Finite Group Databasebeta

← #13273483

Subgroups — #13273483

Order 1792.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2809C211yesmaximal, characteristic
2#11727426C212yesmaximal
3#11727426C211yesmaximal, characteristic
4#11725450C211yesmaximal, characteristic
5#11725450C212yesmaximal
6#90503C2211yescharacteristic
7C7×#400—21no—
8C7×#398—22no—
9C7×#391C2211yescharacteristic
10C7×#391C2212yes—
11C7×#389—21no—
12C7×#387—22no—
13C2×#6410C2212yes—
14C2×#6410C2211yescharacteristic
15#89736—22no—
16#89736—21no—
17#89318—22no—
18#89318—21no—
19#13465—71nomaximal, Sylow
20C22×C7×Q8C2311yescharacteristic
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4D412yes—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C42×C7D412yes—
28C2×C42×C7D411yescharacteristic
29Q8×Dic7—41no—
30Q8×Dic7—41no—
31#6489—22no—
32#6489—21no—
33C2×#1136—22no—
34C2×#1136—21no—
35C7×#95—21no—
36C7×#95—22no—
37C7×#95—22no—
38C7×#95—22no—
39C7×#95—22no—
40#6410—22no—
41#6410—21no—
42#6410Q814yes—
43#6410Q812yes—
44#6382—42no—
45#6382—41no—
46#6381—42no—
47#6381—41no—
48#2809D711yescharacteristic
49#1857—72no—
50#1857—71no—
51#1394—71no—
52#1394—72no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8C2×Q812yes—
56C2×C7×Q8C2×Q811yescharacteristic
57C2×C7×Q8#4111yescharacteristic
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C2×D412yes—
63C22×C4×C7C2×D411yescharacteristic
64C7×C4⋊C4—42no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—44no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—81no—
69C7×C4⋊C4—81no—
70C42×C7—41no—
71C42×C7—42no—
72C22⋊Dic7—41no—
73C4⋊Dic7—21no—
74C4⋊Dic7—22no—
75C4⋊Dic7—22no—
76C4⋊Dic7—21no—
77C4×Dic7—42no—
78C4×Dic7—41no—
79#1136—24no—
80#1136—22no—
81#1136—41no—
82#1136—42no—
83C2×C7⋊C8—42no—
84C2×C7⋊C8—41no—
85#519—71no—
86#428—142no—
87#428—141no—
88C2×#101—72no—
89C2×#101—71no—
90#400—21no—
91#398—22no—
92#391C2×D711yescharacteristic
93#391C2×D712yes—
94#389—21no—
95#387—22no—
96#339—142no—
97#339—141no—
98C23×C7#12011yescharacteristic
99C7×Q8—42no—
100C7×Q8—41no—
101C7×Q8—41no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7C22⋊Q812yes—
118C2×C4×C7C22⋊Q811yescharacteristic
119C2×C4×C7C4⋊Q812yes—
120C2×C4×C7C4⋊Q811yescharacteristic
121C2×Dic7—41no—
122C2×Dic7—41no—
123C7⋊C8—44no—
124C7⋊C8—42no—
125C22×Q8C22×D711yescharacteristic
126C2×M16—142no—
127C2×M16—141no—
128C4×Q8—281no—
129C4×Q8—281no—
130#117—142no—
131#117—141no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C22⋊D712yes—
137C2×C4⋊C4C22⋊D711yescharacteristic
138C2×C42C22⋊D712yes—
139C2×C42C22⋊D711yescharacteristic
140#107—282no—
141#107—281no—
142#106—282no—
143#106—281no—
144#101—74no—
145#101—72no—
146#101—142no—
147#101—141no—
148#95—21no—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—22no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39511yescharacteristic
156C22×C7B(4,𝔽2)11yescharacteristic
157C22×C7#45811yescharacteristic
158C4×C7—22no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—42no—
163C4×C7—82no—
164C4×C7—81no—
165C4×C7—82no—
166Dic7—81no—
167Dic7—81no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8Q8×D711yescharacteristic
171C2×Q8Q8×D712yes—
172C2×Q8#116111yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4C2×C22⋊D712yes—
178C22×C4C2×C22⋊D711yescharacteristic
179M16—144no—
180M16—142no—
181M16—281no—
182M16—282no—
183C2×C8—282no—
184C2×C8—281no—
185C4⋊C4—42no—
186C4⋊C4—41no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—141no—
192C4⋊C4—142no—
193C4⋊C4—142no—
194C4⋊C4—141no—
195C22⋊C4—281no—
196C42—41no—
197C42—42no—
198C42—282no—
199C42—281no—
200C2×C7—21no—
201C2×C7—21no—
202C2×C7#206211yescharacteristic
203C2×C7#220311yescharacteristic
204C2×C7#220411yescharacteristic
205C23#651611yescharacteristic
206Q8—41no—
207Q8—42no—
208Q8—41no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—281no—
225C2×C4—281no—
226C2×C4#650712yes—
227C2×C4#650711yescharacteristic
228C2×C4#650912yes—
229C2×C4#650911yescharacteristic
230C8—282no—
231C8—284no—
232C7#1346511yescharacteristic, complemented, Sylow
233C22—21no—
234C22—21no—
235C22#9000111yescharacteristic
236C22#9001511yescharacteristic
237C22#9001611yescharacteristic
238C4—21no—
239C4—22no—
240C4—21no—
241C4—22no—
242C4—42no—
243C4—82no—
244C4—82no—
245C4—81no—
246C4—561no—
247C4—561no—
248C2—21no—
249C2—21no—
250C2#1172964911yescharacteristic
251C2#1172968411yescharacteristic
252C2#1172971311yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC22×C7×Q8C23
FrattiniC22×Q8C22×D7
FittingC7×#2809C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2809D7
hypocenterC7#13465
LCSG ▷ C22×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

Complements

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

NH ≅ G/Ntype
232. C7#13465semidirect

Maximal subgroups

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

subgroupindex
C7×#28092
#117274262
#117254502
#134657