Finite Group Databasebeta

← #13267957

Subgroups — #13267957

Order 1792.

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

normal subgroups only

This group has 234 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
1#11729141C211yesmaximal, characteristic
2#11729114C211yesmaximal, characteristic, complemented
3#11727041C211yesmaximal, characteristic
4#11726310C211yesmaximal, characteristic
5C7×#1364C211yesmaximal, characteristic, complemented
6#11725360C211yesmaximal, characteristic, complemented
7#11725280C211yesmaximal, characteristic, complemented
8C7×#500C2211yescharacteristic
9#89853—21no—
10#89851—21no—
11C2×#6374—21no—
12#89606—21no—
13#89471C2211yescharacteristic
14C4×D56C2211yescharacteristic
15C7×C4⋊C16C2211yescharacteristic
16C7×#335C2211yescharacteristic
17#89298—21no—
18#89286—21no—
19#89284—21no—
20#89278—21no—
21C4⋊(C7⋊C16)C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12422—71nomaximal, Sylow
24C7×C4⋊Q8—21no—
25C2×Dic28—21no—
26C2×Dic28—21no—
27C2×D56D411yescharacteristic
28C2×C8×D7—41no—
29C4×D28—21no—
30#6432—21no—
31C2×C7×C16—21no—
32C7×#107—21no—
33C7×#107—21no—
34C7×C4⋊C8—21no—
35C4×C7×C8C2311yescharacteristic
36#6409—41no—
37#6409—41no—
38#6395—21no—
39(C2×D7)⋊C8—41no—
40#6392D411yescharacteristic
41C2×C7⋊C16—21no—
42#6382—42no—
43C4×C7⋊C8D411yescharacteristic
44C4×C7⋊C8D411yescharacteristic
45#6374—42no—
46#2224—71no—
47#2208—71no—
48#1707—71no—
49C8×D8—71no—
50#1367—71no—
51#1364D711yescharacteristic, complemented
52#1362—71no—
53C2×C7×Q8—41no—
54C2×D28—21no—
55C2×C4×D7—41no—
56C2×Dic14—41no—
57C2×C7×C8—41no—
58C2×C7×C8C2×D411yescharacteristic
59C2×C7×C8#4111yescharacteristic
60C7×C4⋊C4—41no—
61C7×C4⋊C4—41no—
62C42×C7C2×D411yescharacteristic
63(C2×D7)⋊C4—41no—
64C4⋊Dic7—21no—
65C4⋊Dic7—41no—
66C4⋊Dic7—41no—
67C2×C7⋊C8—21no—
68C2×C7⋊C8D811yescharacteristic
69C2×C7⋊C8D811yescharacteristic
70Dic28—41no—
71Dic28—41no—
72Dic28—81no—
73D56—21no—
74D56SD1612yes—
75C8×D7—82no—
76C7×C16—41no—
77C7⋊C16—41no—
78C2×SD32—141no—
79#500C2×D711yescharacteristic
80#498—141no—
81#494—71no—
82#487—141no—
83C4×D8—71no—
84C8×D4—141no—
85#366—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—81no—
107Dic14—81no—
108C7×C8—21no—
109C7×C8—81no—
110C7×C8C2×SD1611yescharacteristic
111C7×C8#13511yescharacteristic
112C7⋊C8—22no—
113C7⋊C8—21no—
114C7⋊C8—21no—
115C2×Q16—141no—
116C2×Q16—141no—
117C2×D8—71no—
118C22×C8—281no—
119C4⋊Q8—21no—
120C4⋊Q8—141no—
121C4×D4—141no—
122SD32—282no—
123C2×C16—21no—
124C2×C16—141no—
125#107—21no—
126#107—21no—
127#107—71no—
128#106—282no—
129C4⋊C8—21no—
130#103—281no—
131#103—281no—
132#102—141no—
133C22⋊C8—281no—
134C4×C8—71no—
135C4×C8—71no—
136C4×C8C22×D711yescharacteristic
137C22×C7#45911yescharacteristic
138C2×D7—42no—
139C2×D7—81no—
140C2×D7—81no—
141C4×C7—21no—
142C4×C7—161no—
143C4×C7#46511yescharacteristic
144C4×C7#47111yescharacteristic
145C4×C7#50811yescharacteristic
146C4×C7#50911yescharacteristic
147Dic7—81no—
148Dic7—161no—
149C2×Q8—41no—
150C2×Q8—281no—
151C2×D4—141no—
152C22×C4—281no—
153Q16—281no—
154Q16—281no—
155Q16—561no—
156D8—72no—
157D8—141no—
158C2×C8—41no—
159C2×C8—71no—
160C2×C8—71no—
161C2×C8—141no—
162C2×C8—562no—
163C2×C8#115711yescharacteristic
164C2×C8D4×D711yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—141no—
168C4⋊C4—281no—
169C4⋊C4—281no—
170C22⋊C4—281no—
171C42D4×D711yescharacteristic
172C16—41no—
173C16—281no—
174C2×C7#166511yescharacteristic
175C2×C7#221011yescharacteristic
176C2×C7#221411yescharacteristic
177D7—82no—
178C23—281no—
179Q8—81no—
180Q8—81no—
181Q8—561no—
182Q8—561no—
183D4—142no—
184D4—281no—
185C2×C4—81no—
186C2×C4—281no—
187C2×C4—561no—
188C2×C4—561no—
189C2×C4—561no—
190C2×C4#644711yescharacteristic
191C2×C4D7×D811yescharacteristic
192C2×C4#647711yescharacteristic
193C8—21no—
194C8—81no—
195C8—142no—
196C8—141no—
197C8—141no—
198C8D7×SD1611yescharacteristic
199C8#648211yescharacteristic
200C7#1242211yescharacteristic, complemented, Sylow
201C22—282no—
202C22—561no—
203C22—561no—
204C22#8955811yescharacteristic
205C4—21no—
206C4—161no—
207C4—561no—
208C4—1121no—
209C4#8952611yescharacteristic
210C4#8957911yescharacteristic
211C4#8968311yescharacteristic
212C4#8968811yescharacteristic
213C2—562no—
214C2#1172619411yescharacteristic
215C2#1172695711yescharacteristic
216C2#1172702711yescharacteristic
subgroupG/H
centerC22#89558
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1364C2
socleC22×C7#459
prime radicalC2×D56D4
hypercenter#1364D7
hypocenterC7#12422
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
2. #11729114C2semidirect
5. C7×#1364C2semidirect
6. #11725360C2semidirect
7. #11725280C2semidirect
51. #1364D7semidirect
200. C7#12422semidirect

Maximal subgroups

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

subgroupindex
#117291412
#117291142
#117270412
#117263102
C7×#13642
#117253602
#117252802
#124227