Finite Group Databasebeta

← #13266880

Subgroups — #13266880

Order 1792.

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

normal subgroups only

This group has 330 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×#2844C211yesmaximal, characteristic
2#11727473C212yesmaximal
3#11727297C211yesmaximal, characteristic
4#11725458C212yesmaximal
5#11725438C211yesmaximal, characteristic
6#90191C2211yescharacteristic
7C7×#400—22no—
8C7×#400C2212yes—
9C7×#395—22no—
10C7×#395—21no—
11C7×#382—21no—
12C7×#382C2211yescharacteristic
13#89742—22no—
14#89736—21no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12261—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×C7×C4⋊C4—21no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C7×C4⋊C4D412yes—
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7D411yescharacteristic
28#6489—21no—
29#6487—22no—
30C2×#1136—21no—
31#6453—41no—
32#6451—41no—
33C7×#95—22no—
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#2844D711yescharacteristic
45#2061—71no—
46#1617—72no—
47#1353—72no—
48#1324—71no—
49C2×C7×Q8—82no—
50C2×C7×Q8—81no—
51C22×C4×C7—21no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—21no—
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—81no—
62C7×C4⋊C4—81no—
63C7×C4⋊C4—82no—
64C42×C7—21no—
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—22no—
80#400C2×D712yes—
81#395—22no—
82#395—21no—
83#382—21no—
84#382C2×D711yescharacteristic
85#343—71no—
86#339—141no—
87#328—142no—
88#324—72no—
89C23×C7#12011yescharacteristic
90C7×Q8—84no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—41no—
96C2×C4×C7—41no—
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—42no—
104C2×C4×C7—41no—
105C2×C4×C7—42no—
106C2×C4×C7C2×Q1611yescharacteristic
107C2×C4×C7#13711yescharacteristic
108C2×Dic7—41no—
109C2×Dic7—41no—
110Dic14—82no—
111C7⋊C8—42no—
112C7⋊C8—82no—
113C22×Q8—21no—
114C2×M16—141no—
115C4⋊Q8—281no—
116#125—281no—
117C22⋊Q8—142no—
118#117—141no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C22⋊D711yescharacteristic
123C2×C4⋊C4C22⋊D712yes—
124C2×C4⋊C4C22×D711yescharacteristic
125C2×C42C22⋊D711yescharacteristic
126#107—281no—
127#107—282no—
128#106—281no—
129#103—282no—
130C22⋊C8—142no—
131#95—22no—
132#95—21no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95C22⋊D712yes—
137C22×C7—21no—
138C22×C7—21no—
139C22×C7#45111yescharacteristic
140C22×C7#45611yescharacteristic
141C22×C7B(4,𝔽2)11yescharacteristic
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—42no—
145C4×C7—42no—
146C4×C7—82no—
147C4×C7—81no—
148C4×C7—82no—
149Dic7—81no—
150Dic7—81no—
151C2×Q8—82no—
152C2×Q8—81no—
153C2×Q8—142no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4C2×C22⋊D711yescharacteristic
159C22×C4C2×C22⋊D712yes—
160M16—142no—
161M16—281no—
162C2×C8—281no—
163C2×C8—282no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—141no—
171C4⋊C4—141no—
172C4⋊C4—142no—
173C4⋊C4—282no—
174C22⋊C4—281no—
175C42—21no—
176C42—281no—
177C42#114412yes—
178C2×C7—21no—
179C2×C7—21no—
180C2×C7#160511yescharacteristic
181C2×C7#220111yescharacteristic
182C2×C7#220211yescharacteristic
183C23#651611yescharacteristic
184Q8—84no—
185Q8—562no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—281no—
202C2×C4—281no—
203C2×C4#650511yescharacteristic
204C2×C4C2×#114411yescharacteristic
205C8—282no—
206C8—562no—
207C7#1226111yescharacteristic, complemented, Sylow
208C22—21no—
209C22—21no—
210C22#9000011yescharacteristic
211C22#9001611yescharacteristic
212C22#9001811yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4—81no—
220C4—561no—
221C4—561no—
222C2—21no—
223C2—21no—
224C2#1172965811yescharacteristic
225C2#1172970511yescharacteristic
226C2#1172971211yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2844C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2844D7
hypocenterC7#12261
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
207. C7#12261semidirect

Maximal subgroups

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

subgroupindex
C7×#28442
#117274732
#117272972
#117254582
#117254382
#122617