Finite Group Databasebeta

← #31908204

Subgroups — #31908204

Order 18176.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588423C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3#22586367C214yesmaximal, complemented
4C71×#2628C211yesmaximal, characteristic
5#17986056—21no—
6#17985963—21no—
7#17985768—21no—
8#17985768C2212yes—
9#17985725C2212yes—
10#17985723—21no—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×#117×C71C2211yescharacteristic
13#400×C71C2212yes—
14#399×C71C412yes—
15#395×C71—21no—
16#393×C71—21no—
17#389×C71—21no—
18#382×C71C411yescharacteristic
19#380×C71—21no—
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—41no—
24C22⋊(C71⋊C8)—41no—
25#14556039—41no—
26C22×Q8×C71—21no—
27C23×C4×C71D411yescharacteristic
28C4⋊Q8×C71—41no—
29C4⋊Q8×C71—41no—
30#117×C71—41no—
31#117×C71—41no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71C2×C412yes—
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C4⋊C4×C71C2311yescharacteristic
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71C2×C411yescharacteristic
40C2×C22⋊C4×C71D411yescharacteristic
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43#95×C71—21no—
44#95×C71—22no—
45#95×C71—22no—
46#95×C71C2×C412yes—
47#11765849—42no—
48#11765849—42no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—42no—
51C24×C71C22⋊C411yescharacteristic
52C2×Q8×C71—42no—
53C2×Q8×C71—82no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—41no—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22×C411yescharacteristic
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—82no—
73C22⋊C4×C71—41no—
74C22⋊C4×C71—41no—
75C22⋊C4×C71—82no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C23×C71—21no—
81C23×C71—21no—
82C23×C71—21no—
83C23×C71—41no—
84C23×C71—41no—
85C23×C71C2×C22⋊C411yescharacteristic
86Q8×C71—84no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71#9911yescharacteristic
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71#13712yes—
110C71⋊C8—82no—
111C71⋊C8—82no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71—42no—
115C22×C71—41no—
116C22×C71—41no—
117C22×C71—41no—
118C22×C71—81no—
119C22×C71C2×#9911yescharacteristic
120C22×C71#41911yescharacteristic
121C22×C71#42111yescharacteristic
122C4×C71—22no—
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—41no—
126C4×C71—41no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71—82no—
130C4×C71—82no—
131#11818—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71—81no—
135C2×C71#151111yescharacteristic
136C2×C71#212311yescharacteristic
137C2×C71#213411yescharacteristic
138#2628D7111yescharacteristic
139#1822—711no—
140#1464—711no—
141#1353—714no—
142C71#1181811yescharacteristic, complemented, Sylow
143C2×C4⋊Q8Dic7111yescharacteristic
144C2×#117C2×D7111yescharacteristic
145#432—1421no—
146#407—1421no—
147#400C2×D7112yes—
148#399Dic7112yes—
149#395—21no—
150#393—21no—
151#389—21no—
152#382Dic7111yescharacteristic
153#380—21no—
154#343—712no—
155#343—1421no—
156#324—712no—
157#323—1421no—
158C22×Q8—21no—
159C23×C4C22⋊D7111yescharacteristic
160C2×M16—1422no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×Dic7112yes—
169C2×C4⋊C4C22⋊D7111yescharacteristic
170C2×C4⋊C4C22×D7111yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4C2×Dic7111yescharacteristic
173C2×C22⋊C4C22⋊D7111yescharacteristic
174C2×C42C2×Dic7111yescharacteristic
175C2×C42C22⋊D7111yescharacteristic
176C4⋊C8—2841no—
177C22⋊C8—2841no—
178C22⋊C8—2841no—
179C22⋊C8—1422no—
180#97—2841no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95C2×Dic7112yes—
185C24C22⋊Dic7111yescharacteristic
186C2×Q8—42no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C22⋊Dic7111yescharacteristic
196C22×C4C22⋊Dic7111yescharacteristic
197C22×C4C22×Dic7111yescharacteristic
198C22×C4C2×C22⋊D7111yescharacteristic
199C22×C4C2×C22⋊D7111yescharacteristic
200M16—2842no—
201M16—2842no—
202C2×C8—2842no—
203C2×C8—2842no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23C2×C22⋊Dic7111yescharacteristic
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4#1455604311yescharacteristic
246C2×C4#1455604311yescharacteristic
247C2×C4#1455615912yes—
248C8—5682no—
249C8—5682no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22C2×#1455604311yescharacteristic
258C22#1798600811yescharacteristic
259C22#1798601611yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4—82no—
269C2—21no—
270C2—21no—
271C2—81no—
272C2#2258687311yescharacteristic
273C2#2258999411yescharacteristic
274C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2628C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2628D71
hypocenterC71#11818
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #22586367C2semidirect
142. C71#11818semidirect

Maximal subgroups

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

subgroupindex
#225884232
#225866742
#225863672
C71×#26282
#1181871