Finite Group Databasebeta

← #13258786

Subgroups — #13258786

Order 1792.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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×#1570C211yesmaximal, characteristic
2#11728524C211yesmaximal, characteristic
3#11728273C211yesmaximal, characteristic
4#11728254C211yesmaximal, characteristic
5#11728227C211yesmaximal, characteristic
6#11728175C211yesmaximal, characteristic
7#11725546C211yesmaximal, characteristic
8C2×C4×Dic14C2211yescharacteristic
9C7×#431C2211yescharacteristic
10C7×#423C2211yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×#6396—22no—
13#89697C2211yescharacteristic
14#89696C2211yescharacteristic
15C2×#6377C2211yescharacteristic
16C7×#334—22no—
17#89345—22no—
18#89279—22no—
19#89275—22no—
20#89251—22no—
21#10663—71nomaximal, Sylow
22C22×Dic14Q811yescharacteristic
23C2×C7×M16—21no—
24C22×C7×C8—21no—
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7Q811yescharacteristic
27C2×Dic7⋊C4—21no—
28C2×C4×Dic7—21no—
29C2×#1136—21no—
30C22×C7⋊C8D412yes—
31C4×Dic14—21no—
32C4×Dic14—21no—
33C4×Dic14—21no—
34C4×Dic14—21no—
35C7×C4⋊C8—21no—
36C7×C4⋊C8—21no—
37C7×C4⋊C8—21no—
38C7×C4⋊C8—21no—
39C7×C4⋊C8—21no—
40C7×C22⋊C8—21no—
41C4×C7×C8D412yes—
42C22⋊(C7⋊C8)—21no—
43#6396—44no—
44C4⋊(C7⋊C8)—21no—
45C4⋊(C7⋊C8)—21no—
46C4⋊(C7⋊C8)—21no—
47#6377—21no—
48#6377—21no—
49C4×C7⋊C8D412yes—
50#1985—71no—
51#1849—71no—
52#1570—71no—
53#1570D711yescharacteristic
54#1487—71no—
55#1487—71no—
56#1275—71no—
57C22×C4×C7C2×D411yescharacteristic
58C22×C4×C7C2×Q811yescharacteristic
59C22×C4×C7#4111yescharacteristic
60C22×Dic7—21no—
61C22×Dic7—21no—
62C2×Dic14—21no—
63C2×Dic14—21no—
64C2×Dic14—42no—
65C7×M16—42no—
66C2×C7×C8—22no—
67C2×C7×C8—41no—
68C2×C7×C8—41no—
69C2×C7×C8—42no—
70C42×C7C2×D411yescharacteristic
71C42×C7C2×D411yescharacteristic
72C42×C7#4111yescharacteristic
73C42×C7#4111yescharacteristic
74C4⋊Dic7—21no—
75C4⋊Dic7—21no—
76Dic7⋊C4—41no—
77Dic7⋊C4—41no—
78C4×Dic7—41no—
79C4×Dic7—41no—
80#1136—42no—
81C2×C7⋊C8—22no—
82C2×C7⋊C8—22no—
83C2×C7⋊C8—22no—
84C2×C7⋊C8—41no—
85C2×C7⋊C8SD1614yes—
86C2×C4×Q8—71no—
87#431—71no—
88#431C2×D711yescharacteristic
89C2×#108—142no—
90#423—71no—
91#423C2×D711yescharacteristic
92C2×C4⋊C8C2×D711yescharacteristic
93C2×#97—71no—
94#336—142no—
95#334—22no—
96#334—142no—
97Q8⋊C8—142no—
98Q8⋊C8—142no—
99C23×C7C22⋊Q811yescharacteristic
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7C22⋊Q811yescharacteristic
103C2×C4×C7C22⋊Q811yescharacteristic
104C2×C4×C7#12311yescharacteristic
105C2×C4×C7#12311yescharacteristic
106C2×C4×C7#12311yescharacteristic
107C2×C4×C7#12711yescharacteristic
108C2×C4×C7C2×SD1612yes—
109C2×C4×C7#13712yes—
110C2×Dic7—22no—
111C2×Dic7—41no—
112C2×Dic7—41no—
113C2×Dic7—42no—
114C2×Dic7—41no—
115Dic14—22no—
116Dic14—22no—
117Dic14—41no—
118Dic14—41no—
119C7×C8—42no—
120C7×C8—81no—
121C7×C8—81no—
122C7⋊C8—24no—
123C7⋊C8—42no—
124C7⋊C8—81no—
125C22×Q8—71no—
126C2×M16—21no—
127C2×M16—141no—
128C22×C8—21no—
129C22×C8—72no—
130C4×Q8—141no—
131C4×Q8—141no—
132C4×Q8—141no—
133C4×Q8—141no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—141no—
136C2×C42—141no—
137C2×C42C22×D711yescharacteristic
138#108—284no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—141no—
145C4⋊C8—141no—
146C4⋊C8—141no—
147C22⋊C8—21no—
148C22⋊C8—141no—
149#97—141no—
150#97—141no—
151C4×C8—72no—
152C4×C8C22⋊D712yes—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39911yescharacteristic
156C22×C7#47512yes—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7#48312yes—
162C4×C7#49211yescharacteristic
163C4×C7#49711yescharacteristic
164Dic7—42no—
165Dic7—81no—
166C2×Q8—141no—
167C2×Q8—141no—
168C2×Q8—282no—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4#115711yescharacteristic
172C22×C4D4×D711yescharacteristic
173C22×C4Q8×D711yescharacteristic
174M16—42no—
175M16—282no—
176C2×C8—22no—
177C2×C8—42no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—74no—
181C2×C8—142no—
182C2×C8—142no—
183C2×C8—142no—
184C2×C8—281no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—281no—
188C4⋊C4—281no—
189C42—281no—
190C42—281no—
191C42#115711yescharacteristic
192C42D4×D711yescharacteristic
193C42#116111yescharacteristic
194C42C2×C22⋊D711yescharacteristic
195C2×C7—22no—
196C2×C7#209111yescharacteristic
197C2×C7#209611yescharacteristic
198C2×C7#209811yescharacteristic
199C23#645911yescharacteristic
200Q8—142no—
201Q8—142no—
202Q8—281no—
203Q8—281no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—142no—
207C2×C4—281no—
208C2×C4—281no—
209C2×C4—282no—
210C2×C4—281no—
211C2×C4#645711yescharacteristic
212C2×C4#645711yescharacteristic
213C2×C4#645911yescharacteristic
214C2×C4D7×SD1612yes—
215C2×C4#647812yes—
216C2×C4#649211yescharacteristic
217C2×C4#650311yescharacteristic
218C2×C4#650911yescharacteristic
219C8—42no—
220C8—81no—
221C8—81no—
222C8—144no—
223C8—282no—
224C8—561no—
225C7#1066311yescharacteristic, complemented, Sylow
226C22—21no—
227C22—21no—
228C22#8962612yes—
229C22#8976411yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—282no—
235C4—561no—
236C4#8963412yes—
237C4#8993911yescharacteristic
238C4#8994711yescharacteristic
239C2—22no—
240C2#1172830311yescharacteristic
241C2#1172834411yescharacteristic
242C2#1172848311yescharacteristic
subgroupG/H
centerC2×C4#6503
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1570C2
socleC22×C7#399
prime radicalC23×C7C22⋊Q8
hypercenter#1570D7
hypocenterC7#10663
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
225. C7#10663semidirect

Maximal subgroups

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

subgroupindex
C7×#15702
#117285242
#117282732
#117282542
#117282272
#117281752
#117255462
#106637