Finite Group Databasebeta

← #31885186

Subgroups — #31885186

Order 18176.

262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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
3C71×#1558C211yesmaximal, characteristic
4#17986477C411yescharacteristic
5#17986008—21no—
6C2×#14556047—21no—
7C2×#14556047C411yescharacteristic
8#17985798—21no—
9#17985798C412yescomplemented
10#17985759—21no—
11C2×C4⋊Q8×C71C2211yescharacteristic
12#432×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
26C4⋊Q8×C71—41no—
27C4⋊Q8×C71C2×C411yescharacteristic
28C4⋊Q8×C71C2×C411yescharacteristic
29C4⋊Q8×C71D411yescharacteristic
30C4⋊Q8×C71Q811yescharacteristic
31C2×C4⋊C4×C71—41no—
32C2×C42×C71D411yescharacteristic
33C4⋊C8×C71—22no—
34C22⋊C8×C71—41no—
35#97×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—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C42×C71—21no—
67C42×C71C22⋊C411yescharacteristic
68C42×C71C4⋊C411yescharacteristic
69C23×C71#9911yescharacteristic
70Q8×C71—41no—
71Q8×C71—41no—
72Q8×C71—41no—
73Q8×C71—41no—
74Q8×C71—42no—
75Q8×C71—81no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—42no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71#9511yescharacteristic
90C2×C4×C71#9911yescharacteristic
91C2×Dic71—42no—
92C8×C71—82no—
93C71⋊C8—81no—
94C71⋊C8—82no—
95C22×C71—41no—
96C22×C71#34211yescharacteristic
97C22×C71AΣL(1,𝔽16)11yescharacteristic
98C22×C71#35211yescharacteristic
99C4×C71—22no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71—41no—
107Dic71—82no—
108#7483—711nomaximal, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71—21no—
112C2×C71#139011yescharacteristic
113C2×#356—711no—
114#1886—711no—
115#1558D7111yescharacteristic
116C71#748311yescharacteristic, complemented, Sylow
117C2×C4⋊Q8C2×D7111yescharacteristic
118#519—711no—
119#432—21no—
120#419—1421no—
121C2×#101—711no—
122C2×#101—1421no—
123#356—712no—
124#356—1421no—
125#339—1421no—
126#333—21no—
127#333Dic7112yescomplemented
128C22×Q8C4×D7111yescharacteristic
129C22×Q8D28411yescharacteristic
130C2×M16—2841no—
131C2×M16—1421no—
132C4⋊Q8—41no—
133C4⋊Q8Dic14211yescharacteristic
134C4⋊Q8C4×D7111yescharacteristic
135C4⋊Q8C2×Dic7111yescharacteristic
136C4⋊Q8C22⋊D7111yescharacteristic
137C4×Q8—2842no—
138#117—2841no—
139#117—1421no—
140C2×C4⋊C4—41no—
141C2×C42C22⋊D7111yescharacteristic
142C4⋊C8—22no—
143#103—2842no—
144#101—2841no—
145#101—2841no—
146#101—1422no—
147C22⋊C8—41no—
148#97—41no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8C4×Dic7111yescharacteristic
156C2×Q8(C2×D71)⋊C411yescharacteristic
157C2×Q8Dic71⋊C411yescharacteristic
158C2×Q8C4⋊Dic7111yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4(C2×D71)⋊C411yescharacteristic
163M16—2842no—
164M16—2841no—
165M16—2841no—
166M16—5681no—
167C2×C8—42no—
168C2×C8—2841no—
169C2×C8—5681no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—2842no—
175C4⋊C4—1422no—
176C22⋊C4—2841no—
177C42—21no—
178C42—2841no—
179C42—2842no—
180C42C22⋊Dic7111yescharacteristic
181C42Dic71⋊C411yescharacteristic
182C23#1455604211yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2842no—
203C2×C4C71⋊#9511yescharacteristic
204C2×C4#1455604211yescharacteristic
205C8—82no—
206C8—5681no—
207C8—5682no—
208C22—41no—
209C22#1798576511yescharacteristic
210C22#1798578711yescharacteristic
211C22#1798579011yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—5682no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2258645911yescharacteristic
subgroupG/H
centerC2#22586459
commutatorC2×Q8×C71C42
FrattiniC2×C4⋊Q8C2×D71
FittingC71×#1558C2
socleC2×C71#1390
prime radicalC23×C71#99
cosocleC2×C4⋊Q8×C71C22
hypercenter#1558D71
hypocenterC71#7483
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1558

Complements

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

NH ≅ G/Ntype
9. #17985798C4semidirect
14. #333×C71C4semidirect
116. C71#7483semidirect
127. #333Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225887342
C71×#15582
#748371