Finite Group Databasebeta

← #13271265

Subgroups — #13271265

Order 1792.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3407C211yesmaximal, characteristic
2#11741252C211yesmaximal, characteristic
3#11729768C211yesmaximal, characteristic
4C2×#89312C212yesmaximal, complemented
5#11725628C212yesmaximal, complemented
6C2×C7×#131C2211yescharacteristic
7#90516—21no—
8C22×Dic28C2211yescharacteristic
9C7×#504—24no—
10C2×C7×M32—22no—
11C2×C7×M32C2212yes—
12#90009—21no—
13#89909C414yes—
14#89892—21no—
15#89888—22no—
16C2×#6393C2212yes—
17#89878C2211yescharacteristic
18C7⋊#338—21no—
19#89312—24no—
20#13078—71nomaximal, Sylow
21C2×C7×#41D411yescharacteristic
22C22×Dic14—21no—
23C7×#131—22no—
24C7×#131—22no—
25C2×C7×M16C2×C412yes—
26C2×C7×M16D411yescharacteristic
27C22×C7×C8D412yes—
28C22×C7×C8C2311yescharacteristic
29#6507—41no—
30#6499—41no—
31#6498—42no—
32#6489—21no—
33#6487—24no—
34C2×#1136—22no—
35C2×Dic28—22no—
36C2×Dic28—22no—
37C2×Dic28—21no—
38C2×Dic28C2×C412yes—
39C7×M32—24no—
40C7×M32—22no—
41C7×M32D414yes—
42C2×C7×C16—24no—
43C2×C7×C16—22no—
44#6410—41no—
45#6407—41no—
46#6393—22no—
47#6393—22no—
48#6392—21no—
49#6391C2×C412yes—
50#6390—24no—
51#3407D711yescharacteristic
52#3059—71no—
53C2×#362—72no—
54#1813—71no—
55#1389—72no—
56C7×#41—22no—
57C7×#41—22no—
58C2×C7×Q8D811yescharacteristic
59C2×C7×D4D811yescharacteristic
60C2×C7×D4SD1612yes—
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C2×D411yescharacteristic
63C22×Dic7—41no—
64C2×Dic14—22no—
65C2×Dic14—42no—
66C2×Dic14—42no—
67C2×Dic14—41no—
68C7×M16—22no—
69C7×M16—24no—
70C2×C7×C8—22no—
71C2×C7×C8—22no—
72C2×C7×C8C22⋊C412yes—
73C2×C7×C8C22×C411yescharacteristic
74C2×C7×C8C2×D412yes—
75C2×C7×C8C2×D411yescharacteristic
76C2×C7×C8#4112yes—
77C22⋊Dic7—44no—
78C22⋊Dic7—41no—
79C4⋊Dic7—22no—
80Dic7⋊C4—44no—
81C4×Dic7—41no—
82#1136—42no—
83#1136—42no—
84C2×C7⋊C8—42no—
85Dic28—24no—
86Dic28—41no—
87Dic28—41no—
88C7×C16—24no—
89C22×Q16—71no—
90C2×#131C2×D711yescharacteristic
91#536—141no—
92#504—24no—
93C2×M32—22no—
94C2×M32C2×D712yes—
95#470—74no—
96#467—142no—
97C2×#108—72no—
98#427—71no—
99#419—141no—
100#413—141no—
101#362—144no—
102#338—141no—
103C23×C7#10212yes—
104C23×C7C2×SD1611yescharacteristic
105C7×Q8—22no—
106C7×D4—24no—
107C7×D4—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7#10212yes—
111C2×C4×C7C2×C22⋊C411yescharacteristic
112C2×C4×C7#12112yes—
113C2×C4×C7C2×D811yescharacteristic
114C2×C4×C7#13512yes—
115C2×Dic7—42no—
116C2×Dic7—42no—
117C2×Dic7—82no—
118Dic14—42no—
119Dic14—42no—
120Dic14—82no—
121C7×C8—22no—
122C7×C8C4×D412yes—
123C7×C8#12011yescharacteristic
124C7×C8#12311yescharacteristic
125C7⋊C8—82no—
126C2×#41D2811yescharacteristic
127C22×Q8—141no—
128C2×Q16—72no—
129C2×Q16—142no—
130C2×Q16—142no—
131C2×Q16—141no—
132#131—22no—
133#131—22no—
134C2×M16—142no—
135C2×M16C4×D712yes—
136C2×M16D2811yescharacteristic
137C22×C8C22⋊D712yes—
138C22×C8C22×D711yescharacteristic
139C4⋊Q8—281no—
140#124—281no—
141#123—282no—
142C22⋊Q8—144no—
143#117—141no—
144M32—24no—
145M32—22no—
146M32C22⋊D714yes—
147C2×C16—24no—
148C2×C16—22no—
149#108—142no—
150#108—142no—
151#107—141no—
152#106—72no—
153#103—144no—
154#101—281no—
155#100—281no—
156C22×C7—22no—
157C22×C7—41no—
158C22×C7—42no—
159C22×C7C2×#10211yescharacteristic
160C22×C7#46512yes—
161C4×C7—22no—
162C4×C7#38611yescharacteristic
163C4×C7#41611yescharacteristic
164C4×C7#46612yes—
165Dic7—82no—
166Dic7—82no—
167#41—22no—
168#41—22no—
169C2×Q8—142no—
170C2×Q8—281no—
171C2×Q8—282no—
172C2×Q8—282no—
173C2×Q8D5611yescharacteristic
174C2×D4#113212yes—
175C2×D4D5611yescharacteristic
176C22×C4—281no—
177C22×C4(C2×D7)⋊C412yes—
178C22×C4C2×D2811yescharacteristic
179Q16—144no—
180Q16—281no—
181Q16—281no—
182M16—22no—
183M16—24no—
184M16—282no—
185M16—282no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—282no—
189C2×C8(C2×D7)⋊C412yes—
190C2×C8C2×C4×D711yescharacteristic
191C2×C8#115712yes—
192C2×C8C2×C22⋊D712yes—
193C2×C8C2×C22⋊D711yescharacteristic
194C4⋊C4—142no—
195C4⋊C4—284no—
196C22⋊C4—281no—
197C22⋊C4—284no—
198C42—281no—
199C16—24no—
200C2×C7—21no—
201C2×C7—22no—
202C2×C7—42no—
203C2×C7#189311yescharacteristic
204C23#639512yes—
205C23C2×#113211yescharacteristic
206Q8—22no—
207Q8—282no—
208Q8—282no—
209Q8—562no—
210D4—24no—
211D4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—282no—
215C2×C4—282no—
216C2×C4—562no—
217C2×C4#639512yes—
218C2×C4C2×D5611yescharacteristic
219C2×C4#646712yes—
220C2×C4C2×(C2×D7)⋊C411yescharacteristic
221C2×C4#649312yes—
222C8—22no—
223C8—562no—
224C8C4×C22⋊D712yes—
225C8#649211yescharacteristic
226C8#651611yescharacteristic
227C7#1307811yescharacteristic, complemented, Sylow
228C22—22no—
229C22—41no—
230C22—42no—
231C22C2×#639511yescharacteristic
232C22#8988612yes—
233C4—22no—
234C4—562no—
235C4—562no—
236C4#8988511yescharacteristic
237C4#8988712yes—
238C4#8998511yescharacteristic
239C2—21no—
240C2—22no—
241C2—42no—
242C2#1172972711yescharacteristic
subgroupG/H
centerC2#11729727
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#3407C2
socleC2×C7#1893
prime radicalC2×C7×#41D4
cosocleC22×C7×C8C23
hypercenter#3407D7
hypocenterC7#13078
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
4. C2×#89312C2semidirect
5. #11725628C2semidirect
227. C7#13078semidirect

Maximal subgroups

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

subgroupindex
C7×#34072
#117412522
#117297682
C2×#893122
#117256282
#130787