—
86#398C2×D712yes—
87#382C2×D711yescharacteristic
88#343—71no—
89#339—141no—
90#328—142no—
91#324—72no—
92C23×C7#12011yescharacteristic
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—41no—
98C2×C4×C7—41no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—41no—
108C2×C4×C7C2×SD1611yescharacteristic
109C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
110C2×Dic7—41no—
111C2×Dic7—41no—
112Dic14—82no—
113C7⋊C8—42no—
114C7⋊C8—82no—
115C2×M16—141no—
116C4⋊Q8—281no—
117#125—42no—
118#125—42no—
119#125—281no—
120C22⋊Q8—142no—
121#117—141no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C22⋊D711yescharacteristic
128C2×C4⋊C4C22⋊D712yes—
129C2×C4⋊C4C22×D711yescharacteristic
130C2×C42C22⋊D711yescharacteristic
131#107—281no—
132#106—281no—
133#106—282no—
134#103—282no—
135C22⋊C8—142no—
136#95—22no—
137#95—21no—
138#95—22no—
139#95—22no—
140#95C22⋊D712yes—
141C22×C7—21no—
142C22×C7—21no—
143C22×C7#45011yescharacteristic
144C22×C7#45611yescharacteristic
145C22×C7B(4,𝔽2)11yescharacteristic
146C4×C7—21no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—42no—
150C4×C7—82no—
151C4×C7—82no—
152C4×C7—81no—
153Dic7—81no—
154Dic7—81no—
155C2×Q8—142no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4C2×C22⋊D711yescharacteristic
161C22×C4C2×C22⋊D712yes—
162M16—142no—
163M16—281no—
164C2×C8—281no—
165C2×C8—282no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—141no—
176C4⋊C4—142no—
177C4⋊C4—141no—
178C4⋊C4—282no—
179C22⋊C4—281no—
180C42—21no—
181C42—281no—
182C42#114212yes—
183C2×C7—21no—
184C2×C7—21no—
185C2×C7#160411yescharacteristic
186C2×C7#220111yescharacteristic
187C2×C7#220311yescharacteristic
188C23#651611yescharacteristic
189Q8—562no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—281no—
206C2×C4—281no—
207C2×C4#649511yescharacteristic
208C2×C4C2×#114211yescharacteristic
209C8—282no—
210C8—562no—
211C7#1225111yescharacteristic, complemented, Sylow
212C22—21no—
213C22—21no—
214C22#8999011yescharacteristic
215C22#9001611yescharacteristic
216C22#9001811yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4—82no—
224C4—561no—
225C4—561no—
226C2—21no—
227C2—21no—
228C2#1172963411yescharacteristic
229C2#1172971211yescharacteristic
230C2#1172971311yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2822C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2822D7
hypocenterC7#12251
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
211. C7#12251semidirect

Maximal subgroups

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

subgroupindex
C7×#28222
#117274672
#117272972
#117254562
#117254382
#122517