Finite Group Databasebeta

← #31889598

Subgroups — #31889598

Order 18176.

746 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 143 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#22589139C211yesmaximal, characteristic
2#22587040C212yesmaximal, complemented
3#22586955C212yesmaximal, complemented
4#22586681C211yesmaximal, characteristic
5C71×#3501C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8C2×C22⋊(C71⋊C8)—22no—
9C2×#14556039C2211yescharacteristic
10#17985745—22no—
11#17985745—22no—
12#17985741—24no—
13#17985723—22no—
14#554×C71—22no—
15#554×C71—24no—
16D4×Q8×C71—22no—
17#544×C71C412yes—
18#543×C71—22no—
19#533×C71C2212yes—
20C2×C22⋊Q8×C71—22no—
21#518×C71C412yes—
22C2×C4×D4×C71C2211yescharacteristic
23C22×C71⋊C8—21no—
24C22×C71⋊C8—21no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—44no—
29#14556039—22no—
30#14556039—22no—
31C22×Q8×C71—22no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71D412yes—
34C4⋊Q8×C71—24no—
35C4⋊Q8×C71D412yes—
36C4⋊Q8×C71D412yes—
37#125×C71—24no—
38#125×C71D414yes—
39C22⋊Q8×C71—24no—
40C22⋊Q8×C71—24no—
41C22⋊Q8×C71—28no—
42C22⋊Q8×C71—28no—
43C22⋊Q8×C71—24no—
44C22⋊Q8×C71—22no—
45C22⋊Q8×C71—22no—
46C4×Q8×C71—22no—
47C4×Q8×C71—22no—
48C4×D4×C71—24no—
49C4×D4×C71—24no—
50C4×D4×C71—22no—
51C4×D4×C71—22no—
52#117×C71C2×C414yes—
53C2×C4⋊C4×C71—22no—
54C2×C4⋊C4×C71—24no—
55C2×C4⋊C4×C71C2×C411yescharacteristic
56C2×C22⋊C4×C71—24no—
57C2×C22⋊C4×C71D412yes—
58C2×C42×C71C2311yescharacteristic
59C2×C71⋊C8—22no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—41no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—41no—
65C24×C71C22⋊C412yes—
66C2×Q8×C71—24no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C2×Q8×C71—82no—
70C2×Q8×C71—82no—
71C2×D4×C71—24no—
72C2×D4×C71—22no—
73C22×C4×C71—24no—
74C22×C4×C71—24no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71C22⋊C412yes—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C4⋊C4×C71—24no—
84C4⋊C4×C71—28no—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—24no—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—24no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71C22⋊C414yes—
91C4⋊C4×C71C22⋊C414yes—
92C22⋊C4×C71—28no—
93C22⋊C4×C71—24no—
94C22⋊C4×C71—24no—
95C42×C71—24no—
96C42×C71C2×D412yes—
97C42×C71C2×D412yes—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—42no—
101C23×C71—42no—
102C23×C71#10114yes—
103C23×C71C2×C22⋊C411yescharacteristic
104Q8×C71—42no—
105Q8×C71—42no—
106Q8×C71—82no—
107Q8×C71—84no—
108D4×C71—28no—
109C2×C4×C71—22no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—24no—
113C2×C4×C71—24no—
114C2×C4×C71—22no—
115C2×C4×C71—41no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—82no—
120C2×C4×C71—84no—
121C2×C4×C71—84no—
122C2×C4×C71C2×C22⋊C412yes—
123C2×C4×C71#12012yes—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12011yescharacteristic
126C71⋊C8—42no—
127C71⋊C8—82no—
128C71⋊C8—81no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—24no—
132C22×C71—44no—
133C22×C71—42no—
134C22×C71—42no—
135C22×C71—82no—
136C22×C71#37911yescharacteristic
137C22×C71C2×#10112yes—
138C4×C71—21no—
139C4×C71—22no—
140C4×C71—22no—
141C4×C71—42no—
142C4×C71—44no—
143C4×C71—44no—
144C4×C71#45311yescharacteristic
145C4×C71#45511yescharacteristic
146#8457—711nomaximal, Sylow
147C2×C71—22no—
148C2×C71—44no—
149C2×C71#178411yescharacteristic
150C2×C71#179211yescharacteristic
151C2×C71#179411yescharacteristic
152#3501D7111yescharacteristic
153#1955—711no—
154#1552—712no—
155#1531—712no—
156#1466—711no—
157C71#845711yescharacteristic, complemented, Sylow
158#554—22no—
159#554—24no—
160D4×Q8—22no—
161#544Dic7112yes—
162#543—22no—
163#533C2×D7112yes—
164C2×C22⋊Q8—22no—
165#518Dic7112yes—
166C2×C4×D4C2×D7111yescharacteristic
167#432—712no—
168C2×C4⋊C8—711no—
169C2×C22⋊C8—1422no—
170C2×#97—711no—
171#332—1422no—
172#332—1422no—
173#330—1424no—
174#323—1422no—
175C22×Q8—22no—
176C22×D4C2×Dic7111yescharacteristic
177C23×C4C22⋊D7112yes—
178C22×C8—1421no—
179C22×C8—1421no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D7112yes—
182C4⋊Q8C22⋊D7112yes—
183#125—24no—
184#125C22⋊D7114yes—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—24no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198#117C2×Dic7114yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4C2×Dic7111yescharacteristic
202C2×C22⋊C4—24no—
203C2×C22⋊C4C22⋊D7112yes—
204C2×C42C22×D7111yescharacteristic
205C4⋊C8—1422no—
206C4⋊C8—1422no—
207C22⋊C8—2844no—
208C22⋊C8—1422no—
209#97—1422no—
210#97—1422no—
211C24C22⋊Dic7112yes—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×D4—24no—
218C2×D4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C22⋊Dic7112yes—
226C22×C4C22×Dic7111yescharacteristic
227C22×C4C2×C22⋊D7111yescharacteristic
228C22×C4C2×C22⋊D7111yescharacteristic
229C2×C8—2841no—
230C2×C8—2841no—
231C2×C8—2842no—
232C2×C8—2842no—
233C2×C8—2841no—
234C2×C8—1422no—
235C4⋊C4—24no—
236C4⋊C4—28no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4C22⋊Dic7114yes—
243C4⋊C4C22⋊Dic7114yes—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C42—24no—
248C42C2×C22⋊D7112yes—
249C42C2×C22⋊D7112yes—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23#1455604714yes—
255C23C2×C22⋊Dic7111yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—82no—
259Q8—84no—
260D4—28no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4C2×C22⋊Dic7112yes—
275C2×C4#1455609712yes—
276C2×C4#1455609711yescharacteristic
277C2×C4#1455609711yescharacteristic
278C8—2842no—
279C8—5682no—
280C8—5681no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#1798585511yescharacteristic
289C22C2×#1455604712yes—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—44no—
295C4—44no—
296C4#1798616711yescharacteristic
297C4#1798617711yescharacteristic
298C2—22no—
299C2—44no—
300C2#2258826411yescharacteristic
301C2#2258829411yescharacteristic
302C2#2258830411yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3501C2
socleC22×C71#379
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#3501D71
hypocenterC71#8457
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #22587040C2semidirect
3. #22586955C2semidirect
157. C71#8457semidirect

Maximal subgroups

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

subgroupindex
#225891392
#225870402
#225869552
#225866812
C71×#35012
#845771