Finite Group Databasebeta

← #13266603

Subgroups — #13266603

Order 1792.

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

normal subgroups only

This group has 456 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×#2842C211yesmaximal, characteristic
2#11727470C212yesmaximal, complemented
3#11727251C211yesmaximal, characteristic
4#11725457C212yesmaximal, complemented
5#11725428C211yesmaximal, characteristic
6#90174C2211yescharacteristic
7C7×#399—21no—
8C7×#399C2212yes—
9C7×#396—22no—
10C7×#396—22no—
11C7×#379—21no—
12C7×#379C2211yescharacteristic
13#90019—21no—
14#89742—22no—
15#89322—21no—
16#89321C2211yescharacteristic
17#89320C2212yes—
18#89247—22no—
19#12218—71nomaximal, Sylow
20C25×C7D411yescharacteristic
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C22⋊C4—21no—
26C2×C7×C22⋊C4—22no—
27C2×C7×C22⋊C4—21no—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4D411yescharacteristic
31C2×C7×C22⋊C4C2311yescharacteristic
32C2×C22⋊Dic7—21no—
33C2×C22⋊Dic7—21no—
34C2×C22⋊Dic7—42no—
35#6487—22no—
36#6441—42no—
37C7×#95—21no—
38C7×#95D412yes—
39C22⋊(C7⋊C8)—22no—
40#6384—42no—
41#6381—42no—
42#2842D711yescharacteristic
43#2029—71no—
44#1614—72no—
45#1352—72no—
46#1316—71no—
47C24×C7—21no—
48C24×C7—21no—
49C24×C7—41no—
50C24×C7—41no—
51C24×C7—82no—
52C24×C7Q1612yes—
53C24×C7C2×D411yescharacteristic
54C22×C4×C7—22no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7C2×D412yes—
58C22×Dic7—22no—
59C22×Dic7—41no—
60C2×Dic14—22no—
61C7×C4⋊C4—42no—
62C7×C4⋊C4—82no—
63C7×C4⋊C4—82no—
64C7×C4⋊C4—81no—
65C7×C22⋊C4—41no—
66C7×C22⋊C4—41no—
67C7×C22⋊C4—82no—
68C7×C22⋊C4—82no—
69C7×C22⋊C4—82no—
70C7×C22⋊C4—81no—
71C7×C22⋊C4—81no—
72C22⋊Dic7—41no—
73C22⋊Dic7—41no—
74C22⋊Dic7—41no—
75C22⋊Dic7—41no—
76C22⋊Dic7—84no—
77C4⋊Dic7—22no—
78Dic7⋊C4—42no—
79C2×C7⋊C8—42no—
80#543—71no—
81#484—142no—
82#399—21no—
83#399C2×D712yes—
84#396—22no—
85#396—22no—
86#379—21no—
87#379—141no—
88#379C2×D711yescharacteristic
89#342—71no—
90#342—141no—
91#328—142no—
92#323—72no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—42no—
98C23×C7—42no—
99C23×C7—42no—
100C23×C7—44no—
101C23×C7—41no—
102C23×C7—41no—
103C23×C7—81no—
104C23×C7—82no—
105C23×C7—82no—
106C23×C7—82no—
107C23×C7—84no—
108C23×C7#12011yescharacteristic
109C23×C7C2×Q1611yescharacteristic
110C23×C7#13711yescharacteristic
111C2×C4×C7—22no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×Dic7—42no—
122C2×Dic7—82no—
123C2×Dic7—81no—
124C2×Dic7—82no—
125Dic14—82no—
126C7⋊C8—82no—
127C25C22⋊D711yescharacteristic
128C22⋊Q8—142no—
129C22⋊Q8—282no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C22⋊D712yes—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—141no—
140C2×C22⋊C4—141no—
141C2×C22⋊C4—282no—
142C2×C22⋊C4C22⋊D711yescharacteristic
143C2×C22⋊C4C22×D711yescharacteristic
144#107—282no—
145#103—282no—
146C22⋊C8—142no—
147#95—21no—
148#95C22⋊D712yes—
149C22×C7—22no—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—44no—
158C22×C7—42no—
159C22×C7—81no—
160C22×C7—82no—
161C22×C7—82no—
162C22×C7—84no—
163C22×C7—82no—
164C22×C7#45111yescharacteristic
165C22×C7#45511yescharacteristic
166C22×C7B(4,𝔽2)11yescharacteristic
167C4×C7—42no—
168C4×C7—81no—
169C4×C7—82no—
170C4×C7—82no—
171Dic7—82no—
172Dic7—161no—
173C24—21no—
174C24—21no—
175C24—41no—
176C24—41no—
177C24—82no—
178C24#114412yes—
179C24C2×C22⋊D711yescharacteristic
180C2×Q8—142no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—142no—
185C22×C4—281no—
186C22×C4C2×C22⋊D712yes—
187C2×C8—282no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—142no—
193C4⋊C4—282no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—281no—
202C22⋊C4—281no—
203C22⋊C4—281no—
204C22⋊C4—281no—
205C22⋊C4—564no—
206C2×C7—21no—
207C2×C7—21no—
208C2×C7—41no—
209C2×C7—41no—
210C2×C7—82no—
211C2×C7#160211yescharacteristic
212C2×C7#220012yes—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—44no—
221C23—41no—
222C23—41no—
223C23—81no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—84no—
228C23#650511yescharacteristic
229C23C2×#114411yescharacteristic
230C23#651611yescharacteristic
231Q8—562no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—282no—
243C2×C4—562no—
244C2×C4—561no—
245C2×C4—562no—
246C8—562no—
247C7#1221811yescharacteristic, complemented, Sylow
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#9000011yescharacteristic
264C22#9001411yescharacteristic
265C22#9001611yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—562no—
271C4—1121no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2#1172964711yescharacteristic
278C2#1172970112yes—
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2842C2
socleC22×C7B(4,𝔽2)
prime radicalC25×C7D4
hypercenter#2842D7
hypocenterC7#12218
LCSG ▷ C2×C7×C22⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2842

Complements

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

NH ≅ G/Ntype
2. #11727470C2semidirect
4. #11725457C2semidirect
247. C7#12218semidirect

Maximal subgroups

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

subgroupindex
C7×#28422
#117274702
#117272512
#117254572
#117254282
#122187