Finite Group Databasebeta

← #13266253

Subgroups — #13266253

Order 1792.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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#11736471C211yesmaximal, characteristic
2C7×#2056C211yesmaximal, characteristic, complemented
3#11729817C212yesmaximal, complemented
4#11725515C211yesmaximal, characteristic, complemented
5#11725291C212yesmaximal, complemented
6C22×C7×Q16C2211yescharacteristic
7#90168—21no—
8C2×C7×C4⋊C8C2211yescharacteristic
9C2×C7×#103—21no—
10C7×#391—21no—
11C2×#6402—22no—
12#89887—21no—
13#89887C2212yes—
14#89618—22no—
15#89615—22no—
16C7×#340C2211yescharacteristic
17#89359—22no—
18C22⋊(C7⋊C16)C2212yes—
19#12161—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×#1157—41no—
22C2×C7×Q16—22no—
23C2×C7×Q16—42no—
24C22×C7×C8D411yescharacteristic
25C22×C7×C8C2311yescharacteristic
26C2×C7×C4⋊C4—21no—
27C2×C42×C7D411yescharacteristic
28#6493—22no—
29#6493—41no—
30C2×D56—21no—
31C2×D56D412yes—
32#6438—41no—
33#6437—21no—
34C4×D28—22no—
35C4×D28—41no—
36#6432—21no—
37C7×C4⋊C8—22no—
38C7×#103—42no—
39C7×#95—41no—
40C7×#95—41no—
41#6402—44no—
42#6395—22no—
43#6395—22no—
44#6392—21no—
45#6392D412yes—
46C2×C7⋊C16—22no—
47#3245—71no—
48#2185—72no—
49#2056D711yescharacteristic, complemented
50#1347—72no—
51#1322—71no—
52C2×C7×Q8—22no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7D811yescharacteristic
61C22×C4×C7D811yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C2×C22⋊D7—42no—
64#1157—82no—
65#1157—82no—
66C2×D28—22no—
67C2×D28—41no—
68C2×D28—41no—
69C2×C4×D7—42no—
70C2×Dic14—41no—
71C7×Q16—24no—
72C7×Q16—42no—
73C2×C7×C8—21no—
74C2×C7×C8—41no—
75C2×C7×C8—41no—
76C2×C7×C8C2×D412yes—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C42×C7—21no—
80C42×C7—21no—
81(C2×D7)⋊C4—42no—
82(C2×D7)⋊C4—42no—
83C4⋊Dic7—22no—
84C4⋊Dic7—41no—
85D56—42no—
86D56—81no—
87C7⋊C16—42no—
88C22×Q16C2×D711yescharacteristic
89#532—141no—
90C2×SD32—142no—
91#474—142no—
92#466—72no—
93#466—141no—
94#459—142no—
95C2×C4⋊C8C2×D711yescharacteristic
96C2×#103—21no—
97#391—21no—
98#358—142no—
99C22⋊C16—72no—
100#340C2×D711yescharacteristic
101C23×C7C2×D811yescharacteristic
102C22×D7—42no—
103C7×Q8—42no—
104C7×Q8—42no—
105C7×Q8—81no—
106C7×Q8—81no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—41no—
113C2×C4×C7—42no—
114C2×C4×C7—81no—
115C2×C4×C7—81no—
116C2×C4×C7—81no—
117C2×C4×C7—81no—
118C2×C4×C7—81no—
119C2×C4×C7SD3212yescomplemented
120C2×C4×C7#12011yescharacteristic
121C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
122C22⋊D7—82no—
123C22⋊D7—82no—
124C2×Dic7—42no—
125D28—42no—
126D28—42no—
127D28—82no—
128D28—82no—
129C4×D7—84no—
130Dic14—82no—
131C7×C8—22no—
132C7×C8—41no—
133C2×#41—281no—
134C22×Q8—21no—
135C2×Q16—22no—
136C2×Q16—42no—
137C2×D8—72no—
138C2×D8—141no—
139C22×C8C22⋊D711yescharacteristic
140C22×C8C22×D711yescharacteristic
141C4⋊Q8—141no—
142#127—141no—
143#124—281no—
144#121—142no—
145#121—281no—
146C4×D4—142no—
147C4×D4—281no—
148C2×C4⋊C4—21no—
149C2×C42C22⋊D711yescharacteristic
150SD32—284no—
151C2×C16—142no—
152#107—72no—
153#107—141no—
154C4⋊C8—22no—
155#103—42no—
156#102—142no—
157#102—142no—
158#95—41no—
159#95—41no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7#44711yescharacteristic
163C22×C7C2×SD3211yescharacteristic
164C22×C7#50911yescharacteristic
165C2×D7—82no—
166C2×D7—82no—
167C2×D7—82no—
168C4×C7—21no—
169C4×C7—22no—
170C4×C7—41no—
171C4×C7—82no—
172C4×C7—161no—
173C4×C7#45411yescharacteristic
174C4×C7B(4,𝔽2)11yescharacteristic
175Dic7—82no—
176#41—562no—
177#41—562no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—281no—
185C2×D4—142no—
186C2×D4—281no—
187C2×D4—281no—
188C2×D4—282no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—282no—
192C22×C4#114111yescharacteristic
193C22×C4#114111yescharacteristic
194C22×C4C2×C22⋊D711yescharacteristic
195Q16—24no—
196Q16—42no—
197D8—282no—
198D8—561no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8D4×D712yes—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—142no—
206C4⋊C4—281no—
207C22⋊C4—282no—
208C22⋊C4—282no—
209C42—21no—
210C42—21no—
211C16—282no—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#159811yescharacteristic
215C2×C7#218711yescharacteristic
216C2×C7#219111yescharacteristic
217D7—162no—
218C23—282no—
219C23C2×#114111yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—81no—
223Q8—81no—
224Q8—562no—
225D4—282no—
226D4—282no—
227D4—562no—
228D4—562no—
229D4—562no—
230D4—562no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—282no—
244C2×C4—564no—
245C2×C4#640212yescomplemented
246C2×C4#650011yescharacteristic
247C2×C4#651211yescharacteristic
248C8—22no—
249C8—41no—
250C7#1216111yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—562no—
254C22—562no—
255C22—562no—
256C22#8980811yescharacteristic
257C22C2×#640211yescharacteristic
258C22#8996511yescharacteristic
259C4—21no—
260C4—22no—
261C4—41no—
262C4—82no—
263C4—161no—
264C4—562no—
265C4#8980411yescharacteristic
266C4#8983411yescharacteristic
267C2—21no—
268C2—21no—
269C2—1122no—
270C2#1172745711yescharacteristic
271C2#1172883911yescharacteristic
272C2#1172884511yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#2056C2
socleC22×C7#447
prime radical#11736471C2
hypercenter#2056D7
hypocenterC7#12161
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. C7×#2056C2semidirect
3. #11729817C2semidirect
4. #11725515C2semidirect
5. #11725291C2semidirect
49. #2056D7semidirect
119. C2×C4×C7SD32semidirect
245. C2×C4#6402semidirect
250. C7#12161semidirect

Maximal subgroups

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

subgroupindex
#117364712
C7×#20562
#117298172
#117255152
#117252912
#121617