Finite Group Databasebeta

← #13266372

Subgroups — #13266372

Order 1792.

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

normal subgroups only

This group has 372 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, complemented
2#11727472C212yesmaximal
3#11727347C211yesmaximal, characteristic
4#11725458C212yesmaximal, complemented
5#11725441C211yesmaximal, characteristic, complemented
6#90201C2211yescharacteristic
7C7×#400—21no—
8C7×#400C2212yes—
9C7×#389—22no—
10C7×#384C2211yescharacteristic
11C7×#382—22no—
12C4×C7×C4⋊C4—21no—
13#89740—22no—
14#89738—21no—
15C2×#6379—21no—
16#89325C2211yescharacteristic
17#89323C2212yes—
18#89246—22no—
19#12181—71nomaximal, Sylow
20C2×#1157—21no—
21C2×C7×C4⋊C4—21no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C4⋊C4D412yes—
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7—22no—
29C2×C42×C7D411yescharacteristic
30#6493—22no—
31C2×#1136—21no—
32#6460—41no—
33#6458—41no—
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#2571D711yescharacteristic, complemented
46#2039—71no—
47#1616—72no—
48#1353—72no—
49#1325—71no—
50C22×C4×C7—21no—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—21no—
54C22×C4×C7C2×D411yescharacteristic
55C22×C4×C7C2×D412yes—
56C2×C22⋊D7—41no—
57#1157—41no—
58#1157—41no—
59#1157—41no—
60#1157—41no—
61C2×D28—21no—
62C2×D28—22no—
63C2×C4×D7—41no—
64C2×Dic14—21no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—42no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—44no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—81no—
72C42×C7—21no—
73C42×C7—44no—
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—21no—
86#400C2×D712yes—
87#389—22no—
88#384C2×D711yescharacteristic
89#382—22no—
90C4×C4⋊C4—21no—
91#343—71no—
92#327—142no—
93#324—72no—
94C23×C7#12011yescharacteristic
95C22×D7—41no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—42no—
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—41no—
110C2×C4×C7—42no—
111C2×C4×C7C2×SD1611yescharacteristic
112C2×C4×C7#13711yescharacteristic
113C22⋊D7—81no—
114C22⋊D7—81no—
115C2×Dic7—41no—
116D28—41no—
117D28—41no—
118D28—82no—
119C4×D7—81no—
120C4×D7—81no—
121Dic14—41no—
122Dic14—41no—
123C7⋊C8—81no—
124C7⋊C8—82no—
125C2×#41—141no—
126C2×M16—141no—
127C22⋊Q8—281no—
128#121—142no—
129#121—281no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C22⋊D711yescharacteristic
135C2×C4⋊C4C22⋊D712yes—
136C2×C4⋊C4C22×D711yescharacteristic
137C2×C42—22no—
138C2×C42C22⋊D711yescharacteristic
139#106—282no—
140#104—282no—
141#103—281no—
142#102—281no—
143#102—282no—
144C22⋊C8—142no—
145#95—21no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95C22⋊D712yes—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7#44811yescharacteristic
153C22×C7#45411yescharacteristic
154C22×C7B(4,𝔽2)11yescharacteristic
155C2×D7—81no—
156C2×D7—81no—
157C2×D7—81no—
158C4×C7—24no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7—42no—
162C4×C7—82no—
163C4×C7—81no—
164C4×C7—82no—
165Dic7—81no—
166#41—281no—
167#41—281no—
168#41—281no—
169#41—281no—
170C2×Q8—141no—
171C2×D4—141no—
172C2×D4—142no—
173C2×D4—281no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—281no—
179C22×C4C2×C22⋊D711yescharacteristic
180C22×C4C2×C22⋊D712yes—
181M16—281no—
182M16—281no—
183C2×C8—282no—
184C2×C8—281no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—142no—
193C22⋊C4—282no—
194C42—21no—
195C42—44no—
196C42#114312yes—
197C2×C7—21no—
198C2×C7—21no—
199C2×C7#159911yescharacteristic
200C2×C7#219811yescharacteristic
201C2×C7#220411yescharacteristic
202D7—161no—
203C23—281no—
204C23#651611yescharacteristic
205Q8—281no—
206Q8—281no—
207D4—281no—
208D4—281no—
209D4—562no—
210D4—561no—
211D4—561no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—281no—
228C2×C4—561no—
229C2×C4—561no—
230C2×C4C2×#114311yescharacteristic
231C2×C4#650511yescharacteristic
232C8—561no—
233C8—562no—
234C7#1218111yescharacteristic, complemented, Sylow
235C22—21no—
236C22—21no—
237C22—561no—
238C22—561no—
239C22—561no—
240C22#8999911yescharacteristic
241C22#9001611yescharacteristic
242C22#9001711yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—82no—
249C4—81no—
250C4—561no—
251C2—21no—
252C2—21no—
253C2—1121no—
254C2#1172965611yescharacteristic
255C2#1172968411yescharacteristic
256C2#1172970011yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2571C2
socleC22×C7B(4,𝔽2)
prime radical#90201C22
hypercenter#2571D7
hypocenterC7#12181
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
1. C7×#2571C2semidirect
4. #11725458C2semidirect
5. #11725441C2semidirect
45. #2571D7semidirect
234. C7#12181semidirect

Maximal subgroups

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

subgroupindex
C7×#25712
#117274722
#117273472
#117254582
#117254412
#121817