Finite Group Databasebeta

← #13263234

Subgroups — #13263234

Order 1792.

280 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 75 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#11735965C211yesmaximal, characteristic
2C7×#2165C211yesmaximal, characteristic, complemented
3#11727618C211yesmaximal, characteristic, complemented
4#11725339C212yesmaximal, complemented
5#11725337C212yesmaximal
6C4×#1157C411yescharacteristic
7C4×C7×M16C2211yescharacteristic
8#89901C411yescharacteristic
9#89901C2211yescharacteristic
10#89891C411yescharacteristic
11#89891C2211yescharacteristic
12C2×#6377C411yescharacteristic
13#89477—21no—
14#89477—21no—
15#89462—21no—
16#89462—21no—
17C7×#364—21no—
18C7×#349C2212yes—
19C7×#347—21no—
20#89348C2212yes—
21#89261—21no—
22#89258—21no—
23#11523—71nomaximal, Sylow
24C2×#1157C2×C411yescharacteristic
25C2×C7×M16C2×C411yescharacteristic
26C2×C7×M16C2311yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C4×C22⋊D7—22no—
29#6489C2×C411yescharacteristic
30C22×C7⋊C8C2×C411yescharacteristic
31C22×C7⋊C8C2×C411yescharacteristic
32C4×D28—21no—
33#6435—21no—
34C42×D7—21no—
35C4×Dic14—21no—
36C7×M32—22no—
37C7×#97D411yescharacteristic
38C7×#97D411yescharacteristic
39C4×C7×C8D411yescharacteristic
40C4×C7×C8D411yescharacteristic
41(C2×D7)⋊C8—21no—
42(C2×D7)⋊C8—21no—
43(C2×D7)⋊C8—21no—
44(C2×D7)⋊C8—21no—
45Dic7⋊C8—21no—
46Dic7⋊C8—21no—
47Dic7⋊C8—21no—
48Dic7⋊C8—21no—
49#6386—22no—
50#6377—21no—
51#6377—21no—
52#2923—71no—
53#2165—71no—
54#2165D711yescharacteristic, complemented
55#1328—72no—
56#1321—72no—
57C22×C4×C7—21no—
58C22×C4×C7C22×C411yescharacteristic
59C2×C22⋊D7—21no—
60#1157—41no—
61#1157—41no—
62C2×D28M1611yescharacteristic
63C2×C4×D7—21no—
64C2×C4×D7—22no—
65C2×Dic14M1611yescharacteristic
66C7×M16—41no—
67C7×M16—41no—
68C2×C7×C8C22⋊C411yescharacteristic
69C2×C7×C8C22⋊C411yescharacteristic
70C2×C7×C8C2×D411yescharacteristic
71C2×C7×C8C2×D411yescharacteristic
72C42×C7—21no—
73C42×C7C22⋊C411yescharacteristic
74C42×C7C22⋊C411yescharacteristic
75C22⋊Dic7M1612yes—
76(C2×D7)⋊C4—22no—
77C4⋊Dic7—21no—
78Dic7⋊C4—22no—
79C4×Dic7—22no—
80C4×Dic7—21no—
81C2×C7⋊C8—21no—
82C2×C7⋊C8—21no—
83C2×C7⋊C8—41no—
84C2×C7⋊C8—41no—
85C7×C16—42no—
86C7⋊C16—42no—
87C4×#41—71no—
88#433—141no—
89#433—141no—
90#432—141no—
91#432—141no—
92#424—71no—
93#424—71no—
94#408—71no—
95#408—71no—
96C4×M16C2×D711yescharacteristic
97C2×#97—71no—
98#364—21no—
99#364—141no—
100#349—72no—
101#349C2×D712yes—
102#347—21no—
103#347—141no—
104C23×C7C2×M1611yescharacteristic
105C22×D7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—81no—
111C2×C4×C7C2×C22⋊C411yescharacteristic
112C2×C4×C7C2×M1611yescharacteristic
113C22⋊D7—42no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116D28—41no—
117C4×D7—41no—
118C4×D7—41no—
119C4×D7—82no—
120Dic14—41no—
121C7×C8—21no—
122C7×C8—21no—
123C7×C8—21no—
124C7×C8—21no—
125C7⋊C8—41no—
126C7⋊C8—41no—
127C2×#41—71no—
128C2×M16C4×D711yescharacteristic
129C2×M16C22×D711yescharacteristic
130C22×C8—71no—
131C22×C8—71no—
132C4×Q8—141no—
133C4×D4—141no—
134C4×D4—142no—
135#117—71no—
136#117—141no—
137C2×C42—141no—
138C2×C42C4×D711yescharacteristic
139M32—22no—
140M32—142no—
141C4⋊C8—141no—
142C4⋊C8—141no—
143C4⋊C8—141no—
144C4⋊C8—141no—
145C22⋊C8—141no—
146C22⋊C8—141no—
147C22⋊C8—141no—
148C22⋊C8—141no—
149#97—141no—
150#97—141no—
151#97D2811yescharacteristic
152#97C22⋊D711yescharacteristic
153C4×C8D2811yescharacteristic
154C4×C8C22⋊D711yescharacteristic
155C22×C7—21no—
156C22×C7—41no—
157C22×C7#40711yescharacteristic
158C2×D7—42no—
159C2×D7—41no—
160C4×C7—21no—
161C4×C7—41no—
162C4×C7—41no—
163C4×C7#41011yescharacteristic
164C4×C7#41311yescharacteristic
165Dic7—42no—
166Dic7—41no—
167#41—281no—
168#41—281no—
169C2×Q8—71no—
170C2×D4—71no—
171C2×D4—141no—
172C22×C4—21no—
173C22×C4—142no—
174C22×C4—141no—
175C22×C4C2×C4×D711yescharacteristic
176M16—41no—
177M16—41no—
178C2×C8—141no—
179C2×C8—141no—
180C2×C8—281no—
181C2×C8—281no—
182C2×C8(C2×D7)⋊C411yescharacteristic
183C2×C8(C2×D7)⋊C411yescharacteristic
184C2×C8C2×D2811yescharacteristic
185C2×C8C2×C22⋊D711yescharacteristic
186C4⋊C4—142no—
187C4⋊C4—141no—
188C22⋊C4—72no—
189C22⋊C4—142no—
190C42—21no—
191C42—142no—
192C42—141no—
193C42(C2×D7)⋊C411yescharacteristic
194C42(C2×D7)⋊C411yescharacteristic
195C16—42no—
196C16—282no—
197C2×C7—21no—
198C2×C7—41no—
199C2×C7#146411yescharacteristic
200D7—81no—
201C23—141no—
202C23D7×M1611yescharacteristic
203Q8—281no—
204D4—281no—
205D4—282no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—81no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4—281no—
214C2×C4—281no—
215C2×C4—562no—
216C2×C4D7×M1611yescharacteristic
217C2×C4C2×(C2×D7)⋊C411yescharacteristic
218C8—21no—
219C8—21no—
220C8—21no—
221C8—21no—
222C8—281no—
223C8—281no—
224C7#1152311yescharacteristic, complemented, Sylow
225C22—21no—
226C22—41no—
227C22—282no—
228C22—281no—
229C22#8989811yescharacteristic
230C4—21no—
231C4—41no—
232C4—41no—
233C4—282no—
234C4—281no—
235C4#8977811yescharacteristic
236C4#8990411yescharacteristic
237C2—21no—
238C2—41no—
239C2—561no—
240C2#1172858011yescharacteristic
subgroupG/H
centerC4#89778
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#2165C2
socleC2×C7#1464
prime radicalC2×#1157C2×C4
cosocleC2×C7×M16C23
hypercenter#2165D7
hypocenterC7#11523
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. C7×#2165C2semidirect
3. #11727618C2semidirect
4. #11725339C2semidirect
54. #2165D7semidirect
224. C7#11523semidirect

Maximal subgroups

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

subgroupindex
#117359652
C7×#21652
#117276182
#117253392
#117253372
#115237