Finite Group Databasebeta

← #31908646

Subgroups — #31908646

Order 18176.

326 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590152C212yesmaximal, complemented
2#22586373C214yesmaximal, complemented
3C71×#3138C211yesmaximal, characteristic
4#17985820—22no—
5#17985781—22no—
6#17985777C2214yescomplemented
7#533×C71C411yescharacteristic
8C2×#127×C71C411yescharacteristic
9#490×C71C412yes—
10#490×C71C2212yes—
11#486×C71—22no—
12#418×C71—22no—
13C4×M16×C71C2211yescharacteristic
14#14556068—24no—
15C22×D4×C71—21no—
16C22×D4×C71—41no—
17C2×M16×C71D412yes—
18C4⋊Q8×C71—21no—
19#127×C71—21no—
20#127×C71—41no—
21#127×C71C2×C412yes—
22#125×C71C2×C412yes—
23C22⋊Q8×C71—41no—
24C4×Q8×C71—41no—
25#117×C71—21no—
26C2×C42×C71C2×C411yescharacteristic
27#102×C71—24no—
28#102×C71—24no—
29#97×C71C2×C411yescharacteristic
30#97×C71C2311yescharacteristic
31C4×C8×C71D412yes—
32C71⋊C16—44no—
33C24×C71—41no—
34C2×Q8×C71—41no—
35C2×D4×C71—22no—
36C2×D4×C71—44no—
37C2×D4×C71—42no—
38C2×D4×C71—42no—
39C2×D4×C71—41no—
40C2×D4×C71—84no—
41C22×C4×C71—21no—
42C22×C4×C71C22⋊C411yescharacteristic
43M16×C71—42no—
44C2×C8×C71C22⋊C412yes—
45C2×C8×C71C2×D412yes—
46C4⋊C4×C71—22no—
47C4⋊C4×C71—42no—
48C4⋊C4×C71—41no—
49C22⋊C4×C71—41no—
50C42×C71—21no—
51C42×C71—41no—
52C42×C71C22⋊C411yescharacteristic
53C42×C71C22×C411yescharacteristic
54C23×C71—42no—
55C23×C71—82no—
56C23×C71—84no—
57C23×C71#10011yescharacteristic
58Q8×C71—82no—
59D4×C71—44no—
60D4×C71—82no—
61D4×C71—84no—
62D4×C71—84no—
63C2×C4×C71—21no—
64C2×C4×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—42no—
67C2×C4×C71—42no—
68C2×C4×C71#10011yescharacteristic
69C2×C4×C71C2×C22⋊C411yescharacteristic
70C8×C71—22no—
71C8×C71—22no—
72C22×C71—21no—
73C22×C71—41no—
74C22×C71—84no—
75C22×C71—82no—
76C22×C71—164no—
77C22×C71C2×#10011yescharacteristic
78C4×C71—22no—
79C4×C71—21no—
80C4×C71—41no—
81C4×C71—82no—
82C4×C71#42112yes—
83#11915—711nomaximal, Sylow
84C2×C71—21no—
85C2×C71—41no—
86C2×C71—162no—
87C2×C71#153311yescharacteristic
88#3138D7111yescharacteristic
89#2165—712no—
90#1355—714no—
91C71#1191511yescharacteristic, complemented, Sylow
92#533Dic7111yescharacteristic
93C2×#127Dic7111yescharacteristic
94#490Dic7112yes—
95#490C2×D7112yes—
96#486—22no—
97#418—22no—
98C4×M16C2×D7111yescharacteristic
99#364—1422no—
100#349—1422no—
101#347—714no—
102C22×D4—21no—
103C22×D4—41no—
104C2×M16C22⋊D7112yes—
105C4⋊Q8—21no—
106#127—21no—
107#127—41no—
108#127C2×Dic7112yes—
109#125C2×Dic7112yes—
110C22⋊Q8—41no—
111C4×Q8—41no—
112#117—21no—
113C2×C42C2×Dic7111yescharacteristic
114M32—1424no—
115#102—24no—
116#102—24no—
117#97C2×Dic7111yescharacteristic
118#97C22×D7111yescharacteristic
119C4×C8C22⋊D7112yes—
120C24—41no—
121C2×Q8—41no—
122C2×D4—22no—
123C2×D4—44no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—41no—
127C2×D4—84no—
128C22×C4—21no—
129C22×C4C22⋊Dic7111yescharacteristic
130M16—42no—
131C2×C8C22⋊Dic7112yes—
132C2×C8C2×C22⋊D7112yes—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—41no—
136C22⋊C4—41no—
137C42—21no—
138C42—41no—
139C42C22⋊Dic7111yescharacteristic
140C42C22×Dic7111yescharacteristic
141C16—2844no—
142C23—42no—
143C23—82no—
144C23—84no—
145C23#1455604511yescharacteristic
146Q8—82no—
147D4—44no—
148D4—82no—
149D4—84no—
150D4—84no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4#1455604511yescharacteristic
157C2×C4C2×C22⋊Dic7111yescharacteristic
158C8—22no—
159C8—22no—
160C22—21no—
161C22—41no—
162C22—84no—
163C22—82no—
164C22—164no—
165C22C2×#1455604511yescharacteristic
166C4—22no—
167C4—21no—
168C4—41no—
169C4—82no—
170C4#1798601612yes—
171C2—21no—
172C2—41no—
173C2—162no—
174C2#2258696411yescharacteristic
subgroupG/H
centerC2#22586964
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#3138C2
socleC2×C71#1533
prime radicalC2×#127×C71C4
cosocle#97×C71C23
hypercenter#3138D71
hypocenterC71#11915
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #22590152C2semidirect
2. #22586373C2semidirect
6. #17985777C22semidirect
91. C71#11915semidirect

Maximal subgroups

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

subgroupindex
#225901522
#225863732
C71×#31382
#1191571