Finite Group Databasebeta

← #13266856

Subgroups — #13266856

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×#2571C211yesmaximal, characteristic
2#11727473C212yesmaximal
3#11727327C211yesmaximal, characteristic
4#11725458C212yesmaximal
5#11725441C211yesmaximal, characteristic
6#90191C2211yescharacteristic
7C7×#400—21no—
8C7×#400C2212yes—
9C7×#389—22no—
10C7×#384C2211yescharacteristic
11C7×#382—22no—
12C4×C7×C4⋊C4—21no—
13#89742—22no—
14#89736—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12257—71nomaximal, Sylow
20C2×C7×C4⋊C4—21no—
21C2×C7×C4⋊C4—21no—
22C2×C7×C4⋊C4—22no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C7×C4⋊C4D412yes—
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7—22no—
28C2×C42×C7D411yescharacteristic
29#6489—21no—
30#6487—22no—
31C2×#1136—21no—
32#6453—41no—
33#6451—41no—
34C7×#95—21no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38C7×#95D412yes—
39C22⋊(C7⋊C8)—22no—
40#6384—42no—
41#6382—41no—
42#6381—41no—
43#6381—42no—
44#2571D711yescharacteristic
45#2060—71no—
46#1617—72no—
47#1353—72no—
48#1325—71no—
49C22×C4×C7—21no—
50C22×C4×C7—22no—
51C22×C4×C7—22no—
52C22×C4×C7—21no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D412yes—
55C2×Dic14—22no—
56C7×C4⋊C4—41no—
57C7×C4⋊C4—41no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—42no—
60C7×C4⋊C4—44no—
61C7×C4⋊C4—42no—
62C7×C4⋊C4—81no—
63C42×C7—21no—
64C42×C7—44no—
65C42×C7Q1612yes—
66C22⋊Dic7—41no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7—21no—
70Dic7⋊C4—42no—
71C4×Dic7—41no—
72#1136—22no—
73#1136—41no—
74C2×C7⋊C8—41no—
75C2×C7⋊C8—42no—
76#544—71no—
77#484—142no—
78#428—141no—
79#400—21no—
80#400C2×D712yes—
81#389—22no—
82#384C2×D711yescharacteristic
83#382—22no—
84C4×C4⋊C4—21no—
85#343—71no—
86#339—141no—
87#328—142no—
88#324—72no—
89C23×C7#12011yescharacteristic
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—42no—
97C2×C4×C7—42no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7C2×Q1611yescharacteristic
106C2×C4×C7#13711yescharacteristic
107C2×Dic7—41no—
108C2×Dic7—41no—
109Dic14—82no—
110C7⋊C8—42no—
111C7⋊C8—82no—
112C2×M16—141no—
113C4⋊Q8—281no—
114#125—281no—
115C22⋊Q8—142no—
116#117—141no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C22⋊D711yescharacteristic
122C2×C4⋊C4C22⋊D712yes—
123C2×C4⋊C4C22×D711yescharacteristic
124C2×C42—22no—
125C2×C42C22⋊D711yescharacteristic
126#107—281no—
127#107—282no—
128#106—281no—
129#103—282no—
130C22⋊C8—142no—
131#95—21no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95C22⋊D712yes—
136C22×C7—21no—
137C22×C7—21no—
138C22×C7#45111yescharacteristic
139C22×C7#45611yescharacteristic
140C22×C7B(4,𝔽2)11yescharacteristic
141C4×C7—24no—
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—42no—
145C4×C7—82no—
146C4×C7—81no—
147C4×C7—82no—
148Dic7—81no—
149Dic7—81no—
150C2×Q8—142no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4C2×C22⋊D711yescharacteristic
156C22×C4C2×C22⋊D712yes—
157M16—142no—
158M16—281no—
159C2×C8—281no—
160C2×C8—282no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—141no—
169C4⋊C4—141no—
170C4⋊C4—142no—
171C4⋊C4—282no—
172C22⋊C4—281no—
173C42—21no—
174C42—44no—
175C42—281no—
176C42#114412yes—
177C2×C7—21no—
178C2×C7—21no—
179C2×C7#160511yescharacteristic
180C2×C7#219811yescharacteristic
181C2×C7#220411yescharacteristic
182C23#651611yescharacteristic
183Q8—562no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—281no—
200C2×C4—281no—
201C2×C4#650511yescharacteristic
202C2×C4C2×#114411yescharacteristic
203C8—282no—
204C8—562no—
205C7#1225711yescharacteristic, complemented, Sylow
206C22—21no—
207C22—21no—
208C22#9000011yescharacteristic
209C22#9001611yescharacteristic
210C22#9001811yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—82no—
217C4—81no—
218C4—561no—
219C4—561no—
220C2—21no—
221C2—21no—
222C2#1172965811yescharacteristic
223C2#1172968411yescharacteristic
224C2#1172970011yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2571C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2571D7
hypocenterC7#12257
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
205. C7#12257semidirect

Maximal subgroups

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

subgroupindex
C7×#25712
#117274732
#117273272
#117254582
#117254412
#122577