Finite Group Databasebeta

← #31906948

Subgroups — #31906948

Order 18176.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#22586684C212yesmaximal
2#22586284C212yesmaximal
3#22586274C212yesmaximal
4C71×#2327C211yesmaximal, characteristic
5#17986056—22no—
6#17985725—22no—
7#17985725C2214yes—
8C2×C4⋊Q8×C71C411yescharacteristic
9C2×C4×Q8×C71C411yescharacteristic
10C2×C4×Q8×C71C2211yescharacteristic
11#391×C71C411yescharacteristic
12#391×C71C2211yescharacteristic
13#387×C71—22no—
14#387×C71—22no—
15#382×C71C2211yescharacteristic
16#376×C71C411yescharacteristic
17C4⋊(C71⋊C8)—42no—
18C22⋊(C71⋊C8)—24no—
19#14556039—42no—
20C22×Q8×C71C2×C411yescharacteristic
21C22×Q8×C71C2×C411yescharacteristic
22C4⋊Q8×C71—22no—
23C4⋊Q8×C71—22no—
24C4×Q8×C71—42no—
25C4×Q8×C71—42no—
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71C2×C411yescharacteristic
29C2×C4⋊C4×C71D412yes—
30C2×C4⋊C4×C71C2311yescharacteristic
31C2×C42×C71—21no—
32C2×C42×C71C2×C411yescharacteristic
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71C2×C411yescharacteristic
35C2×C42×C71D412yes—
36#95×C71—21no—
37#95×C71—21no—
38#95×C71—22no—
39#95×C71—22no—
40#95×C71—22no—
41C2×C71⋊C8—44no—
42C2×Q8×C71—21no—
43C2×Q8×C71—21no—
44C2×Q8×C71—21no—
45C2×Q8×C71—41no—
46C2×Q8×C71—41no—
47C2×Q8×C71—41no—
48C2×Q8×C71—41no—
49C2×Q8×C71M1612yes—
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71C22⋊C412yes—
55C22×C4×C71C22⋊C411yescharacteristic
56C22×C4×C71C22⋊C411yescharacteristic
57C22×C4×C71C22×C411yescharacteristic
58C22×C4×C71C2×D412yes—
59C4⋊C4×C71—28no—
60C4⋊C4×C71—42no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—42no—
64C42×C71—22no—
65C42×C71—22no—
66C42×C71—22no—
67C42×C71—21no—
68C42×C71—41no—
69C42×C71—42no—
70C42×C71M1612yes—
71C23×C71C2×C22⋊C411yescharacteristic
72Q8×C71—42no—
73Q8×C71—42no—
74Q8×C71—42no—
75Q8×C71—42no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—42no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—42no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71#9912yes—
98C2×C4×C71C2×M1611yescharacteristic
99C2×C4×C71C2×M1611yescharacteristic
100C71⋊C8—84no—
101C22×C71—21no—
102C22×C71—21no—
103C22×C71#40711yescharacteristic
104C22×C71C2×#9911yescharacteristic
105C22×C71#41311yescharacteristic
106C4×C71—22no—
107C4×C71—21no—
108C4×C71—21no—
109C4×C71—42no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C71—42no—
113C4×C71—42no—
114C4×C71—81no—
115C4×C71—81no—
116#11536—711nomaximal, Sylow
117C2×C71—21no—
118C2×C71—21no—
119C2×C71#146511yescharacteristic
120C2×C71#212911yescharacteristic
121C2×C71#213311yescharacteristic
122#2327D7111yescharacteristic
123#1467—712no—
124#1324—712no—
125#1319—712no—
126C71#1153611yescharacteristic, complemented, Sylow
127C2×C4⋊Q8Dic7111yescharacteristic
128C2×C4×Q8Dic7111yescharacteristic
129C2×C4×Q8C2×D7111yescharacteristic
130#432—1422no—
131#391Dic7111yescharacteristic
132#391C2×D7111yescharacteristic
133#387—22no—
134#387—22no—
135#382C2×D7111yescharacteristic
136#376Dic7111yescharacteristic
137#324—714no—
138#324—1422no—
139C22×Q8C2×Dic7111yescharacteristic
140C22×Q8C2×Dic7111yescharacteristic
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4×Q8—42no—
144C4×Q8—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic7111yescharacteristic
148C2×C4⋊C4C22⋊D7112yes—
149C2×C4⋊C4C22×D7111yescharacteristic
150C2×C42—21no—
151C2×C42C2×Dic7111yescharacteristic
152C2×C42C2×Dic7111yescharacteristic
153C2×C42C2×Dic7111yescharacteristic
154C2×C42C22⋊D7112yes—
155C4⋊C8—2842no—
156C22⋊C8—1424no—
157#97—2842no—
158#95—21no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8#1176584912yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C22⋊Dic7112yes—
176C22×C4C22⋊Dic7111yescharacteristic
177C22×C4C22⋊Dic7111yescharacteristic
178C22×C4C22×Dic7111yescharacteristic
179C22×C4C2×C22⋊D7112yes—
180C2×C8—2844no—
181C4⋊C4—28no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—21no—
190C42—41no—
191C42—42no—
192C42#1176584912yes—
193C23C2×C22⋊Dic7111yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4#1455604312yes—
220C2×C4C2×#1176584911yescharacteristic
221C2×C4C2×#1176584911yescharacteristic
222C8—5684no—
223C22—21no—
224C22—21no—
225C22#1798596311yescharacteristic
226C22C2×#1455604311yescharacteristic
227C22#1798598311yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2#2258667811yescharacteristic
241C2#2259001511yescharacteristic
242C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2327C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2327D71
hypocenterC71#11536
LCSG ▷ C22×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
126. C71#11536semidirect

Maximal subgroups

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

subgroupindex
#225866842
#225862842
#225862742
C71×#23272
#1153671