Finite Group Databasebeta

← #31885209

Subgroups — #31885209

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
2C2×#17985748C211yesmaximal, characteristic
3C71×#1886C211yesmaximal, characteristic
4C2×C4⋊(C71⋊C8)—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—
16C22×C71⋊C8—41no—
17C4⋊(C71⋊C8)—22no—
18C4⋊(C71⋊C8)—41no—
19#14556047—22no—
20#14556047—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—82no—
40C2×C71⋊C8—81no—
41C2×Q8×C71—21no—
42C2×Q8×C71—21no—
43C2×Q8×C71—41no—
44C2×Q8×C71—41no—
45C2×Q8×C71—41no—
46C2×Q8×C71—41no—
47C2×Q8×C71C4211yescharacteristic
48C2×Q8×C71C22⋊C411yescharacteristic
49C2×Q8×C71C4⋊C411yescharacteristic
50C2×Q8×C71C4⋊C411yescharacteristic
51C22×C4×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71C22⋊C411yescharacteristic
55M16×C71—41no—
56M16×C71—41no—
57C2×C8×C71—41no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—41no—
60C4⋊C4×C71—41no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—42no—
64C22⋊C4×C71—41no—
65C42×C71—21no—
66C42×C71—41no—
67C42×C71—42no—
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—42no—
92C2×C4×C71#9511yescharacteristic
93C2×C4×C71#9911yescharacteristic
94C8×C71—81no—
95C71⋊C8—82no—
96C71⋊C8—82no—
97C22×C71—22no—
98C22×C71#34211yescharacteristic
99C22×C71AΣL(1,𝔽16)11yescharacteristic
100C22×C71#35211yescharacteristic
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—22no—
104C4×C71—21no—
105C4×C71—21no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—41no—
109C4×C71—41no—
110C4×C71—82no—
111#7491—711nomaximal, Sylow
112C2×C71—21no—
113C2×C71—21no—
114C2×C71#139011yescharacteristic
115C2×C71#140512yes—
116C2×#356—711no—
117#1886D7111yescharacteristic
118C2×#333—711no—
119C71#749111yescharacteristic, complemented, Sylow
120C2×C4⋊Q8C2×D7111yescharacteristic
121#519Dic7111yescharacteristic
122C2×C4⋊C8—1421no—
123#419—21no—
124C2×#101—1421no—
125C2×#101Dic7111yescharacteristic
126#356—712no—
127#356—1421no—
128#339—21no—
129#333—712no—
130#333—1421no—
131C22×Q8C2×Dic7111yescharacteristic
132C22×Q8C22⋊D7111yescharacteristic
133C2×M16—21no—
134C2×M16—2841no—
135C22×C8—2841no—
136C4⋊Q8—41no—
137C4⋊Q8Dic14211yescharacteristic
138C4⋊Q8C4×D7111yescharacteristic
139C4⋊Q8C4×D7111yescharacteristic
140C4⋊Q8D28411yescharacteristic
141C4×Q8—42no—
142#117—21no—
143#117—41no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D7111yescharacteristic
146C4⋊C8—2841no—
147C4⋊C8—1422no—
148#103—42no—
149#101—41no—
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×C4C22⋊Dic7111yescharacteristic
166M16—41no—
167M16—41no—
168M16—2842no—
169M16—5681no—
170C2×C8—41no—
171C2×C8—2842no—
172C2×C8—5682no—
173C2×C8—5681no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42—42no—
184C42(C2×D71)⋊C411yescharacteristic
185C42Dic71⋊C411yescharacteristic
186C23#1455604311yescharacteristic
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—42no—
208C2×C4C71⋊#9511yescharacteristic
209C2×C4#1455604311yescharacteristic
210C8—81no—
211C8—5682no—
212C8—5682no—
213C22—22no—
214C22#1798576611yescharacteristic
215C22#1798578611yescharacteristic
216C22#1798578911yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—82no—
227C2—21no—
228C2—21no—
229C2#2258645811yescharacteristic
230C2#2258649412yes—
subgroupG/H
centerC22#17985786
commutatorC2×Q8×C71C42
FrattiniC2×C4⋊Q8C2×D71
FittingC71×#1886C2
socleC22×C71AΣL(1,𝔽16)
prime radicalC23×C71#99
cosocleC2×C4⋊Q8×C71C22
hypercenter#1886D71
hypocenterC71#7491
LCSG ▷ C2×Q8×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ 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
119. C71#7491semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
C2×#179857482
C71×#18862
#749171