Finite Group Databasebeta

← #13264994

Subgroups — #13264994

Order 1792.

280 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 61 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#11739161C211yesmaximal, characteristic
2#11726842C212yesmaximal, complemented
3C7×#1355C211yesmaximal, characteristic, complemented
4#11725589C211yesmaximal, characteristic, complemented
5#11725327C212yesmaximal, complemented
6#90407C411yescharacteristic
7D7×#125C411yescharacteristic
8C7×#490C2211yescharacteristic
9#89843C411yescharacteristic
10#89841C411yescharacteristic
11#89543—22no—
12#89540—22no—
13#89479C2211yescharacteristic
14#89475C2211yescharacteristic
15C7×#347C2212yes—
16#89309—22no—
17#89306—22no—
18#89261C2212yes—
19#11917—71nomaximal, Sylow
20C2×#1159—21no—
21C7×#127C2×C411yescharacteristic
22C7×#125C2×C411yescharacteristic
23#6501—41no—
24#6499—21no—
25D4×Dic7—41no—
26C2×#1131D412yes—
27D7×C4⋊C4—21no—
28D7×C4⋊C4—41no—
29#6453—21no—
30#6452—41no—
31C42×D7C2×C411yescharacteristic
32#6433C2×C411yescharacteristic
33#6432C2×C411yescharacteristic
34C7×M32—22no—
35C7×#102—22no—
36C7×#97C2311yescharacteristic
37#6406—22no—
38#6390—22no—
39C8×Dic7D412yes—
40#6386—22no—
41#6384—22no—
42#6377C2×C411yescharacteristic
43#3142—71no—
44#2165—72no—
45#1356—72no—
46#1355—71no—
47#1355D711yescharacteristic, complemented
48C2×C7×D4—21no—
49C2×C7×D4—41no—
50C2×C22⋊D7—41no—
51C22×Dic7—41no—
52#1159—44no—
53C2×C4×D7—21no—
54C2×C4×D7—41no—
55C2×C4×D7C22⋊C411yescharacteristic
56C2×Dic14—21no—
57C2×C7×C8C2×D412yes—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—41no—
60C42×C7C22×C411yescharacteristic
61C22⋊Dic7—42no—
62C4⋊Dic7—21no—
63C4⋊Dic7—41no—
64C4⋊Dic7—41no—
65Dic7⋊C4—41no—
66Dic7⋊C4—41no—
67Dic7⋊C4—41no—
68C4×Dic7—21no—
69C4×Dic7C22⋊C411yescharacteristic
70C2×C7⋊C8C22⋊C412yes—
71#1131—42no—
72C7×C16—42no—
73C7⋊C16—42no—
74#532—71no—
75C2×#125—71no—
76#491—71no—
77#491—71no—
78#490—71no—
79#490C2×D711yescharacteristic
80#488—142no—
81#417—142no—
82C4×M16—71no—
83#364—142no—
84#349—142no—
85#347—72no—
86#347C2×D712yes—
87C23×C7—41no—
88C22×D7#10011yescharacteristic
89C7×D4—42no—
90C7×D4—82no—
91C2×C4×C7—21no—
92C2×C4×C7—41no—
93C2×C4×C7C2×C22⋊C411yescharacteristic
94C22⋊D7—82no—
95C2×Dic7—21no—
96C2×Dic7—41no—
97C2×Dic7—41no—
98C2×Dic7—82no—
99C2×Dic7#10011yescharacteristic
100C4×D7—22no—
101C4×D7—81no—
102C4×D7—82no—
103Dic14—42no—
104C7×C8—22no—
105C7⋊C8—22no—
106C2×#41—141no—
107C2×M16—72no—
108C4⋊Q8—71no—
109#127C4×D711yescharacteristic
110#125—71no—
111#125—141no—
112#125—281no—
113#125C4×D711yescharacteristic
114#124—141no—
115#121—281no—
116C4×D4—281no—
117C2×C4⋊C4—141no—
118C2×C4⋊C4—281no—
119C2×C42—71no—
120M32—22no—
121M32—142no—
122#103—142no—
123#103—142no—
124#102—22no—
125#102—142no—
126#97—71no—
127#97C22×D711yescharacteristic
128C4×C8—72no—
129C22×C7—82no—
130C22×C7—81no—
131C22×C7C2×#10011yescharacteristic
132C2×D7—21no—
133C2×D7—41no—
134C4×C7—41no—
135C4×C7—81no—
136C4×C7#42112yes—
137Dic7—21no—
138Dic7—41no—
139Dic7—81no—
140Dic7—81no—
141#41—284no—
142C2×Q8—141no—
143C2×D4—21no—
144C2×D4—41no—
145C2×D4—281no—
146C22×C4—71no—
147C22×C4—141no—
148C22×C4—281no—
149C22×C4—281no—
150M16—282no—
151C2×C8—72no—
152C2×C8D4×D712yes—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—141no—
156C4⋊C4—281no—
157C4⋊C4—281no—
158C4⋊C4—281no—
159C4⋊C4—281no—
160C4⋊C4—281no—
161C22⋊C4—282no—
162C42—71no—
163C42—141no—
164C42C2×C4×D711yescharacteristic
165C16—42no—
166C16—282no—
167C2×C7—21no—
168C2×C7—161no—
169C2×C7#153311yescharacteristic
170D7—41no—
171C23—41no—
172C23—71no—
173Q8—282no—
174D4—42no—
175D4—82no—
176D4—562no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—71no—
180C2×C4—142no—
181C2×C4—141no—
182C2×C4—281no—
183C2×C4—281no—
184C2×C4—561no—
185C2×C4—562no—
186C2×C4—562no—
187C2×C4D7×C22⋊C411yescharacteristic
188C8—22no—
189C8—142no—
190C7#1191711yescharacteristic, complemented, Sylow
191C22—82no—
192C22—81no—
193C22—141no—
194C22—281no—
195C22D7×#10011yescharacteristic
196C4—41no—
197C4—81no—
198C4—141no—
199C4—281no—
200C4—561no—
201C4—561no—
202C4#8959312yes—
203C2—21no—
204C2—161no—
205C2—281no—
206C2#1172610111yescharacteristic
subgroupG/H
centerC2#11726101
commutatorC42×C7C22×C4
Frattini#97C22×D7
FittingC7×#1355C2
socleC2×C7#1533
prime radical#90407C4
cosocleC7×#97C23
hypercenter#1355D7
hypocenterC7#11917
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #11726842C2semidirect
3. C7×#1355C2semidirect
4. #11725589C2semidirect
5. #11725327C2semidirect
47. #1355D7semidirect
190. C7#11917semidirect

Maximal subgroups

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

subgroupindex
#117391612
#117268422
C7×#13552
#117255892
#117253272
#119177