Finite Group Databasebeta

← #13246787

Subgroups — #13246787

Order 1792.

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

normal subgroups only

This group has 388 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
1#11736486C211yesmaximal, characteristic
2#11728565C212yesmaximal
3#11728548C212yesmaximal, complemented
4C7×#1307C211yesmaximal, characteristic, complemented
5#11725522C211yesmaximal, characteristic, complemented
6C2×#6470C412yes—
7#90168C2211yescharacteristic
8C7×#432C2212yes—
9C7×#423C2211yescharacteristic
10#89901C412yes—
11#89698C2212yes—
12C2×C4×C7⋊C8C2211yescharacteristic
13#89610—22no—
14#89606—22no—
15#89275—22no—
16#89274—22no—
17#89259—24no—
18#8630—71nomaximal, Sylow
19C2×#1157C2×C412yes—
20C2×C7×M16C2×C412yes—
21C2×C42×C7C2311yescharacteristic
22#6493—22no—
23C22×C7⋊C8C2×C411yescharacteristic
24C22×C7⋊C8C2×C411yescharacteristic
25#6470—44no—
26C2×#1131—22no—
27C2×C8×D7—22no—
28#6438D412yes—
29#6437Q811yescharacteristic
30C4×D28—22no—
31#6432Q811yescharacteristic
32C7×C4⋊C8—22no—
33C7×C4⋊C8—22no—
34C7×C22⋊C8—22no—
35C7×#97—22no—
36C22⋊(C7⋊C8)—22no—
37(C2×D7)⋊C8—24no—
38C4⋊(C7⋊C8)—22no—
39#6377—22no—
40C4×C7⋊C8D412yes—
41C4×C7⋊C8D412yes—
42#2931—71no—
43#1568—72no—
44#1536—72no—
45#1307—71no—
46#1307D711yescharacteristic, complemented
47C22×C4×C7C22×C411yescharacteristic
48C22×C4×C7#4112yes—
49C2×C22⋊D7—22no—
50#1157—42no—
51#1157—42no—
52C2×D28—22no—
53C2×D28C4⋊C412yes—
54C2×C4×D7—22no—
55C2×Dic14C4⋊C412yes—
56C7×M16—42no—
57C2×C7×C8—22no—
58C2×C7×C8—42no—
59C42×C7C2×D412yes—
60C42×C7C2×D411yescharacteristic
61C42×C7C2×Q811yescharacteristic
62(C2×D7)⋊C4—24no—
63C4⋊Dic7—22no—
64C2×C7⋊C8—22no—
65C2×C7⋊C8—41no—
66C2×C7⋊C8—42no—
67C2×C7⋊C8C22⋊C411yescharacteristic
68C2×C7⋊C8C22⋊C412yes—
69C2×C7⋊C8C22⋊C411yescharacteristic
70#1131—44no—
71C8×D7—44no—
72C2×#131—72no—
73#532—71no—
74#436—142no—
75C8×D4—142no—
76#432—72no—
77#432C2×D712yes—
78#423C2×D711yescharacteristic
79#408—72no—
80C2×C4×C8—71no—
81#335—142no—
82#334—142no—
83#329—144no—
84C23×C7#11711yescharacteristic
85C22×D7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—42no—
89C2×C4×C7C2×C22⋊C411yescharacteristic
90C2×C4×C7C2×C4⋊C411yescharacteristic
91C2×C4×C7#12112yes—
92C2×C4×C7C22⋊Q812yes—
93C22⋊D7—44no—
94C2×Dic7—22no—
95D28—42no—
96D28—44no—
97C4×D7—42no—
98C4×D7—42no—
99Dic14—42no—
100C7×C8—42no—
101C7×C8—82no—
102C7⋊C8—21no—
103C7⋊C8—22no—
104C7⋊C8—82no—
105C7⋊C8#10012yes—
106C2×#41—72no—
107#131—284no—
108C2×M16—142no—
109C2×M16C4×D712yes—
110C22×C8—71no—
111C22×C8—71no—
112C22×C8—142no—
113C4⋊Q8—71no—
114#127—71no—
115#124—72no—
116#121—142no—
117C4×D4—142no—
118C2×C42C22×D711yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—142no—
122C22⋊C8—22no—
123C22⋊C8—144no—
124C22⋊C8—142no—
125#97—22no—
126#97—142no—
127C4×C8—72no—
128C4×C8—72no—
129C22×C7—21no—
130C22×C7—22no—
131C22×C7#38011yescharacteristic
132C2×D7—42no—
133C2×D7—44no—
134C4×C7—21no—
135C4×C7—22no—
136C4×C7—22no—
137C4×C7C2×#10011yescharacteristic
138C4×C7#41311yescharacteristic
139Dic7—42no—
140#41—282no—
141#41—282no—
142C2×Q8—72no—
143C2×D4—72no—
144C2×D4—142no—
145C2×D4—142no—
146C22×C4—142no—
147C22×C4C2×C4×D711yescharacteristic
148C22×C4#115712yes—
149M16—42no—
150M16—284no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—71no—
154C2×C8—72no—
155C2×C8—71no—
156C2×C8—142no—
157C2×C8—284no—
158C2×C8—281no—
159C2×C8—282no—
160C4⋊C4—142no—
161C22⋊C4—144no—
162C42D4×D712yes—
163C42D4×D711yescharacteristic
164C42Q8×D711yescharacteristic
165C2×C7—41no—
166C2×C7#180011yescharacteristic
167C2×C7#181412yes—
168D7—82no—
169C23—142no—
170C23#643511yescharacteristic
171Q8—282no—
172D4—282no—
173D4—284no—
174D4—284no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—142no—
179C2×C4—282no—
180C2×C4—282no—
181C2×C4D7×C22⋊C411yescharacteristic
182C2×C4#644712yes—
183C2×C4D7×C4⋊C411yescharacteristic
184C2×C4#645912yes—
185C8—42no—
186C8—72no—
187C8—82no—
188C8—141no—
189C8—142no—
190C8—562no—
191C7#863011yescharacteristic, complemented, Sylow
192C22—21no—
193C22—22no—
194C22—282no—
195C22—284no—
196C22#8943911yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—282no—
201C4D7×#10011yescharacteristic
202C4#8952311yescharacteristic
203C2—41no—
204C2—562no—
205C2#1172674811yescharacteristic
206C2#1172678612yes—
subgroupG/H
centerC22#89439
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1307C2
socleC22×C7#380
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#1307D7
hypocenterC7#8630
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
3. #11728548C2semidirect
4. C7×#1307C2semidirect
5. #11725522C2semidirect
46. #1307D7semidirect
191. C7#8630semidirect

Maximal subgroups

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

subgroupindex
#117364862
#117285652
#117285482
C7×#13072
#117255222
#86307