Finite Group Databasebeta

← #31885185

Subgroups — #31885185

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#22587066C211yesmaximal, characteristic
3C71×#1886C211yesmaximal, characteristic
4#17986056—21no—
5C2×#14556047—21no—
6#17985798—21no—
7#17985798C412yescomplemented
8#17985748—21no—
9#17985748C412yescomplemented
10C2×C4⋊Q8×C71C2211yescharacteristic
11#519×C71C411yescharacteristic
12#419×C71—21no—
13C2×#101×C71C411yescharacteristic
14#339×C71—21no—
15C2×#11765849—41no—
16C4⋊(C71⋊C8)—22no—
17#14556047—22no—
18#14556047—41no—
19C22⋊(C71⋊C8)—41no—
20#14556039—41no—
21C22×Q8×C71C2×C411yescharacteristic
22C22×Q8×C71D411yescharacteristic
23C2×M16×C71—21no—
24C4⋊Q8×C71—41no—
25C4⋊Q8×C71C2×C411yescharacteristic
26C4⋊Q8×C71C2×C411yescharacteristic
27C4⋊Q8×C71D411yescharacteristic
28C4⋊Q8×C71Q811yescharacteristic
29C4×Q8×C71—42no—
30#117×C71—21no—
31#117×C71—41no—
32C2×C4⋊C4×C71—41no—
33C2×C42×C71D411yescharacteristic
34#103×C71—42no—
35#101×C71—41no—
36#11765849—42no—
37#11765849—81no—
38C2×C71⋊C8—42no—
39C2×C71⋊C8—81no—
40C2×Q8×C71—21no—
41C2×Q8×C71—21no—
42C2×Q8×C71—41no—
43C2×Q8×C71—41no—
44C2×Q8×C71—41no—
45C2×Q8×C71—41no—
46C2×Q8×C71C4211yescharacteristic
47C2×Q8×C71C22⋊C411yescharacteristic
48C2×Q8×C71C4⋊C411yescharacteristic
49C2×Q8×C71C4⋊C411yescharacteristic
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71C22⋊C411yescharacteristic
54M16×C71—41no—
55M16×C71—41no—
56C2×C8×C71—41no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—41no—
59C4⋊C4×C71—41no—
60C4⋊C4×C71—41no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—42no—
63C22⋊C4×C71—41no—
64C42×C71—21no—
65C42×C71—41no—
66C42×C71—42no—
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—42no—
90C2×C4×C71#9511yescharacteristic
91C2×C4×C71#9911yescharacteristic
92C8×C71—81no—
93C71⋊C8—82no—
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—
107C4×C71—82no—
108#7483—711nomaximal, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71—21no—
112C2×C71#139011yescharacteristic
113C2×#356—711no—
114#1886D7111yescharacteristic
115#1558—711no—
116C71#748311yescharacteristic, complemented, Sylow
117C2×C4⋊Q8C2×D7111yescharacteristic
118#519Dic7111yescharacteristic
119#432—1421no—
120#419—21no—
121C2×#101—1421no—
122C2×#101Dic7111yescharacteristic
123#356—712no—
124#356—1421no—
125#339—21no—
126#333—712no—
127#333—1421no—
128C22×Q8C2×Dic7111yescharacteristic
129C22×Q8C22⋊D7111yescharacteristic
130C2×M16—21no—
131C2×M16—2841no—
132C4⋊Q8—41no—
133C4⋊Q8Dic14211yescharacteristic
134C4⋊Q8C4×D7111yescharacteristic
135C4⋊Q8C4×D7111yescharacteristic
136C4⋊Q8D28411yescharacteristic
137C4×Q8—42no—
138#117—21no—
139#117—41no—
140C2×C4⋊C4—41no—
141C2×C42C22⋊D7111yescharacteristic
142C4⋊C8—1422no—
143#103—42no—
144#101—41no—
145#101—2841no—
146#101—1422no—
147C22⋊C8—2841no—
148#97—2841no—
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×C4C22⋊Dic7111yescharacteristic
163M16—41no—
164M16—41no—
165M16—2842no—
166M16—5681no—
167C2×C8—41no—
168C2×C8—2842no—
169C2×C8—5681no—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C22⋊C4—41no—
177C42—21no—
178C42—41no—
179C42—42no—
180C42(C2×D71)⋊C411yescharacteristic
181C42Dic71⋊C411yescharacteristic
182C23#1455604311yescharacteristic
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—42no—
203C2×C4C71⋊#9511yescharacteristic
204C2×C4#1455604311yescharacteristic
205C8—81no—
206C8—5682no—
207C8—5682no—
208C22—41no—
209C22#1798576611yescharacteristic
210C22#1798578611yescharacteristic
211C22#1798578911yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2258645811yescharacteristic
subgroupG/H
centerC2#22586458
commutatorC2×Q8×C71C42
FrattiniC2×C4⋊Q8C2×D71
FittingC71×#1886C2
socleC2×C71#1390
prime radicalC23×C71#99
cosocleC2×C4⋊Q8×C71C22
hypercenter#1886D71
hypocenterC71#7483
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
7. #17985798C4semidirect
9. #17985748C4semidirect
116. C71#7483semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225870662
C71×#18862
#748371