Finite Group Databasebeta

← C71×#11196 #31832068

Subgroups — C71×#11196 #31832068

Order 18176.

2470 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 2470 conjugacy classes of subgroups — too many to draw. Only its 254 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×#379×C71C212yesmaximal
2C2×#379×C71C211yesmaximal, characteristic
3C2×#342×C71C214yesmaximal, complemented
4C26×C71C2211yescharacteristic
5C22×C22⋊C4×C71—24no—
6C22×C22⋊C4×C71C414yescomplemented
7C22×C22⋊C4×C71C2212yes—
8C22×C22⋊C4×C71C2214yes—
9#379×C71—24no—
10#379×C71—28no—
11#342×C71—28no—
12C25×C71—24no—
13C25×C71—24no—
14C25×C71—24no—
15C25×C71—48no—
16C25×C71C2×C412yes—
17C25×C71D414yes—
18C25×C71C2311yescharacteristic
19C23×C4×C71—24no—
20C23×C4×C71C2×C414yes—
21C2×C22⋊C4×C71—24no—
22C2×C22⋊C4×C71—24no—
23C2×C22⋊C4×C71—24no—
24C2×C22⋊C4×C71—24no—
25C2×C22⋊C4×C71—216no—
26C2×C22⋊C4×C71—24no—
27C2×C22⋊C4×C71—24no—
28C2×C22⋊C4×C71—24no—
29C2×C22⋊C4×C71—22no—
30C2×C22⋊C4×C71—22no—
31C2×C22⋊C4×C71—24no—
32C2×C22⋊C4×C71—24no—
33C2×C22⋊C4×C71—416no—
34C2×C22⋊C4×C71—48no—
35C2×C22⋊C4×C71D418yes—
36C24×C71—28no—
37C24×C71—216no—
38C24×C71—24no—
39C24×C71—24no—
40C24×C71—24no—
41C24×C71—28no—
42C24×C71—28no—
43C24×C71—28no—
44C24×C71—44no—
45C24×C71—48no—
46C24×C71—44no—
47C24×C71—432no—
48C24×C71—432no—
49C24×C71—432no—
50C24×C71—44no—
51C24×C71—44no—
52C24×C71—44no—
53C24×C71—44no—
54C24×C71C22⋊C414yes—
55C24×C71C22×C411yescharacteristic
56C24×C71C2×D412yes—
57C24×C71C2×D414yes—
58C24×C71#4114yes—
59C24×C71#4114yes—
60C22×C4×C71—24no—
61C22×C4×C71—28no—
62C22×C4×C71—28no—
63C22×C4×C71—416no—
64C22×C4×C71—44no—
65C22×C4×C71—44no—
66C22×C4×C71—48no—
67C22⋊C4×C71—28no—
68C22⋊C4×C71—28no—
69C22⋊C4×C71—432no—
70C22⋊C4×C71—48no—
71C22⋊C4×C71—48no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—44no—
74C22⋊C4×C71—44no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—44no—
77C23×C71—24no—
78C23×C71—28no—
79C23×C71—216no—
80C23×C71—216no—
81C23×C71—216no—
82C23×C71—24no—
83C23×C71—28no—
84C23×C71—24no—
85C23×C71—28no—
86C23×C71—48no—
87C23×C71—48no—
88C23×C71—48no—
89C23×C71—432no—
90C23×C71—432no—
91C23×C71—432no—
92C23×C71—48no—
93C23×C71—48no—
94C23×C71—48no—
95C23×C71—432no—
96C23×C71—432no—
97C23×C71—432no—
98C23×C71—432no—
99C23×C71—48no—
100C23×C71—48no—
101C23×C71—48no—
102C23×C71#9918yes—
103C23×C71C2×C22⋊C411yescharacteristic
104C23×C71#11712yes—
105C23×C71C4×D414yes—
106C23×C71#12012yes—
107C23×C71#12114yes—
108C23×C71#12114yes—
109C23×C71#12314yes—
110C23×C71#12312yes—
111C23×C71#12314yes—
112C23×C71#12414yes—
113C23×C71#12614yes—
114C2×C4×C71—416no—
115C2×C4×C71—44no—
116C2×C4×C71—48no—
117C2×C4×C71—48no—
118C2×C4×C71—88no—
119C2×C4×C71—88no—
120C2×C4×C71—88no—
121C22×C71—28no—
122C22×C71—24no—
123C22×C71—24no—
124C22×C71—28no—
125C22×C71—28no—
126C22×C71—24no—
127C22×C71—28no—
128C22×C71—216no—
129C22×C71—44no—
130C22×C71—44no—
131C22×C71—44no—
132C22×C71—44no—
133C22×C71—432no—
134C22×C71—432no—
135C22×C71—432no—
136C22×C71—44no—
137C22×C71—48no—
138C22×C71—44no—
139C22×C71#38111yescharacteristic
140C22×C71#38612yes—
141C22×C71#38814yes—
142C22×C71#39414yes—
143C22×C71#39714yes—
144C22×C71C2×#9914yes—
145C4×C71—88no—
146C4×C71—84no—
147#11196C7111yesmaximal, characteristic, direct product factor, Sylow
148C2×C71—24no—
149C2×C71—24no—
150C2×C71—24no—
151C2×C71—48no—
152C2×C71#144011yescharacteristic
153C2×C71#181912yes—
154C2×C71#189814yes—
155C2×#379C2×C7112yes—
156C2×#379C2×C7111yescharacteristic
157C2×#342C2×C7114yescomplemented
158C71#1119611yescharacteristic, direct product factor, Sylow
159C26C22×C7111yescharacteristic
160C22×C22⋊C4—24no—
161C22×C22⋊C4C4×C7114yescomplemented
162C22×C22⋊C4C22×C7112yes—
163C22×C22⋊C4C22×C7114yes—
164#379—24no—
165#379—28no—
166#342—28no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—48no—
171C25C2×C4×C7112yes—
172C25D4×C7114yes—
173C25C23×C7111yescharacteristic
174C23×C4—24no—
175C23×C4C2×C4×C7114yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—216no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—416no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4D4×C7118yes—
191C24—28no—
192C24—216no—
193C24—24no—
194C24—24no—
195C24—24no—
196C24—28no—
197C24—28no—
198C24—28no—
199C24—44no—
200C24—48no—
201C24—44no—
202C24—432no—
203C24—432no—
204C24—432no—
205C24—44no—
206C24—44no—
207C24—44no—
208C24—44no—
209C24C22⋊C4×C7114yes—
210C24C22×C4×C7111yescharacteristic
211C24C2×D4×C7112yes—
212C24C2×D4×C7114yes—
213C24#41×C7114yes—
214C24#41×C7114yes—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—416no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—48no—
222C22⋊C4—28no—
223C22⋊C4—28no—
224C22⋊C4—432no—
225C22⋊C4—48no—
226C22⋊C4—48no—
227C22⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C23—24no—
233C23—28no—
234C23—216no—
235C23—216no—
236C23—216no—
237C23—24no—
238C23—28no—
239C23—24no—
240C23—28no—
241C23—48no—
242C23—48no—
243C23—48no—
244C23—432no—
245C23—432no—
246C23—432no—
247C23—48no—
248C23—48no—
249C23—48no—
250C23—432no—
251C23—432no—
252C23—432no—
253C23—432no—
254C23—48no—
255C23—48no—
256C23—48no—
257C23#99×C7118yes—
258C23C2×C22⋊C4×C7111yescharacteristic
259C23#117×C7112yes—
260C23C4×D4×C7114yes—
261C23#120×C7112yes—
262C23#121×C7114yes—
263C23#121×C7114yes—
264C23#123×C7114yes—
265C23#123×C7112yes—
266C23#123×C7114yes—
267C23#124×C7114yes—
268C23#126×C7114yes—
269C2×C4—416no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—48no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4—88no—
276C22—28no—
277C22—24no—
278C22—24no—
279C22—28no—
280C22—28no—
281C22—24no—
282C22—28no—
283C22—216no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—432no—
289C22—432no—
290C22—432no—
291C22—44no—
292C22—48no—
293C22—44no—
294C22#381×C7111yescharacteristic
295C22#386×C7112yes—
296C22#388×C7114yes—
297C22#394×C7114yes—
298C22#397×C7114yes—
299C22C2×#99×C7114yes—
300C4—88no—
301C4—84no—
302C2—24no—
303C2—24no—
304C2—24no—
305C2—48no—
306C2C71×#144011yescharacteristic
307C2C71×#181912yes—
308C2C71×#189814yes—
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC24C22×C4×C71
Frattini, cosocleC25C23×C71
prime radicalC26×C71C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C71 ◁ C26×C71 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#342×C71C2semidirect
6. C22×C22⋊C4×C71C4semidirect
147. #11196C71direct (N × H)
157. C2×#342C2×C71semidirect
158. C71#11196direct (N × H)
161. C22×C22⋊C4C4×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#379×C712
C2×#342×C712
#1119671