Finite Group Databasebeta

← #31891862

Subgroups — #31891862

Order 18176.

336 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 103 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#22597446C211yesmaximal, characteristic
2#22589322C211yesmaximal, characteristic
3#22587107C212yesmaximal
4#22586234C212yesmaximal
5C71×#1526C211yesmaximal, characteristic
6#17986130—22no—
7#17986130—22no—
8#17986060—22no—
9#17986056C2212yes—
10#17986023C2211yescharacteristic
11#17985845C2211yescharacteristic
12#17985758—22no—
13#17985752—22no—
14#17985750—22no—
15#533×C71C2211yescharacteristic
16C4×M16×C71C2212yes—
17#330×C71C411yescharacteristic, complemented
18#330×C71C412yescomplemented
19#330×C71C411yescharacteristic, complemented
20C2×#11765849—21no—
21C2×C4×Dic71—21no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)D412yes—
26C4⋊(C71⋊C8)D412yes—
27C22⋊(C71⋊C8)—22no—
28#14556039—22no—
29#14556039—22no—
30C4×C71⋊C8—22no—
31C71⋊#95—22no—
32C2×M16×C71—22no—
33C4⋊Q8×C71Q812yes—
34#125×C71C2×C411yescharacteristic
35#125×C71C2×C411yescharacteristic
36C22⋊Q8×C71—22no—
37C4×Q8×C71—22no—
38#117×C71D412yes—
39C2×C42×C71C2311yescharacteristic
40#97×C71C2×C412yes—
41#97×C71C2×C412yes—
42C4×C8×C71—22no—
43C22×Dic71—41no—
44#11765849—42no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—41no—
48C4×Dic71—22no—
49C4×Dic71—41no—
50C2×Q8×C71—22no—
51C22×C4×C71C2×D411yescharacteristic
52C22×C4×C71#4112yes—
53M16×C71—44no—
54C2×C8×C71—22no—
55C2×C8×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71C4⋊C412yes—
60C4⋊C4×C71C4⋊C412yes—
61C22⋊C4×C71—22no—
62C42×C71—22no—
63C42×C71C22×C411yescharacteristic
64C42×C71C2×D412yes—
65C42×C71C2×Q811yescharacteristic
66C23×C71#12311yescharacteristic
67Q8×C71—82no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—42no—
73C2×C4×C71C2×C4⋊C411yescharacteristic
74C2×C4×C71C4×D412yes—
75C2×C4×C71#12011yescharacteristic
76C2×C4×C71C22⋊Q812yes—
77C2×Dic71—42no—
78C2×Dic71—82no—
79C8×C71—24no—
80C8×C71—42no—
81C71⋊C8—42no—
82C71⋊C8—42no—
83C71⋊C8—81no—
84C22×C71—21no—
85C22×C71—22no—
86C22×C71#38511yescharacteristic
87C4×C71—21no—
88C4×C71—22no—
89C4×C71—22no—
90C4×C71—42no—
91C4×C71—42no—
92C4×C71#45411yescharacteristic
93C4×C71#45611yescharacteristic
94Dic71—82no—
95#8919—711nomaximal, Sylow
96C2×C71—41no—
97C2×C71#185711yescharacteristic
98C2×C71#186112yes—
99#2987—711no—
100#1995—711no—
101#1569—712no—
102#1526D7111yescharacteristic
103#1300—712no—
104C71#891911yescharacteristic, complemented, Sylow
105#533C2×D7111yescharacteristic
106#446—1422no—
107#446—1422no—
108#433—1422no—
109#432—712no—
110#423—711no—
111C4×M16C2×D7112yes—
112#376—711no—
113#339—1422no—
114#335—1422no—
115#334—1422no—
116#330Dic7111yescharacteristic, complemented
117#330Dic7112yescomplemented
118#330Dic7111yescharacteristic, complemented
119C2×M16—22no—
120C2×M16—1421no—
121C4⋊Q8Dic14212yes—
122#125C2×Dic7111yescharacteristic
123#125C2×Dic7111yescharacteristic
124C22⋊Q8—22no—
125C4×Q8—22no—
126#117D28412yes—
127C2×C42—1421no—
128C2×C42C22×D7111yescharacteristic
129C4⋊C8—712no—
130C4⋊C8—712no—
131C4⋊C8—1422no—
132C4⋊C8—1422no—
133C4⋊C8—1422no—
134C22⋊C8—1422no—
135#97—1422no—
136#97—1422no—
137#97C2×Dic7112yes—
138#97C2×Dic7112yes—
139C4×C8—22no—
140C4×C8—1422no—
141#95—1422no—
142C2×Q8—22no—
143C22×C4—2841no—
144C22×C4C2×D28411yescharacteristic
145C22×C4#1176586412yes—
146M16—44no—
147M16—2842no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—2841no—
151C2×C8—1422no—
152C2×C8—1422no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4C4⋊Dic7112yes—
157C4⋊C4C4⋊Dic7112yes—
158C22⋊C4—22no—
159C42—22no—
160C42—2841no—
161C42—1422no—
162C42C22×Dic7111yescharacteristic
163C42D4×D7112yes—
164C42C2×Dic14211yescharacteristic
165C23#1455611211yescharacteristic
166Q8—82no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—2842no—
173C2×C4—5682no—
174C2×C4C2×C4⋊Dic7111yescharacteristic
175C2×C4D4×Dic7112yes—
176C2×C4#1455609811yescharacteristic
177C2×C4#1455610512yes—
178C8—24no—
179C8—42no—
180C8—2842no—
181C8—2842no—
182C8—5681no—
183C22—21no—
184C22—22no—
185C22#1798587411yescharacteristic
186C4—21no—
187C4—22no—
188C4—22no—
189C4—42no—
190C4—42no—
191C4—5682no—
192C4#1798617411yescharacteristic
193C4#1798618411yescharacteristic
194C2—41no—
195C2#2258855711yescharacteristic
196C2#2258858212yes—
subgroupG/H
centerC2×C4#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1526C2
socleC22×C71#385
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#1526D71
hypocenterC71#8919
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
17. #330×C71C4semidirect
18. #330×C71C4semidirect
19. #330×C71C4semidirect
104. C71#8919semidirect
116. #330Dic71semidirect
117. #330Dic71semidirect
118. #330Dic71semidirect

Maximal subgroups

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

subgroupindex
#225974462
#225893222
#225871072
#225862342
C71×#15262
#891971