Finite Group Databasebeta

← #13263237

Subgroups — #13263237

Order 1792.

288 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 69 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#11736093C211yesmaximal, characteristic
2#11726868C212yesmaximal, complemented
3C7×#1321C211yesmaximal, characteristic
4#11725496C211yesmaximal, characteristic
5#11725339C212yesmaximal, complemented
6#41×Dic7C411yescharacteristic
7C7×#408C2211yescharacteristic
8#90006C411yescharacteristic
9#89891C2211yescharacteristic
10M16×Dic7C2211yescharacteristic
11#89874C411yescharacteristic
12C2×C8×Dic7C411yescharacteristic
13#89492—22no—
14#89490—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11524—71nomaximal, Sylow
20C2×C7×#41C2×C411yescharacteristic
21C2×C7×M16C2311yescharacteristic
22C22×C7×C8C2×C411yescharacteristic
23Q8×Dic7—21no—
24D4×Dic7—21no—
25D4×Dic7—22no—
26#6489—21no—
27#6489C2×C411yescharacteristic
28C2×C4×Dic7—21no—
29C2×C4×Dic7C2×C411yescharacteristic
30C2×#1136C2×C411yescharacteristic
31C22×C7⋊C8C2×C411yescharacteristic
32C7×M32—22no—
33C7×C22⋊C8—22no—
34C22⋊(C7⋊C8)—22no—
35#6389D412yes—
36Dic7⋊C8—22no—
37Dic7⋊C8—22no—
38C8×Dic7—21no—
39C8×Dic7—21no—
40C8×Dic7D412yes—
41#6386—22no—
42#2922—71no—
43#2165—72no—
44#1328—72no—
45#1321—71no—
46#1321D711yescharacteristic
47C7×#41—41no—
48C7×#41—41no—
49C2×C7×Q8M1611yescharacteristic
50C2×C7×D4—21no—
51C2×C7×D4M1611yescharacteristic
52C22×C4×C7—21no—
53C22×C4×C7C22×C411yescharacteristic
54C22×Dic7—22no—
55C22×Dic7—21no—
56C7×M16—41no—
57C2×C7×C8—21no—
58C2×C7×C8—41no—
59C2×C7×C8C2×D412yes—
60C22⋊Dic7—22no—
61C22⋊Dic7M1612yes—
62C4⋊Dic7—22no—
63C4⋊Dic7—21no—
64C4×Dic7—22no—
65C4×Dic7—21no—
66C4×Dic7—21no—
67C4×Dic7C22⋊C411yescharacteristic
68C4×Dic7C22⋊C411yescharacteristic
69#1136—41no—
70C2×C7⋊C8—21no—
71C2×C7⋊C8—41no—
72C2×C7⋊C8C22⋊C412yes—
73C7×C16—42no—
74C7⋊C16—42no—
75C4×#41—71no—
76#433—142no—
77#431—142no—
78#424—71no—
79#424—71no—
80#408—71no—
81#408C2×D711yescharacteristic
82C4×M16—71no—
83C2×C4×C8—71no—
84#364—142no—
85#349—72no—
86#349C2×D712yes—
87#347—142no—
88C23×C7—21no—
89C23×C7C2×M1611yescharacteristic
90C7×Q8—41no—
91C7×D4—41no—
92C7×D4—42no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—41no—
97C2×C4×C7—41no—
98C2×C4×C7C2×C22⋊C411yescharacteristic
99C2×C4×C7C2×M1611yescharacteristic
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—42no—
104C2×Dic7—44no—
105C7×C8—22no—
106C7×C8—41no—
107C7⋊C8—22no—
108C7⋊C8—41no—
109C2×#41C4×D711yescharacteristic
110C2×M16—71no—
111C2×M16C22×D711yescharacteristic
112C22×C8—71no—
113C22×C8C4×D711yescharacteristic
114C4×Q8—141no—
115C4×D4—142no—
116C4×D4—141no—
117#117—71no—
118#117—141no—
119C2×C42—71no—
120C2×C42—141no—
121M32—22no—
122M32—142no—
123C4⋊C8—142no—
124C4⋊C8—142no—
125C22⋊C8—22no—
126C22⋊C8—142no—
127#97—72no—
128C4×C8—72no—
129C4×C8—141no—
130C4×C8—141no—
131C22×C7—21no—
132C22×C7—41no—
133C22×C7—42no—
134C22×C7—41no—
135C22×C7#40711yescharacteristic
136C4×C7—21no—
137C4×C7—41no—
138C4×C7#41011yescharacteristic
139C4×C7#41311yescharacteristic
140Dic7—22no—
141Dic7—42no—
142Dic7—41no—
143#41—41no—
144#41—41no—
145C2×Q8#113111yescharacteristic
146C2×D4—21no—
147C2×D4#113111yescharacteristic
148C22×C4—21no—
149C22×C4—142no—
150C22×C4—141no—
151C22×C4C2×C4×D711yescharacteristic
152M16—41no—
153M16—281no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—72no—
157C2×C8—141no—
158C2×C8—281no—
159C2×C8D4×D712yes—
160C4⋊C4—142no—
161C4⋊C4—141no—
162C22⋊C4—72no—
163C22⋊C4—142no—
164C42—71no—
165C42—71no—
166C42—142no—
167C42—141no—
168C42—141no—
169C16—42no—
170C16—282no—
171C2×C7—21no—
172C2×C7—41no—
173C2×C7—81no—
174C2×C7#146411yescharacteristic
175C23—21no—
176C23D7×M1611yescharacteristic
177Q8—41no—
178D4—41no—
179D4—42no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—141no—
186C2×C4—141no—
187C2×C4—142no—
188C2×C4—282no—
189C2×C4—284no—
190C2×C4D7×C22⋊C411yescharacteristic
191C2×C4C2×#113111yescharacteristic
192C8—22no—
193C8—41no—
194C8—142no—
195C8—281no—
196C7#1152411yescharacteristic, complemented, Sylow
197C22—21no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22#8949811yescharacteristic
202C4—21no—
203C4—41no—
204C4—142no—
205C4—281no—
206C4—282no—
207C4#8951211yescharacteristic
208C4#8951711yescharacteristic
209C2—21no—
210C2—41no—
211C2—81no—
212C2#1172585611yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1321C2
socleC2×C7#1464
prime radicalC2×C7×#41C2×C4
cosocleC2×C7×M16C23
hypercenter#1321D7
hypocenterC7#11524
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #11726868C2semidirect
5. #11725339C2semidirect
196. C7#11524semidirect

Maximal subgroups

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

subgroupindex
#117360932
#117268682
C7×#13212
#117254962
#117253392
#115247