Finite Group Databasebeta

← #31884801

Subgroups — #31884801

Order 18176.

356 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 43 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#22586270C212yesmaximal, complemented
2C71×#1782C211yesmaximal, characteristic
3#17985723—22no—
4C22×#41×C71C411yescharacteristic
5C2×#99×C71—22no—
6#381×C71C411yescharacteristic
7#381×C71C2211yescharacteristic
8C22⋊(C71⋊C8)—42no—
9C2×#41×C71—41no—
10C2×#41×C71—41no—
11C2×#41×C71—44no—
12C22×Q8×C71C811yescharacteristic, complemented
13C22×D4×C71—21no—
14C22×D4×C71C811yescharacteristic, complemented
15C23×C4×C71—21no—
16C23×C4×C71C2×C411yescharacteristic
17C2×C22⋊C4×C71—21no—
18C2×C22⋊C4×C71D412yes—
19#99×C71—44no—
20#95×C71—41no—
21#95×C71—41no—
22C2×C71⋊C8—82no—
23C24×C71—21no—
24C24×C71C22⋊C411yescharacteristic
25#41×C71—44no—
26#41×C71—44no—
27#41×C71—44no—
28#41×C71—44no—
29C2×Q8×C71—42no—
30C2×Q8×C71—41no—
31C2×D4×C71—24no—
32C2×D4×C71—41no—
33C2×D4×C71—42no—
34C2×D4×C71—41no—
35C2×D4×C71—41no—
36C2×D4×C71—42no—
37C22×C4×C71—21no—
38C22×C4×C71—21no—
39C22×C4×C71—21no—
40C22×C4×C71—22no—
41C22×C4×C71—42no—
42C22×C4×C71—42no—
43C22×C4×C71—42no—
44C22×C4×C71—42no—
45C22×C4×C71—41no—
46C22×C4×C71C2×C811yescharacteristic
47C22×C4×C71M1611yescharacteristic
48C22⋊C4×C71—42no—
49C22⋊C4×C71—42no—
50C23×C71—22no—
51C23×C71—22no—
52C23×C71—41no—
53C23×C71—41no—
54C23×C71—81no—
55C23×C71—82no—
56C23×C71C22⋊C811yescharacteristic
57C23×C71#9911yescharacteristic
58C23×C71#10011yescharacteristic
59Q8×C71—42no—
60Q8×C71—42no—
61D4×C71—44no—
62D4×C71—42no—
63D4×C71—42no—
64D4×C71—44no—
65C2×C4×C71—22no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—41no—
69C2×C4×C71—42no—
70C2×C4×C71—48no—
71C2×C4×C71—41no—
72C2×C4×C71—42no—
73C2×C4×C71—41no—
74C2×C4×C71—41no—
75C2×C4×C71—42no—
76C2×C4×C71—82no—
77C2×C4×C71—81no—
78C2×C4×C71—82no—
79C71⋊C8—162no—
80C22×C71—21no—
81C22×C71—21no—
82C22×C71—41no—
83C22×C71—41no—
84C22×C71—41no—
85C22×C71—42no—
86C22×C71—82no—
87C22×C71—82no—
88C22×C71—82no—
89C22×C71—82no—
90C22×C71#32311yescharacteristic
91C22×C71AΣL(1,𝔽16)12yes—
92C4×C71—41no—
93C4×C71—42no—
94C4×C71—41no—
95C4×C71—82no—
96C4×C71—161no—
97#7339—711nomaximal, Sylow
98C2×C71—22no—
99C2×C71—21no—
100C2×C71—41no—
101C2×C71—41no—
102C2×C71—82no—
103C2×C71#131611yescharacteristic
104#1782D7111yescharacteristic
105#1317—712no—
106C71#733911yescharacteristic, complemented, Sylow
107C22×#41Dic7111yescharacteristic
108C2×#99—22no—
109#381Dic7111yescharacteristic
110#381C2×D7111yescharacteristic
111#323—1422no—
112C2×#41—41no—
113C2×#41—41no—
114C2×#41—44no—
115C22×Q8C71⋊C811yescharacteristic, complemented
116C22×D4—21no—
117C22×D4C71⋊C811yescharacteristic, complemented
118C23×C4—21no—
119C23×C4C2×Dic7111yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4C22⋊D7112yes—
122#99—44no—
123C22⋊C8—2842no—
124#95—41no—
125#95—41no—
126C24—21no—
127C24C22⋊Dic7111yescharacteristic
128#41—44no—
129#41—44no—
130#41—44no—
131#41—44no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×D4—24no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4C2×C71⋊C811yescharacteristic
150C22×C4#1176584911yescharacteristic
151C2×C8—5682no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—82no—
160C23C22⋊(C71⋊C8)11yescharacteristic
161C23#1455604311yescharacteristic
162C23#1455604511yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—44no—
166D4—42no—
167D4—42no—
168D4—44no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—48no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—81no—
182C2×C4—82no—
183C8—11362no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1798572311yescharacteristic
195C22#1798578612yes—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—82no—
200C4—161no—
201C2—22no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—82no—
206C2#2258626811yescharacteristic
subgroupG/H
centerC2#22586268
commutatorC22×C4×C71C2×C8
Frattini#381C2×D71
FittingC71×#1782C2
socleC2×C71#1316
prime radicalC22×#41×C71C4
cosocle#381×C71C22
hypercenter#1782D71
hypocenterC71#7339
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #22586270C2semidirect
12. C22×Q8×C71C8semidirect
14. C22×D4×C71C8semidirect
106. C71#7339semidirect
115. C22×Q8C71⋊C8semidirect
117. C22×D4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225862702
C71×#17822
#733971