Finite Group Databasebeta

← #31912961

Subgroups — #31912961

Order 18176.

1018 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17985798C214yesmaximal, complemented
2#22588264C212yesmaximal
3C22×C4⋊Q8×C71C211yesmaximal, characteristic
4C2×#14556047—22no—
5C2×#14556047C2214yes—
6#17985963—22no—
7#17985798—28no—
8C23×Q8×C71C411yescharacteristic
9C23×Q8×C71C2211yescharacteristic
10C2×C4⋊Q8×C71—22no—
11C2×C4⋊Q8×C71—28no—
12C2×C4⋊Q8×C71C412yes—
13C2×C4⋊Q8×C71C2212yes—
14C22×C4⋊C4×C71—22no—
15C22×C42×C71C411yescharacteristic
16C2×#11765849—24no—
17#14556047—28no—
18#14556047—44no—
19C22⋊(C71⋊C8)—44no—
20C22×Q8×C71—24no—
21C22×Q8×C71—24no—
22C22×Q8×C71—21no—
23C22×Q8×C71—42no—
24C22×Q8×C71—42no—
25C22×Q8×C71—42no—
26C22×Q8×C71—42no—
27C22×Q8×C71C2×C412yes—
28C22×Q8×C71C2×C411yescharacteristic
29C22×Q8×C71D412yes—
30C22×Q8×C71C2311yescharacteristic
31C23×C4×C71—21no—
32C23×C4×C71—21no—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D412yes—
35C4⋊Q8×C71—216no—
36C4⋊Q8×C71—24no—
37C4⋊Q8×C71—24no—
38C4⋊Q8×C71—24no—
39C4⋊Q8×C71D418yes—
40C2×C4⋊C4×C71—216no—
41C2×C4⋊C4×C71—24no—
42C2×C4⋊C4×C71—24no—
43C2×C42×C71—21no—
44C2×C42×C71—24no—
45C2×C42×C71C2×C412yes—
46#11765849—44no—
47#11765849—44no—
48C2×C71⋊C8—44no—
49C24×C71C22⋊C411yescharacteristic
50C2×Q8×C71—24no—
51C2×Q8×C71—24no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—24no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57C2×Q8×C71—42no—
58C2×Q8×C71—41no—
59C2×Q8×C71—41no—
60C2×Q8×C71—416no—
61C2×Q8×C71—416no—
62C2×Q8×C71—42no—
63C2×Q8×C71—42no—
64C2×Q8×C71—42no—
65C2×Q8×C71—42no—
66C2×Q8×C71C22⋊C414yes—
67C2×Q8×C71C2×D414yes—
68C22×C4×C71—24no—
69C22×C4×C71—28no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—24no—
75C22×C4×C71—24no—
76C22×C4×C71—22no—
77C22×C4×C71—24no—
78C22×C4×C71—42no—
79C22×C4×C71—42no—
80C22×C4×C71—42no—
81C22×C4×C71C22⋊C412yes—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C4⋊C4×C71—216no—
86C4⋊C4×C71—216no—
87C42×C71—24no—
88C42×C71—22no—
89C42×C71C22⋊C414yes—
90C23×C71—24no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71#9912yes—
94C23×C71C2×C22⋊C411yescharacteristic
95Q8×C71—44no—
96Q8×C71—48no—
97Q8×C71—44no—
98Q8×C71—48no—
99Q8×C71—44no—
100Q8×C71—44no—
101C2×C4×C71—216no—
102C2×C4×C71—28no—
103C2×C4×C71—28no—
104C2×C4×C71—21no—
105C2×C4×C71—24no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—24no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—48no—
116C2×C4×C71—48no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71#9912yes—
120C2×C4×C71C2×C22⋊C412yes—
121C2×C4×C71#12014yes—
122C71⋊C8—84no—
123C22×C71—24no—
124C22×C71—24no—
125C22×C71—22no—
126C22×C71—22no—
127C22×C71—22no—
128C22×C71#35614yescomplemented
129C22×C71#37911yescharacteristic
130C22×C71C2×#9911yescharacteristic
131C22×C71C2×#9911yescharacteristic
132C4×C71—28no—
133C4×C71—24no—
134C4×C71—22no—
135C4×C71—22no—
136C4×C71—42no—
137C4×C71—44no—
138#12700—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—24no—
141C2×C71—21no—
142C2×C71#178111yescharacteristic
143C2×C71C2×#35612yes—
144C22×C4⋊Q8D7111yescharacteristic
145C2×#356—714no—
146#1784—712no—
147C71#1270011yescharacteristic, complemented, Sylow
148C23×Q8Dic7111yescharacteristic
149C23×Q8C2×D7111yescharacteristic
150C2×C4⋊Q8—22no—
151C2×C4⋊Q8—28no—
152C2×C4⋊Q8Dic7112yes—
153C2×C4⋊Q8C2×D7112yes—
154C22×C4⋊C4—22no—
155C22×C42Dic7111yescharacteristic
156C2×#101—714no—
157C2×#101—1422no—
158#407—1422no—
159#356—1428no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8C2×Dic7112yes—
168C22×Q8C2×Dic7111yescharacteristic
169C22×Q8C22⋊D7112yes—
170C22×Q8C22×D7111yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×Dic7111yescharacteristic
174C23×C4C22⋊D7112yes—
175C2×M16—1424no—
176C4⋊Q8—216no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8C22⋊D7118yes—
181C2×C4⋊C4—216no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42C2×Dic7112yes—
187#101—2844no—
188#101—1428no—
189C22⋊C8—2844no—
190C24C22⋊Dic7111yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊Dic7114yes—
208C2×Q8C2×C22⋊D7114yes—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22⋊Dic7112yes—
223C22×C4C22⋊Dic7111yescharacteristic
224C22×C4C22×Dic7111yescharacteristic
225C22×C4C2×C22⋊D7112yes—
226M16—2844no—
227M16—2844no—
228C2×C8—2844no—
229C4⋊C4—216no—
230C4⋊C4—216no—
231C42—24no—
232C42—22no—
233C42C22⋊Dic7114yes—
234C23—24no—
235C23—21no—
236C23—21no—
237C23#1455604312yes—
238C23C2×C22⋊Dic7111yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—48no—
243Q8—44no—
244Q8—44no—
245C2×C4—28no—
246C2×C4—216no—
247C2×C4—28no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4#1455604312yes—
264C2×C4C2×C22⋊Dic7112yes—
265C2×C4#1455609714yes—
266C8—5684no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#1798579814yescomplemented
273C22#1798585511yescharacteristic
274C22C2×#1455604311yescharacteristic
275C22C2×#1455604311yescharacteristic
276C4—28no—
277C4—24no—
278C4—22no—
279C4—22no—
280C4—42no—
281C4—44no—
282C2—21no—
283C2—24no—
284C2—21no—
285C2#2258825511yescharacteristic
286C2C2×#1798579812yes—
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC22×C4⋊Q8×C71C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC22×Q8×C71C23
hypercenterC22×C4⋊Q8D71
hypocenterC71#12700
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17985798C2semidirect
128. C22×C71#356semidirect
147. C71#12700semidirect
272. C22#17985798semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225882642
C22×C4⋊Q8×C712
#1270071