Finite Group Databasebeta

← #13267939

Subgroups — #13267939

Order 1792.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#11729069C211yesmaximal, characteristic
2#11729050C211yesmaximal, characteristic, complemented
3#11727044C211yesmaximal, characteristic
4#11726310C211yesmaximal, characteristic
5C7×#1362C211yesmaximal, characteristic, complemented
6#11725360C211yesmaximal, characteristic, complemented
7#11725282C211yesmaximal, characteristic, complemented
8C7×#494C2211yescharacteristic
9C2×#6402—21no—
10#89865—21no—
11#89851—21no—
12#89606—21no—
13C4×D56C2211yescharacteristic
14#89454C2211yescharacteristic
15C7×C4⋊C16C2211yescharacteristic
16C7×#335C2211yescharacteristic
17#89359—21no—
18#89304—21no—
19#89300—21no—
20#89278—21no—
21C4⋊(C7⋊C16)C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12420—71nomaximal, Sylow
24C2×C7×Q16—21no—
25C2×C7×Q16—21no—
26C7×C4⋊Q8—21no—
27C2×D56D411yescharacteristic
28C2×C8×D7—41no—
29C4×D28—21no—
30#6432—21no—
31C2×C7×C16—21no—
32C7×C4⋊C8—21no—
33C4×C7×C8C2311yescharacteristic
34#6402—42no—
35#6395—21no—
36(C2×D7)⋊C8—41no—
37#6392—21no—
38#6392—21no—
39#6392D411yescharacteristic
40C2×C7⋊C16—21no—
41#6384—42no—
42#6382—41no—
43#6382—41no—
44C4×C7⋊C8D411yescharacteristic
45C4×C7⋊C8D411yescharacteristic
46#2226—71no—
47#2208—71no—
48#1707—71no—
49C8×D8—71no—
50#1367—71no—
51#1365—71no—
52#1362D711yescharacteristic, complemented
53C2×C7×Q8—41no—
54C2×D28—21no—
55C2×C4×D7—41no—
56C2×Dic14—41no—
57C7×Q16—41no—
58C7×Q16—41no—
59C7×Q16—81no—
60C2×C7×C8—41no—
61C2×C7×C8C2×D411yescharacteristic
62C2×C7×C8#4111yescharacteristic
63C7×C4⋊C4—41no—
64C7×C4⋊C4—41no—
65C42×C7C2×D411yescharacteristic
66(C2×D7)⋊C4—41no—
67C4⋊Dic7—21no—
68C4⋊Dic7—41no—
69C4⋊Dic7—41no—
70C2×C7⋊C8—21no—
71C2×C7⋊C8D811yescharacteristic
72C2×C7⋊C8D811yescharacteristic
73D56—21no—
74D56SD1612yes—
75C8×D7—82no—
76C7×C16—41no—
77C7⋊C16—41no—
78C2×SD32—141no—
79#500—71no—
80#498—141no—
81#494C2×D711yescharacteristic
82#487—141no—
83C4×D8—71no—
84C8×D4—141no—
85#367—141no—
86C4⋊C16—71no—
87C4⋊C16C2×D711yescharacteristic
88#358—141no—
89#357—141no—
90#335C2×D711yescharacteristic
91D4⋊C8—141no—
92C82—71no—
93C22×D7—41no—
94C7×Q8—81no—
95C7×Q8—81no—
96C2×C4×C7—81no—
97C2×C4×C7#12111yescharacteristic
98C2×C4×C7C2×D811yescharacteristic
99C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
100C2×Dic7—41no—
101C2×Dic7—81no—
102D28—22no—
103D28—41no—
104C4×D7—81no—
105C4×D7—81no—
106Dic14—82no—
107C7×C8—21no—
108C7×C8—81no—
109C7×C8C2×SD1611yescharacteristic
110C7×C8#13511yescharacteristic
111C7⋊C8—21no—
112C7⋊C8—41no—
113C7⋊C8SD3212yescomplemented
114C2×Q16—21no—
115C2×Q16—21no—
116C2×D8—71no—
117C22×C8—281no—
118C4⋊Q8—21no—
119C4⋊Q8—141no—
120C4×D4—141no—
121SD32—282no—
122C2×C16—21no—
123C2×C16—141no—
124#107—71no—
125#107—141no—
126#107—141no—
127#106—281no—
128#106—281no—
129C4⋊C8—21no—
130#103—282no—
131#102—141no—
132C22⋊C8—281no—
133C4×C8—71no—
134C4×C8—71no—
135C4×C8C22×D711yescharacteristic
136C22×C7#45911yescharacteristic
137C2×D7—42no—
138C2×D7—81no—
139C2×D7—81no—
140C4×C7—21no—
141C4×C7—161no—
142C4×C7#46511yescharacteristic
143C4×C7#47111yescharacteristic
144C4×C7C2×SD3211yescharacteristic
145C4×C7#51011yescharacteristic
146Dic7—81no—
147Dic7—161no—
148C2×Q8—41no—
149C2×Q8—281no—
150C2×D4—141no—
151C22×C4—281no—
152Q16—41no—
153Q16—41no—
154Q16—81no—
155D8—72no—
156D8—141no—
157C2×C8—41no—
158C2×C8—71no—
159C2×C8—71no—
160C2×C8—141no—
161C2×C8—562no—
162C2×C8#115711yescharacteristic
163C2×C8D4×D711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—141no—
167C4⋊C4—281no—
168C4⋊C4—281no—
169C22⋊C4—281no—
170C42D4×D711yescharacteristic
171C16—41no—
172C16—281no—
173C2×C7#166511yescharacteristic
174C2×C7#220811yescharacteristic
175C2×C7#221211yescharacteristic
176D7—82no—
177C23—281no—
178Q8—81no—
179Q8—81no—
180Q8—562no—
181D4—142no—
182D4—281no—
183C2×C4—81no—
184C2×C4—281no—
185C2×C4—561no—
186C2×C4—561no—
187C2×C4—561no—
188C2×C4#644711yescharacteristic
189C2×C4D7×D811yescharacteristic
190C2×C4#647711yescharacteristic
191C8—21no—
192C8—72no—
193C8—81no—
194C8—141no—
195C8—281no—
196C8#647511yescharacteristic
197C8D7×SD1611yescharacteristic
198C7#1242011yescharacteristic, complemented, Sylow
199C22—282no—
200C22—561no—
201C22—561no—
202C22#8955811yescharacteristic
203C4—21no—
204C4—161no—
205C4—561no—
206C4—1121no—
207C4#8952011yescharacteristic
208C4#8955011yescharacteristic
209C4D7×SD3211yescharacteristic
210C4#8969011yescharacteristic
211C2—562no—
212C2#1172614811yescharacteristic
213C2#1172698811yescharacteristic
214C2#1172702611yescharacteristic
subgroupG/H
centerC22#89558
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1362C2
socleC22×C7#459
prime radicalC2×D56D4
hypercenter#1362D7
hypocenterC7#12420
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
2. #11729050C2semidirect
5. C7×#1362C2semidirect
6. #11725360C2semidirect
7. #11725282C2semidirect
52. #1362D7semidirect
113. C7⋊C8SD32semidirect
198. C7#12420semidirect

Maximal subgroups

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

subgroupindex
#117290692
#117290502
#117270442
#117263102
C7×#13622
#117253602
#117252822
#124207