Finite Group Databasebeta

← #13267346

Subgroups — #13267346

Order 1792.

306 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3395C211yesmaximal, characteristic
2#11729066C211yesmaximal, characteristic
3#11728696C211yesmaximal, characteristic
4#11728670C211yesmaximal, characteristic
5#11725599C211yesmaximal, characteristic
6Q8⋊(C7⋊C16)C211yesmaximal, characteristic
7#11725571C211yesmaximal, characteristic
8C7×#557—21no—
9C7×#494—21no—
10C7×#494C2211yescharacteristic
11C7×#462—22no—
12C7×#462—21no—
13C7×C8×Q8C2211yescharacteristic
14C4×C7×Q16—21no—
15C4×C7×Q16C2211yescharacteristic
16C2×#6403—21no—
17C2×#6403—22no—
18#89792—21no—
19#89471C2211yescharacteristic
20#89287—21no—
21#89277—21no—
22C4⋊(C7⋊C16)C2211yescharacteristic
23C4×C7⋊C16C2211yescharacteristic
24#89253C2211yescharacteristic
25#12334—71nomaximal, Sylow
26C2×C7×Q16—22no—
27C2×C7×Q16—21no—
28C2×C7×Q16D411yescharacteristic
29C2×C7×Q16D412yes—
30C7×C4⋊Q8—22no—
31C7×C4⋊Q8—21no—
32C7×#125—42no—
33C7×#125—41no—
34C4×C7×Q8—22no—
35C4×C7×Q8—21no—
36C4×C7×Q8—41no—
37C4×C7×Q8D411yescharacteristic
38C2×Dic28—21no—
39C2×Dic28—21no—
40#6432—21no—
41C7×#107D411yescharacteristic
42C7×C4⋊C8—21no—
43C7×C4⋊C8D411yescharacteristic
44C7×#103—22no—
45C7×#103—21no—
46C4×C7×C8—21no—
47C4×C7×C8C2311yescharacteristic
48#6409—41no—
49#6403—44no—
50#6403—42no—
51C2×C7⋊C16—21no—
52C2×C7⋊C16—21no—
53#6382—41no—
54C4⋊(C7⋊C8)—21no—
55#3395D711yescharacteristic
56#2247—71no—
57#2209—71no—
58#1670—71no—
59#1362—71no—
60Q8⋊C16—71no—
61#1332—71no—
62C2×C7×Q8—21no—
63C2×C7×Q8—22no—
64C2×C7×Q8D811yescharacteristic
65C2×Dic14—41no—
66C7×Q16—21no—
67C7×Q16—21no—
68C7×Q16—42no—
69C7×Q16—41no—
70C7×Q16—41no—
71C7×Q16—82no—
72C2×C7×C8—21no—
73C2×C7×C8C2×D411yescharacteristic
74C2×C7×C8C2×D411yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4D811yescharacteristic
84C42×C7—21no—
85C42×C7—42no—
86C42×C7—41no—
87C42×C7C2×D411yescharacteristic
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90C2×C7⋊C8—41no—
91Dic28—42no—
92Dic28—41no—
93Dic28—41no—
94C7⋊C16—22no—
95C7⋊C16—41no—
96#557—21no—
97C2×Q32—141no—
98C2×Q32—142no—
99#494—21no—
100#494—71no—
101#494C2×D711yescharacteristic
102#475—141no—
103#462—22no—
104#462—21no—
105C8×Q8C2×D711yescharacteristic
106C4×Q16—21no—
107C4×Q16C2×D711yescharacteristic
108C4⋊C16—71no—
109#358—141no—
110C4×C16—71no—
111#335—71no—
112#332—141no—
113C7×Q8—21no—
114C7×Q8—41no—
115C7×Q8—42no—
116C7×Q8—42no—
117C7×Q8—41no—
118C7×Q8Q3212yescomplemented
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—41no—
124C2×C4×C7#12011yescharacteristic
125C2×C4×C7C2×D811yescharacteristic
126C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
127C2×Dic7—81no—
128Dic14—81no—
129Dic14—81no—
130C7×C8—21no—
131C7×C8—41no—
132C7×C8#13511yescharacteristic
133C7×C8AGL(1,ℤ/8)11yescharacteristic
134C7⋊C8—81no—
135C2×Q16—22no—
136C2×Q16—21no—
137C2×Q16—141no—
138C2×Q16—141no—
139C2×Q16C22⋊D711yescharacteristic
140C2×Q16C22⋊D712yes—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—141no—
144#125—42no—
145#125—41no—
146C4×Q8—22no—
147C4×Q8—21no—
148C4×Q8—41no—
149C4×Q8C22⋊D711yescharacteristic
150Q32—284no—
151Q32—282no—
152C2×C16—141no—
153C2×C16—141no—
154#107C22⋊D711yescharacteristic
155#106—281no—
156C4⋊C8—21no—
157C4⋊C8—141no—
158C4⋊C8C22⋊D711yescharacteristic
159#103—22no—
160#103—21no—
161#103—281no—
162C4×C8—21no—
163C4×C8C22×D711yescharacteristic
164C22×C7#44711yescharacteristic
165C4×C7—22no—
166C4×C7—21no—
167C4×C7—41no—
168C4×C7—81no—
169C4×C7—82no—
170C4×C7—81no—
171C4×C7#44911yescharacteristic
172C4×C7#45311yescharacteristic
173C4×C7C2×Q3211yescharacteristic
174C4×C7#51011yescharacteristic
175Dic7—161no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—281no—
179C2×Q8#114111yescharacteristic
180Q16—21no—
181Q16—21no—
182Q16—42no—
183Q16—41no—
184Q16—41no—
185Q16—82no—
186Q16—282no—
187Q16—281no—
188Q16—281no—
189C2×C8—21no—
190C2×C8—281no—
191C2×C8C2×C22⋊D711yescharacteristic
192C2×C8C2×C22⋊D711yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—281no—
202C4⋊C4—281no—
203C4⋊C4#114111yescharacteristic
204C42—21no—
205C42—42no—
206C42—41no—
207C42C2×C22⋊D711yescharacteristic
208C16—142no—
209C16—281no—
210C2×C7#162111yescharacteristic
211C2×C7#218811yescharacteristic
212C2×C7#219511yescharacteristic
213Q8—21no—
214Q8—41no—
215Q8—42no—
216Q8—42no—
217Q8—41no—
218Q8—561no—
219Q8—561no—
220Q8#640312yescomplemented
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—561no—
227C2×C4C2×#114111yescharacteristic
228C2×C4#649511yescharacteristic
229C2×C4#651611yescharacteristic
230C8—21no—
231C8—41no—
232C8—561no—
233C8#651211yescharacteristic
234C8#651311yescharacteristic
235C7#1233411yescharacteristic, complemented, Sylow
236C22#8998911yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—81no—
241C4—82no—
242C4—81no—
243C4—1121no—
244C4#8995111yescharacteristic
245C4C2×#640311yescharacteristic
246C4#9001011yescharacteristic
247C4#9001311yescharacteristic
248C2#1172967211yescharacteristic
249C2#1172981811yescharacteristic
250C2#1172987911yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#3395C2
socle, prime radicalC22×C7#447
hypercenter#3395D7
hypocenterC7#12334
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
118. C7×Q8Q32semidirect
220. Q8#6403semidirect
235. C7#12334semidirect

Maximal subgroups

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

subgroupindex
C7×#33952
#117290662
#117286962
#117286702
#117255992
Q8⋊(C7⋊C16)2
#117255712
#123347