Finite Group Databasebeta

← #31885043

Subgroups — #31885043

Order 18176.

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

normal subgroups only

This group has 236 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#22586465C211yesmaximal, characteristic
2#22586161C211yesmaximal, characteristic
3C71×#2027C211yesmaximal, characteristic
4C2×#14556043—21no—
5#17985754—21no—
6#544×C71C411yescharacteristic, complemented
7#418×C71—21no—
8#390×C71C2211yescharacteristic
9#343×C71C411yescharacteristic, complemented
10#339×C71—21no—
11C22×C71⋊C8—41no—
12C2×C22⋊Dic71—41no—
13#14556043—22no—
14#14556043—41no—
15C22×D4×C71D411yescharacteristic
16C2×M16×C71—21no—
17C4⋊Q8×C71—41no—
18#125×C71—41no—
19C22⋊Q8×C71—41no—
20#117×C71—21no—
21C2×C4⋊C4×C71C2×C411yescharacteristic
22C2×C22⋊C4×C71—41no—
23C2×C42×C71D411yescharacteristic
24#102×C71—41no—
25#102×C71—41no—
26C22×Dic71—81no—
27C2×C71⋊C8—42no—
28C2×C71⋊C8—81no—
29C2×C71⋊C8—81no—
30C22⋊Dic71—42no—
31C22⋊Dic71—81no—
32C24×C71—21no—
33C2×Q8×C71—41no—
34C2×D4×C71—21no—
35C2×D4×C71—41no—
36C2×D4×C71—41no—
37C2×D4×C71SD1611yescharacteristic
38C2×D4×C71Q1611yescharacteristic
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71C22⋊C411yescharacteristic
42M16×C71—41no—
43M16×C71—41no—
44C2×C8×C71—41no—
45C4⋊C4×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—41no—
48C4⋊C4×C71—41no—
49C4⋊C4×C71—42no—
50C22⋊C4×C71—41no—
51C22⋊C4×C71—81no—
52C22⋊C4×C71—81no—
53C42×C71—41no—
54C42×C71—41no—
55C23×C71—21no—
56C23×C71—21no—
57C23×C71—41no—
58C23×C71—41no—
59C23×C71—81no—
60C23×C71—81no—
61C23×C71#9911yescharacteristic
62Q8×C71—161no—
63D4×C71—41no—
64D4×C71—41no—
65D4×C71—81no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71—41no—
74C2×C4×C71—41no—
75C2×C4×C71—81no—
76C2×C4×C71#10311yescharacteristic
77C2×C4×C71#10411yescharacteristic
78C2×Dic71—82no—
79C2×Dic71—161no—
80C2×Dic71—161no—
81C8×C71—81no—
82C71⋊C8—82no—
83C22×C71—21no—
84C22×C71—21no—
85C22×C71—41no—
86C22×C71—41no—
87C22×C71—81no—
88C22×C71—81no—
89C22×C71—81no—
90C22×C71—81no—
91C22×C71—161no—
92C22×C71#32811yescharacteristic
93C22×C71AΣL(1,𝔽16)11yescharacteristic
94C22×C71#35411yescharacteristic
95C4×C71—21no—
96C4×C71—21no—
97C4×C71—41no—
98C4×C71—41no—
99C4×C71—81no—
100C4×C71—81no—
101C4×C71—81no—
102Dic71—162no—
103#7430—711nomaximal, Sylow
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—81no—
107C2×C71—81no—
108C2×C71#135111yescharacteristic
109C2×C71#140211yescharacteristic
110C2×C71#140411yescharacteristic
111#2027D7111yescharacteristic
112#1393—711no—
113#1270—711no—
114C71#743011yescharacteristic, complemented, Sylow
115#544Dic7111yescharacteristic, complemented
116#418—21no—
117C2×#99—1421no—
118#390C2×D7111yescharacteristic
119#343Dic7111yescharacteristic, complemented
120#339—21no—
121#336—1421no—
122C22×D4C22⋊D7111yescharacteristic
123C2×M16—21no—
124C22×C8—2841no—
125C4⋊Q8—41no—
126#125—41no—
127C22⋊Q8—41no—
128#117—21no—
129C2×C4⋊C4C2×Dic7111yescharacteristic
130C2×C22⋊C4—41no—
131C2×C22⋊C4—2841no—
132C2×C42C22⋊D7111yescharacteristic
133#102—41no—
134#102—41no—
135#99—2841no—
136#99—1422no—
137C24—21no—
138C2×Q8—41no—
139C2×D4—21no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4#1176585211yescharacteristic
143C2×D4#1176585511yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—5681no—
147C22×C4C22⋊Dic7111yescharacteristic
148M16—41no—
149M16—41no—
150C2×C8—41no—
151C2×C8—2842no—
152C2×C8—5681no—
153C2×C8—5681no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C22⋊C4—41no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C22⋊C4—2842no—
163C22⋊C4—5681no—
164C42—41no—
165C42—41no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23—81no—
171C23—81no—
172C23#1455604311yescharacteristic
173Q8—161no—
174D4—41no—
175D4—41no—
176D4—81no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—81no—
187C2×C4—5682no—
188C2×C4—11361no—
189C2×C4—11361no—
190C2×C4#1455605211yescharacteristic
191C2×C4#1455605511yescharacteristic
192C8—81no—
193C8—5682no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#1798573611yescharacteristic
204C22#1798578611yescharacteristic
205C22#1798579411yescharacteristic
206C4—21no—
207C4—21no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—11362no—
214C2—21no—
215C2—21no—
216C2—81no—
217C2—81no—
218C2#2258636111yescharacteristic
219C2#2258648511yescharacteristic
220C2#2258649111yescharacteristic
subgroupG/H
centerC22#17985786
commutatorC2×C4⋊C4×C71C2×C4
Frattini#390C2×D71
FittingC71×#2027C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC22×D4×C71D4
cosocle#390×C71C22
hypercenter#2027D71
hypocenterC71#7430
LCSG ▷ C2×C4⋊C4×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2027

Complements

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

NH ≅ G/Ntype
6. #544×C71C4semidirect
9. #343×C71C4semidirect
114. C71#7430semidirect
115. #544Dic71semidirect
119. #343Dic71semidirect

Maximal subgroups

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

subgroupindex
#225864652
#225861612
C71×#20272
#743071