Finite Group Databasebeta

← #13241530

Subgroups — #13241530

Order 1792.

262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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×#1558C211yesmaximal, characteristic
2C2×#89343C211yesmaximal, characteristic
3#11729538C211yesmaximal, characteristic
4C2×C7×C4⋊Q8C2211yescharacteristic
5#90503C411yescharacteristic
6C7×#432—21no—
7C2×#6410—21no—
8C2×#6410C411yescharacteristic
9#89997—21no—
10C7×#333—21no—
11C7×#333C412yescomplemented
12#89343—21no—
13#89343C412yescomplemented
14#89318—21no—
15#7483—71nomaximal, Sylow
16C22×C7×Q8C2×C411yescharacteristic
17C22×C7×Q8D411yescharacteristic
18C7×C4⋊Q8—41no—
19C7×C4⋊Q8C2×C411yescharacteristic
20C7×C4⋊Q8C2×C411yescharacteristic
21C7×C4⋊Q8D411yescharacteristic
22C7×C4⋊Q8Q811yescharacteristic
23C2×C7×C4⋊C4—41no—
24C2×C42×C7D411yescharacteristic
25Q8×Dic7—42no—
26#6489—21no—
27#6489—41no—
28C2×#1136—21no—
29C2×#1136—41no—
30C7×C4⋊C8—22no—
31C7×C22⋊C8—41no—
32C7×#97—41no—
33#6410—22no—
34#6410—41no—
35#6410—41no—
36#6409—42no—
37C2×#356—71no—
38#1886—71no—
39#1558D711yescharacteristic
40C2×C7×Q8—21no—
41C2×C7×Q8—21no—
42C2×C7×Q8—41no—
43C2×C7×Q8—41no—
44C2×C7×Q8—41no—
45C2×C7×Q8—41no—
46C2×C7×Q8C4211yescharacteristic
47C2×C7×Q8C22⋊C411yescharacteristic
48C2×C7×Q8C4⋊C411yescharacteristic
49C2×C7×Q8C4⋊C411yescharacteristic
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7C22⋊C411yescharacteristic
54C2×C7×C8—42no—
55C7×C4⋊C4—41no—
56C7×C4⋊C4—41no—
57C7×C4⋊C4—41no—
58C7×C4⋊C4—41no—
59C42×C7—21no—
60C42×C7C22⋊C411yescharacteristic
61C42×C7C4⋊C411yescharacteristic
62C22⋊Dic7—41no—
63C4⋊Dic7—22no—
64C4⋊Dic7—42no—
65C4×Dic7—41no—
66C4×Dic7—42no—
67#1136—42no—
68#1136—41no—
69#1136—41no—
70#1136—81no—
71C2×C7⋊C8—41no—
72C2×C7⋊C8—81no—
73C2×C4⋊Q8C2×D711yescharacteristic
74#519—71no—
75#432—21no—
76#419—141no—
77C2×#101—71no—
78C2×#101—141no—
79#356—72no—
80#356—141no—
81#339—141no—
82#333—21no—
83#333Dic712yescomplemented
84C23×C7#9911yescharacteristic
85C7×Q8—41no—
86C7×Q8—41no—
87C7×Q8—41no—
88C7×Q8—41no—
89C7×Q8—42no—
90C7×Q8—81no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—42no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7#9511yescharacteristic
105C2×C4×C7#9911yescharacteristic
106C2×Dic7—42no—
107C7×C8—82no—
108C7⋊C8—81no—
109C7⋊C8—82no—
110C22×Q8C4×D711yescharacteristic
111C22×Q8D2811yescharacteristic
112C2×M16—141no—
113C2×M16—281no—
114C4⋊Q8—41no—
115C4⋊Q8Dic1411yescharacteristic
116C4⋊Q8C4×D711yescharacteristic
117C4⋊Q8C2×Dic711yescharacteristic
118C4⋊Q8C22⋊D711yescharacteristic
119C4×Q8—282no—
120#117—141no—
121#117—281no—
122C2×C4⋊C4—41no—
123C2×C42C22⋊D711yescharacteristic
124C4⋊C8—22no—
125#103—282no—
126#101—142no—
127#101—281no—
128#101—281no—
129C22⋊C8—41no—
130#97—41no—
131C22×C7—41no—
132C22×C7#34211yescharacteristic
133C22×C7AΣL(1,𝔽16)11yescharacteristic
134C22×C7#35211yescharacteristic
135C4×C7—22no—
136C4×C7—21no—
137C4×C7—21no—
138C4×C7—41no—
139C4×C7—41no—
140C4×C7—41no—
141C4×C7—41no—
142C4×C7—41no—
143Dic7—82no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8C4×Dic711yescharacteristic
151C2×Q8Dic7⋊C411yescharacteristic
152C2×Q8C4⋊Dic711yescharacteristic
153C2×Q8(C2×D7)⋊C411yescharacteristic
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4(C2×D7)⋊C411yescharacteristic
158M16—282no—
159M16—281no—
160M16—281no—
161M16—561no—
162C2×C8—42no—
163C2×C8—281no—
164C2×C8—561no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—142no—
170C4⋊C4—282no—
171C22⋊C4—281no—
172C42—21no—
173C42—281no—
174C42—282no—
175C42Dic7⋊C411yescharacteristic
176C42C22⋊Dic711yescharacteristic
177C2×C7—21no—
178C2×C7—21no—
179C2×C7—21no—
180C2×C7#139011yescharacteristic
181C23#638011yescharacteristic
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—42no—
187Q8—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—282no—
202C2×C4#638011yescharacteristic
203C2×C4C7⋊#9511yescharacteristic
204C8—82no—
205C8—561no—
206C8—562no—
207C7#748311yescharacteristic, complemented, Sylow
208C22—41no—
209C22#8927011yescharacteristic
210C22#8927111yescharacteristic
211C22#8932211yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—562no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#1172550411yescharacteristic
subgroupG/H
centerC2#11725504
commutatorC2×C7×Q8C42
FrattiniC2×C4⋊Q8C2×D7
FittingC7×#1558C2
socleC2×C7#1390
prime radicalC23×C7#99
cosocleC2×C7×C4⋊Q8C22
hypercenter#1558D7
hypocenterC7#7483
LCSG ▷ C2×C7×Q8 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1558

Complements

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

NH ≅ G/Ntype
11. C7×#333C4semidirect
13. #89343C4semidirect
83. #333Dic7semidirect
207. C7#7483semidirect

Maximal subgroups

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

subgroupindex
C7×#15582
C2×#893432
#117295382
#74837