Finite Group Databasebeta

← #31912638

Subgroups — #31912638

Order 18176.

464 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 135 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×#17985798C212yesmaximal
2C4×#14556047C212yesmaximal
3#22586467C212yesmaximal
4C71×#2327C211yesmaximal, characteristic
5C2×#14556047C2214yes—
6C4×#11765849—22no—
7#17985798C418yescomplemented
8#17985768—22no—
9C2×C4⋊Q8×C71C2211yescharacteristic
10C2×C4×Q8×C71C411yescharacteristic
11C2×C4×Q8×C71C2211yescharacteristic
12#391×C71C411yescharacteristic
13#391×C71C2211yescharacteristic
14#387×C71—22no—
15#387×C71—22no—
16#382×C71C411yescharacteristic
17#376×C71C411yescharacteristic
18C2×#11765849—24no—
19#14556047—24no—
20#14556047C2×C418yes—
21#14556039—42no—
22C4×C71⋊C8—42no—
23C22×Q8×C71C2×C411yescharacteristic
24C22×Q8×C71C2311yescharacteristic
25C4⋊Q8×C71—22no—
26C4⋊Q8×C71C2×C414yes—
27C4×Q8×C71—42no—
28C4×Q8×C71—42no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71D412yes—
34C2×C42×C71—21no—
35C2×C42×C71C2×C411yescharacteristic
36C2×C42×C71C2×C411yescharacteristic
37C2×C42×C71C2×C411yescharacteristic
38C2×C42×C71D412yes—
39#95×C71—21no—
40#95×C71—21no—
41#95×C71—22no—
42#95×C71—22no—
43#95×C71—22no—
44#11765849—28no—
45#11765849—44no—
46C2×C71⋊C8—44no—
47C2×Q8×C71—21no—
48C2×Q8×C71—41no—
49C2×Q8×C71—41no—
50C2×Q8×C71—41no—
51C2×Q8×C71—41no—
52C2×Q8×C71C4212yes—
53C2×Q8×C71C22×C412yes—
54C2×Q8×C71#4112yes—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71C22⋊C412yes—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22×C411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—28no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—42no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C42×C71—22no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—21no—
73C42×C71—41no—
74C42×C71—42no—
75C42×C71C4212yes—
76C23×C71C2×C22⋊C411yescharacteristic
77Q8×C71—42no—
78Q8×C71—42no—
79Q8×C71—42no—
80Q8×C71—42no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—42no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71#9912yes—
102C2×C4×C71C2×C4211yescharacteristic
103C2×C4×C71#11711yescharacteristic
104C2×C4×C71C4×D414yes—
105C71⋊C8—48no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71C4×C22⋊C411yescharacteristic
109C22×C71C2×#9911yescharacteristic
110C22×C71#41011yescharacteristic
111C4×C71—22no—
112C4×C71—24no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—42no—
116C4×C71—41no—
117C4×C71—42no—
118C4×C71—42no—
119C4×C71—81no—
120C4×C71—81no—
121#12631—711nomaximal, Sylow
122C2×C71—21no—
123C2×C71—21no—
124C2×C71C4×#9911yescharacteristic
125C2×C71#213311yescharacteristic
126C2×C71#213511yescharacteristic
127#2327D7111yescharacteristic
128C2×#356—712no—
129C4×#101—712no—
130#1394—712no—
131C71#1263111yescharacteristic, complemented, Sylow
132C2×C4⋊Q8C2×D7111yescharacteristic
133C2×C4×Q8Dic7111yescharacteristic
134C2×C4×Q8C2×D7111yescharacteristic
135C2×#101—714no—
136C4×M16—1422no—
137#391Dic7111yescharacteristic
138#391C2×D7111yescharacteristic
139#387—22no—
140#387—22no—
141#382Dic7111yescharacteristic
142#376Dic7111yescharacteristic
143#356—718no—
144#343—1422no—
145C22×Q8C2×Dic7111yescharacteristic
146C22×Q8C22×D7111yescharacteristic
147C2×M16—1424no—
148C4⋊Q8—22no—
149C4⋊Q8C4×D7114yes—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×Dic7111yescharacteristic
155C2×C4⋊C4C2×Dic7111yescharacteristic
156C2×C4⋊C4C22⋊D7112yes—
157C2×C42—21no—
158C2×C42C2×Dic7111yescharacteristic
159C2×C42C2×Dic7111yescharacteristic
160C2×C42C2×Dic7111yescharacteristic
161C2×C42C22⋊D7112yes—
162#101—718no—
163#101—1424no—
164#97—2842no—
165C4×C8—2842no—
166#95—21no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95—22no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C4×Dic7112yes—
177C2×Q8C2×C4×D7112yes—
178C2×Q8#1176586512yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4C22⋊Dic7112yes—
184C22×C4C22⋊Dic7111yescharacteristic
185C22×C4C22⋊Dic7111yescharacteristic
186C22×C4C22×Dic7111yescharacteristic
187C22×C4C2×C22⋊D7112yes—
188M16—2844no—
189M16—1428no—
190C2×C8—2844no—
191C4⋊C4—28no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—41no—
201C42—42no—
202C42C4×Dic7112yes—
203C23C2×C22⋊Dic7111yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4#1455604312yes—
229C2×C4C2×C4×Dic7111yescharacteristic
230C2×C4#1455608711yescharacteristic
231C2×C4C4×C22⋊D7114yes—
232C8—2848no—
233C22—21no—
234C22—21no—
235C22C4×C22⋊Dic7111yescharacteristic
236C22C2×#1455604311yescharacteristic
237C22#1798597311yescharacteristic
238C4—22no—
239C4—24no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—81no—
247C4—81no—
248C2—21no—
249C2—21no—
250C2C4×#1455604311yescharacteristic
251C2#2259002911yescharacteristic
252C2#2259003511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC2×C4×C71C2×C42
FrattiniC22×Q8C22×D71
FittingC71×#2327C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC22×Q8×C71C23
hypercenter#2327D71
hypocenterC71#12631
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2327

Complements

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

NH ≅ G/Ntype
7. #17985798C4semidirect
131. C71#12631semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
C4×#145560472
#225864672
C71×#23272
#1263171