Finite Group Databasebeta

← #13263239

Subgroups — #13263239

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#11736515C211yesmaximal, characteristic
2#11726868C212yesmaximal, complemented
3C7×#1328C211yesmaximal, characteristic, complemented
4#11725525C211yesmaximal, characteristic, complemented
5#11725337C212yesmaximal
6#90213C411yescharacteristic
7C7×#424C2211yescharacteristic
8#89901C2211yescharacteristic
9M16×Dic7C2211yescharacteristic
10#89882C411yescharacteristic
11C2×C8×Dic7C411yescharacteristic
12#89771C411yescharacteristic
13#89612—22no—
14#89605—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11524—71nomaximal, Sylow
20C2×#1157C2×C411yescharacteristic
21C2×C7×M16C2311yescharacteristic
22C22×C7×C8C2×C411yescharacteristic
23C7×#117C2×C411yescharacteristic
24C2×C4×Dic7C2×C411yescharacteristic
25C2×#1136C2×C411yescharacteristic
26C22×C7⋊C8C2×C411yescharacteristic
27#6456—21no—
28#6450—21no—
29(C22⋊D7)⋊C4—22no—
30#6435—21no—
31C42×D7—21no—
32C7×M32—22no—
33C7×C4⋊C8—22no—
34(C2×D7)⋊C8—22no—
35(C2×D7)⋊C8—22no—
36#6389D412yes—
37C8×Dic7—21no—
38C8×Dic7—21no—
39C8×Dic7D412yes—
40#6386—22no—
41C4⋊(C7⋊C8)—22no—
42#2922—71no—
43#2165—72no—
44#1328—71no—
45#1328D711yescharacteristic, complemented
46#1321—72no—
47C22×C4×C7C22×C411yescharacteristic
48C2×C22⋊D7—21no—
49C22×Dic7—21no—
50#1157—41no—
51#1157—41no—
52C2×D28M1611yescharacteristic
53C2×C4×D7—22no—
54C2×C4×D7—21no—
55C2×Dic14M1611yescharacteristic
56C7×M16—41no—
57C2×C7×C8—21no—
58C2×C7×C8—41no—
59C2×C7×C8C2×D412yes—
60C7×C4⋊C4—21no—
61C7×C22⋊C4M1612yes—
62C42×C7—21no—
63(C2×D7)⋊C4—22no—
64Dic7⋊C4—22no—
65C4×Dic7—22no—
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#424C2×D711yescharacteristic
80#408—71no—
81#408—71no—
82C4×M16—71no—
83C2×C4×C8—71no—
84#364—142no—
85#349—72no—
86#349C2×D712yes—
87#347—142no—
88C23×C7C2×M1611yescharacteristic
89C22×D7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7C2×C22⋊C411yescharacteristic
94C2×C4×C7C2×M1611yescharacteristic
95C22⋊D7—42no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—42no—
100D28—41no—
101C4×D7—44no—
102C4×D7—41no—
103C4×D7—41no—
104Dic14—41no—
105C7×C8—22no—
106C7×C8—41no—
107C7⋊C8—22no—
108C7⋊C8—41no—
109C2×#41—71no—
110C2×M16—71no—
111C2×M16C22×D711yescharacteristic
112C22×C8—71no—
113C22×C8C4×D711yescharacteristic
114C4×Q8—141no—
115C4×D4—141no—
116C4×D4—142no—
117#117—141no—
118#117C4×D711yescharacteristic
119C2×C42—71no—
120C2×C42—141no—
121M32—22no—
122M32—142no—
123C4⋊C8—22no—
124C4⋊C8—142no—
125C22⋊C8—142no—
126C22⋊C8—142no—
127#97—72no—
128C4×C8—72no—
129C4×C8—141no—
130C4×C8—141no—
131C22×C7—21no—
132C22×C7—41no—
133C22×C7#40711yescharacteristic
134C2×D7—42no—
135C2×D7—41no—
136C4×C7—21no—
137C4×C7—42no—
138C4×C7#41011yescharacteristic
139C4×C7#41311yescharacteristic
140Dic7—22no—
141Dic7—41no—
142Dic7—41no—
143#41—281no—
144#41—281no—
145C2×Q8—71no—
146C2×D4—71no—
147C2×D4—141no—
148C22×C4—142no—
149C22×C4—141no—
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—21no—
161C4⋊C4—142no—
162C22⋊C4—142no—
163C22⋊C4#113112yes—
164C42—21no—
165C42—71no—
166C42—71no—
167C42—142no—
168C42—141no—
169C16—42no—
170C16—282no—
171C2×C7—21no—
172C2×C7—41no—
173C2×C7#146411yescharacteristic
174D7—81no—
175C23—141no—
176C23C2×#113111yescharacteristic
177Q8—281no—
178D4—281no—
179D4—282no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—141no—
184C2×C4—141no—
185C2×C4—141no—
186C2×C4—284no—
187C2×C4—282no—
188C2×C4—281no—
189C2×C4—281no—
190C2×C4D7×C22⋊C411yescharacteristic
191C2×C4D7×M1611yescharacteristic
192C8—22no—
193C8—41no—
194C8—142no—
195C8—281no—
196C7#1152411yescharacteristic, complemented, Sylow
197C22—21no—
198C22—41no—
199C22—282no—
200C22—281no—
201C22#8949811yescharacteristic
202C4—21no—
203C4—42no—
204C4—142no—
205C4—281no—
206C4—281no—
207C4#8951211yescharacteristic
208C4#8952311yescharacteristic
209C2—21no—
210C2—41no—
211C2—561no—
212C2#1172588011yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1328C2
socleC2×C7#1464
prime radicalC2×#1157C2×C4
cosocleC2×C7×M16C23
hypercenter#1328D7
hypocenterC7#11524
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
2. #11726868C2semidirect
3. C7×#1328C2semidirect
4. #11725525C2semidirect
45. #1328D7semidirect
196. C7#11524semidirect

Maximal subgroups

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

subgroupindex
#117365152
#117268682
C7×#13282
#117255252
#117253372
#115247