Finite Group Databasebeta

← #13266395

Subgroups — #13266395

Order 1792.

360 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#11731663C211yesmaximal, characteristic, complemented
2C7×#2040C211yesmaximal, characteristic
3#11729629C212yesmaximal
4#11725438C211yesmaximal, characteristic, complemented
5#11725302C212yesmaximal, complemented
6C7×#537C2211yescharacteristic
7C2×C7×#104—21no—
8C7×#417—21no—
9C7×#382C2211yescharacteristic
10#89806—22no—
11#89758C2212yes—
12#89751—21no—
13#89708—21no—
14#89432—22no—
15#89428—22no—
16C7×#343C2211yescharacteristic
17#89332—22no—
18#89323C2212yes—
19#12185—71nomaximal, Sylow
20C2×C7×#41—21no—
21C22×Dic14—21no—
22C2×C7×M16—21no—
23C7×C22⋊Q8—41no—
24C7×#121—22no—
25C7×#121—41no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7D412yes—
32C2×Dic7⋊C4—22no—
33C7×#104—42no—
34C7×#103—41no—
35C7×#102—41no—
36C7×#95—22no—
37#6406—42no—
38C7⋊#95—22no—
39C7⋊#95—21no—
40C7⋊#95—22no—
41C7⋊#95—22no—
42C7⋊#95D412yes—
43C22⋊(C7⋊C8)—22no—
44#6382—42no—
45#2844—71no—
46#2040D711yescharacteristic
47#1616—72no—
48#1353—72no—
49#1324—71no—
50C7×#41—41no—
51C7×#41—41no—
52C7×#41—41no—
53C7×#41—41no—
54C2×C7×Q8—21no—
55C2×C7×D4—21no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—41no—
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D412yes—
63C22×Dic7—22no—
64C22×Dic7—22no—
65C2×Dic14—82no—
66C2×Dic14—81no—
67C7×M16—41no—
68C7×M16—41no—
69C2×C7×C8—41no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C22⋊C4—42no—
74C42×C7—21no—
75C42×C7SD1612yes—
76C4⋊Dic7—42no—
77C4⋊Dic7—81no—
78C4⋊Dic7—81no—
79Dic7⋊C4—82no—
80C2×C7⋊C8—42no—
81#537C2×D711yescharacteristic
82#481—142no—
83C2×#104—21no—
84#417—21no—
85#400—72no—
86#400—142no—
87#395—142no—
88#395—141no—
89#382—141no—
90#382C2×D711yescharacteristic
91#343C2×D711yescharacteristic
92#327—142no—
93#324—72no—
94C23×C7—41no—
95C23×C7#12011yescharacteristic
96C7×Q8—41no—
97C7×Q8—41no—
98C7×D4—41no—
99C7×D4—41no—
100C7×D4—82no—
101C7×D4—81no—
102C7×D4—81no—
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—41no—
112C2×C4×C7—41no—
113C2×C4×C7—81no—
114C2×C4×C7—81no—
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7#13711yescharacteristic
117C2×Dic7—42no—
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—42no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123Dic14—84no—
124C7×C8—81no—
125C7⋊C8—82no—
126C2×#41—21no—
127C22×Q8—141no—
128C2×M16—21no—
129C22⋊Q8—41no—
130#121—22no—
131#121—41no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—142no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4—141no—
136C2×C4⋊C4C22⋊D711yescharacteristic
137C2×C4⋊C4C22×D711yescharacteristic
138C2×C42C22⋊D711yescharacteristic
139#106—282no—
140#104—42no—
141#103—41no—
142#102—41no—
143#102—282no—
144C22⋊C8—142no—
145#95—22no—
146#95—72no—
147#95—142no—
148#95—141no—
149#95—142no—
150#95—142no—
151C22×C7—21no—
152C22×C7—21no—
153C22×C7—81no—
154C22×C7—81no—
155C22×C7—81no—
156C22×C7#44811yescharacteristic
157C22×C7#45411yescharacteristic
158C22×C7B(4,𝔽2)11yescharacteristic
159C4×C7—21no—
160C4×C7—21no—
161C4×C7—42no—
162C4×C7—42no—
163C4×C7—81no—
164C4×C7—81no—
165Dic7—82no—
166Dic7—82no—
167#41—41no—
168#41—41no—
169#41—41no—
170#41—41no—
171C2×Q8—21no—
172C2×Q8—562no—
173C2×Q8—561no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—41no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—142no—
181C22×C4—142no—
182C22×C4D4×D712yes—
183C22×C4C2×C22⋊D711yescharacteristic
184M16—41no—
185M16—41no—
186C2×C8—41no—
187C2×C8—282no—
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—282no—
192C4⋊C4—562no—
193C4⋊C4—561no—
194C4⋊C4—561no—
195C22⋊C4—42no—
196C42—21no—
197C42#114212yes—
198C2×C7—21no—
199C2×C7—21no—
200C2×C7—161no—
201C2×C7#159911yescharacteristic
202C2×C7#220111yescharacteristic
203C2×C7#220211yescharacteristic
204C23—41no—
205C23#650011yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—564no—
209D4—41no—
210D4—41no—
211D4—82no—
212D4—81no—
213D4—81no—
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—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—282no—
227C2×C4—282no—
228C2×C4—282no—
229C2×C4—282no—
230C2×C4—282no—
231C2×C4—282no—
232C2×C4C2×#114211yescharacteristic
233C2×C4#651411yescharacteristic
234C8—81no—
235C8—562no—
236C7#1218511yescharacteristic, complemented, Sylow
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#8980411yescharacteristic
243C22#8981211yescharacteristic
244C22#8983411yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—562no—
252C4—562no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#1172745811yescharacteristic
257C2#1172874111yescharacteristic
258C2#1172874611yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2040C2
socleC22×C7B(4,𝔽2)
prime radicalC7×#537C22
hypercenter#2040D7
hypocenterC7#12185
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #11731663C2semidirect
4. #11725438C2semidirect
5. #11725302C2semidirect
236. C7#12185semidirect

Maximal subgroups

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

subgroupindex
#117316632
C7×#20402
#117296292
#117254382
#117253022
#121857