Finite Group Databasebeta

← #31885210

Subgroups — #31885210

Order 18176.

276 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#17985798C211yesmaximal, characteristic
2#22588734C211yesmaximal, characteristic
3C2×#333×C71C211yesmaximal, characteristic
4#17986477C411yescharacteristic
5#17986008—21no—
6C2×#14556047—21no—
7C2×#14556047C411yescharacteristic
8#17985798—21no—
9#17985798C412yescomplemented
10#17985759—21no—
11C2×C4⋊Q8×C71C2211yescharacteristic
12C2×C4⋊C8×C71—21no—
13#333×C71—21no—
14#333×C71C412yescomplemented
15C2×#11765849—21no—
16C2×#11765849—41no—
17Q8×Dic71—42no—
18#14556087—21no—
19#14556087—41no—
20#14556052—42no—
21#14556047—22no—
22#14556047—41no—
23#14556047—41no—
24C22×Q8×C71C2×C411yescharacteristic
25C22×Q8×C71D411yescharacteristic
26C22×C8×C71—41no—
27C4⋊Q8×C71—41no—
28C4⋊Q8×C71C2×C411yescharacteristic
29C4⋊Q8×C71C2×C411yescharacteristic
30C4⋊Q8×C71D411yescharacteristic
31C4⋊Q8×C71Q811yescharacteristic
32C2×C4⋊C4×C71—41no—
33C2×C42×C71D411yescharacteristic
34C4⋊C8×C71—22no—
35C4⋊C8×C71—41no—
36#11765849—42no—
37#11765849—41no—
38#11765849—41no—
39#11765849—81no—
40C2×C71⋊C8—41no—
41C2×C71⋊C8—81no—
42C4⋊Dic71—22no—
43C4⋊Dic71—42no—
44C22⋊Dic71—41no—
45C4×Dic71—41no—
46C4×Dic71—42no—
47C2×Q8×C71—21no—
48C2×Q8×C71—21no—
49C2×Q8×C71—41no—
50C2×Q8×C71—41no—
51C2×Q8×C71—41no—
52C2×Q8×C71—41no—
53C2×Q8×C71C4211yescharacteristic
54C2×Q8×C71C22⋊C411yescharacteristic
55C2×Q8×C71C4⋊C411yescharacteristic
56C2×Q8×C71C4⋊C411yescharacteristic
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71C22⋊C411yescharacteristic
61C2×C8×C71—42no—
62C2×C8×C71—82no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C42×C71—21no—
68C42×C71C22⋊C411yescharacteristic
69C42×C71C4⋊C411yescharacteristic
70C23×C71#9911yescharacteristic
71Q8×C71—41no—
72Q8×C71—41no—
73Q8×C71—41no—
74Q8×C71—41no—
75Q8×C71—42no—
76Q8×C71—81no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71#9511yescharacteristic
92C2×C4×C71#9911yescharacteristic
93C2×Dic71—42no—
94C8×C71—82no—
95C71⋊C8—81no—
96C71⋊C8—82no—
97C22×C71—22no—
98C22×C71#34211yescharacteristic
99C22×C71AΣL(1,𝔽16)11yescharacteristic
100C22×C71#35211yescharacteristic
101C4×C71—21no—
102C4×C71—22no—
103C4×C71—21no—
104C4×C71—21no—
105C4×C71—21no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—41no—
109C4×C71—41no—
110Dic71—82no—
111#7491—711nomaximal, Sylow
112C2×C71—21no—
113C2×C71—21no—
114C2×C71#139011yescharacteristic
115C2×C71#140512yes—
116C2×#356—711no—
117#1886—711no—
118C2×#333D7111yescharacteristic
119C71#749111yescharacteristic, complemented, Sylow
120C2×C4⋊Q8C2×D7111yescharacteristic
121#519—711no—
122C2×C4⋊C8—21no—
123#419—1421no—
124C2×#101—711no—
125C2×#101—1421no—
126#356—712no—
127#356—1421no—
128#339—1421no—
129#333—21no—
130#333Dic7112yescomplemented
131C22×Q8C4×D7111yescharacteristic
132C22×Q8D28411yescharacteristic
133C2×M16—2841no—
134C2×M16—1421no—
135C22×C8—41no—
136C4⋊Q8—41no—
137C4⋊Q8Dic14211yescharacteristic
138C4⋊Q8C4×D7111yescharacteristic
139C4⋊Q8C2×Dic7111yescharacteristic
140C4⋊Q8C22⋊D7111yescharacteristic
141C4×Q8—2842no—
142#117—2841no—
143#117—1421no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D7111yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—41no—
148#103—2842no—
149#101—2841no—
150#101—2841no—
151#101—1422no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C4×Dic7111yescharacteristic
159C2×Q8(C2×D71)⋊C411yescharacteristic
160C2×Q8Dic71⋊C411yescharacteristic
161C2×Q8C4⋊Dic7111yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4(C2×D71)⋊C411yescharacteristic
166M16—2842no—
167M16—2841no—
168M16—2841no—
169M16—5681no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8—2841no—
173C2×C8—5681no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—2842no—
179C4⋊C4—1422no—
180C22⋊C4—2841no—
181C42—21no—
182C42—2841no—
183C42—2842no—
184C42C22⋊Dic7111yescharacteristic
185C42Dic71⋊C411yescharacteristic
186C23#1455604211yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—2842no—
208C2×C4C71⋊#9511yescharacteristic
209C2×C4#1455604211yescharacteristic
210C8—82no—
211C8—5681no—
212C8—5682no—
213C22—22no—
214C22#1798576511yescharacteristic
215C22#1798578711yescharacteristic
216C22#1798579011yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—5682no—
227C2—21no—
228C2—21no—
229C2#2258645911yescharacteristic
230C2#2258649512yes—
subgroupG/H
centerC22#17985787
commutatorC2×Q8×C71C42
FrattiniC2×C4⋊Q8C2×D71
FittingC2×#333×C71C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocleC2×C4⋊Q8×C71C22
hypercenterC2×#333D71
hypocenterC71#7491
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
9. #17985798C4semidirect
14. #333×C71C4semidirect
119. C71#7491semidirect
130. #333Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225887342
C2×#333×C712
#749171