Finite Group Databasebeta

← #13266232

Subgroups — #13266232

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×#2822C211yesmaximal, characteristic, complemented
2#11727466C212yesmaximal
3#11727317C211yesmaximal, characteristic
4#11725456C212yesmaximal, complemented
5#11725438C211yesmaximal, characteristic, complemented
6C2×C7×#125—21no—
7#90201C2211yescharacteristic
8C7×#400—22no—
9C7×#398—21no—
10C7×#398—22no—
11C7×#398C2212yes—
12C7×#382C2211yescharacteristic
13#89739—22no—
14#89738—21no—
15C2×#6379—21no—
16#89325C2211yescharacteristic
17#89323C2212yes—
18#89246—22no—
19#12157—71nomaximal, Sylow
20C2×#1157—21no—
21C7×#125—42no—
22C7×#125—42no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7D411yescharacteristic
32#6493—22no—
33C2×#1136—21no—
34#6460—41no—
35#6458—41no—
36C7×#95—22no—
37C7×#95—21no—
38C7×#95—22no—
39C7×#95—22no—
40C7×#95D412yes—
41C22⋊(C7⋊C8)—22no—
42#6384—41no—
43#6383—41no—
44#6383—42no—
45#6381—42no—
46#6379—42no—
47#2822D711yescharacteristic, complemented
48#2040—71no—
49#1610—72no—
50#1346—72no—
51#1324—71no—
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
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—42no—
71C7×C4⋊C4—44no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—81no—
76C42×C7—21no—
77C42×C7D812yes—
78(C2×D7)⋊C4—42no—
79C4⋊Dic7—22no—
80#1136—41no—
81#1136—41no—
82C2×C7⋊C8—42no—
83C2×C7⋊C8—41no—
84#537—71no—
85C2×#125—21no—
86#480—142no—
87C2×#104—141no—
88#417—141no—
89#400—22no—
90#398—21no—
91#398—22no—
92#398C2×D712yes—
93#382C2×D711yescharacteristic
94#343—71no—
95#327—142no—
96#324—72no—
97C23×C7#12011yescharacteristic
98C22×D7—41no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—41no—
104C2×C4×C7—41no—
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—42no—
112C2×C4×C7—42no—
113C2×C4×C7—41no—
114C2×C4×C7C2×D811yescharacteristic
115C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
116C22⋊D7—81no—
117C22⋊D7—81no—
118C2×Dic7—41no—
119D28—41no—
120D28—41no—
121D28—82no—
122C4×D7—81no—
123C4×D7—81no—
124Dic14—41no—
125Dic14—41no—
126C7⋊C8—81no—
127C7⋊C8—82no—
128C2×#41—141no—
129C2×M16—141no—
130#125—42no—
131#125—42no—
132C22⋊Q8—281no—
133#121—142no—
134#121—281no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C22⋊D711yescharacteristic
141C2×C4⋊C4C22⋊D712yes—
142C2×C4⋊C4C22×D711yescharacteristic
143C2×C42C22⋊D711yescharacteristic
144#107—282no—
145#104—282no—
146#103—281no—
147#102—281no—
148#102—282no—
149C22⋊C8—142no—
150#95—22no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95C22⋊D712yes—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7#44711yescharacteristic
158C22×C7#45411yescharacteristic
159C22×C7B(4,𝔽2)11yescharacteristic
160C2×D7—81no—
161C2×D7—81no—
162C2×D7—81no—
163C4×C7—21no—
164C4×C7—21no—
165C4×C7—42no—
166C4×C7—42no—
167C4×C7—82no—
168C4×C7—82no—
169C4×C7—81no—
170Dic7—81no—
171#41—281no—
172#41—281no—
173#41—281no—
174#41—281no—
175C2×Q8—141no—
176C2×D4—141no—
177C2×D4—142no—
178C2×D4—281no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—281no—
184C22×C4C2×C22⋊D711yescharacteristic
185C22×C4C2×C22⋊D712yes—
186M16—281no—
187M16—281no—
188C2×C8—282no—
189C2×C8—281no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—142no—
200C22⋊C4—282no—
201C42—21no—
202C42#114112yes—
203C2×C7—21no—
204C2×C7—21no—
205C2×C7#159811yescharacteristic
206C2×C7#220111yescharacteristic
207C2×C7#220311yescharacteristic
208D7—161no—
209C23—281no—
210C23#651611yescharacteristic
211Q8—281no—
212Q8—281no—
213D4—281no—
214D4—281no—
215D4—562no—
216D4—561no—
217D4—561no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—281no—
234C2×C4—561no—
235C2×C4—561no—
236C2×C4C2×#114111yescharacteristic
237C2×C4#649511yescharacteristic
238C8—561no—
239C8—562no—
240C7#1215711yescharacteristic, complemented, Sylow
241C22—21no—
242C22—21no—
243C22—561no—
244C22—561no—
245C22—561no—
246C22#8998911yescharacteristic
247C22#9001611yescharacteristic
248C22#9001711yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—561no—
257C2—21no—
258C2—21no—
259C2—1121no—
260C2#1172963211yescharacteristic
261C2#1172971211yescharacteristic
262C2#1172971311yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2822C2
socleC22×C7B(4,𝔽2)
prime radical#90201C22
hypercenter#2822D7
hypocenterC7#12157
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
1. C7×#2822C2semidirect
4. #11725456C2semidirect
5. #11725438C2semidirect
47. #2822D7semidirect
240. C7#12157semidirect

Maximal subgroups

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

subgroupindex
C7×#28222
#117274662
#117273172
#117254562
#117254382
#121577