Finite Group Databasebeta

← #13266396

Subgroups — #13266396

Order 1792.

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

normal subgroups only

This group has 360 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, complemented
2#11727472C212yesmaximal
3#11727317C211yesmaximal, characteristic
4#11725458C212yesmaximal, complemented
5#11725438C211yesmaximal, characteristic, complemented
6#90201C2211yescharacteristic
7C7×#400—22no—
8C7×#400C2212yes—
9C7×#395—22no—
10C7×#395—21no—
11C7×#382—21no—
12C7×#382C2211yescharacteristic
13#89740—22no—
14#89738—21no—
15C2×#6379—21no—
16#89325C2211yescharacteristic
17#89323C2212yes—
18#89246—22no—
19#12185—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×#1157—21no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C4⋊C4D412yes—
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7D411yescharacteristic
29#6493—22no—
30C2×#1136—21no—
31#6460—41no—
32#6458—41no—
33C7×#95—22no—
34C7×#95—21no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38C7×#95D412yes—
39C22⋊(C7⋊C8)—22no—
40#6384—41no—
41#6383—41no—
42#6383—42no—
43#6382—42no—
44#6379—42no—
45#2844D711yescharacteristic, complemented
46#2040—71no—
47#1616—72no—
48#1353—72no—
49#1324—71no—
50C2×C7×Q8—82no—
51C2×C7×Q8—81no—
52C22×C4×C7—21no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—21no—
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D412yes—
58C2×C22⋊D7—41no—
59#1157—41no—
60#1157—41no—
61#1157—41no—
62#1157—41no—
63C2×D28—21no—
64C2×D28—22no—
65C2×C4×D7—41no—
66C2×Dic14—21no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—81no—
71C7×C4⋊C4—81no—
72C7×C4⋊C4—82no—
73C42×C7—21no—
74C42×C7SD1612yes—
75(C2×D7)⋊C4—42no—
76C4⋊Dic7—22no—
77#1136—41no—
78#1136—41no—
79C2×C7⋊C8—42no—
80C2×C7⋊C8—41no—
81#537—71no—
82#481—142no—
83C2×#104—141no—
84#417—141no—
85#400—22no—
86#400C2×D712yes—
87#395—22no—
88#395—21no—
89#382—21no—
90#382C2×D711yescharacteristic
91#343—71no—
92#327—142no—
93#324—72no—
94C23×C7#12011yescharacteristic
95C22×D7—41no—
96C7×Q8—84no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—41no—
112C2×C4×C7C2×SD1611yescharacteristic
113C2×C4×C7#13711yescharacteristic
114C22⋊D7—81no—
115C22⋊D7—81no—
116C2×Dic7—41no—
117D28—41no—
118D28—41no—
119D28—82no—
120C4×D7—81no—
121C4×D7—81no—
122Dic14—41no—
123Dic14—41no—
124C7⋊C8—81no—
125C7⋊C8—82no—
126C2×#41—141no—
127C22×Q8—21no—
128C2×M16—141no—
129C22⋊Q8—281no—
130#121—142no—
131#121—281no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C22⋊D711yescharacteristic
136C2×C4⋊C4C22⋊D712yes—
137C2×C4⋊C4C22×D711yescharacteristic
138C2×C42C22⋊D711yescharacteristic
139#106—282no—
140#104—282no—
141#103—281no—
142#102—281no—
143#102—282no—
144C22⋊C8—142no—
145#95—22no—
146#95—21no—
147#95—22no—
148#95—22no—
149#95—22no—
150#95C22⋊D712yes—
151C22×C7—21no—
152C22×C7—21no—
153C22×C7#44811yescharacteristic
154C22×C7#45411yescharacteristic
155C22×C7B(4,𝔽2)11yescharacteristic
156C2×D7—81no—
157C2×D7—81no—
158C2×D7—81no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7—42no—
162C4×C7—42no—
163C4×C7—82no—
164C4×C7—82no—
165C4×C7—81no—
166Dic7—81no—
167#41—281no—
168#41—281no—
169#41—281no—
170#41—281no—
171C2×Q8—82no—
172C2×Q8—81no—
173C2×Q8—141no—
174C2×D4—141no—
175C2×D4—142no—
176C2×D4—281no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—281no—
182C22×C4C2×C22⋊D711yescharacteristic
183C22×C4C2×C22⋊D712yes—
184M16—281no—
185M16—281no—
186C2×C8—282no—
187C2×C8—281no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—82no—
194C4⋊C4—142no—
195C22⋊C4—282no—
196C42—21no—
197C42#114312yes—
198C2×C7—21no—
199C2×C7—21no—
200C2×C7#159911yescharacteristic
201C2×C7#220111yescharacteristic
202C2×C7#220211yescharacteristic
203D7—161no—
204C23—281no—
205C23#651611yescharacteristic
206Q8—84no—
207Q8—281no—
208Q8—281no—
209D4—281no—
210D4—281no—
211D4—562no—
212D4—561no—
213D4—561no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—281no—
230C2×C4—561no—
231C2×C4—561no—
232C2×C4C2×#114311yescharacteristic
233C2×C4#650511yescharacteristic
234C8—561no—
235C8—562no—
236C7#1218511yescharacteristic, complemented, Sylow
237C22—21no—
238C22—21no—
239C22—561no—
240C22—561no—
241C22—561no—
242C22#8999911yescharacteristic
243C22#9001611yescharacteristic
244C22#9001711yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—561no—
253C2—21no—
254C2—21no—
255C2—1121no—
256C2#1172965611yescharacteristic
257C2#1172970511yescharacteristic
258C2#1172971211yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2844C2
socleC22×C7B(4,𝔽2)
prime radical#90201C22
hypercenter#2844D7
hypocenterC7#12185
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
1. C7×#2844C2semidirect
4. #11725458C2semidirect
5. #11725438C2semidirect
45. #2844D7semidirect
236. C7#12185semidirect

Maximal subgroups

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

subgroupindex
C7×#28442
#117274722
#117273172
#117254582
#117254382
#121857