Finite Group Databasebeta

← #31917825

Subgroups — #31917825

Order 18176.

326 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 79 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#22590025C211yesmaximal, characteristic
2#22590005C211yesmaximal, characteristic
3#22586515C214yesmaximal, complemented
4C71×#3131C211yesmaximal, characteristic
5#17985983—21no—
6#17985973—21no—
7#17985798—21no—
8#17985796C2212yes—
9#17985794C2212yes—
10#17985792—21no—
11#533×C71C411yescharacteristic, complemented
12#532×C71C2211yescharacteristic
13#488×C71C412yescomplemented
14#488×C71C2212yes—
15#487×C71—21no—
16#486×C71—21no—
17#416×C71—21no—
18#416×C71—21no—
19C2×C4×C8×C71C411yescharacteristic, complemented
20C2×#11765849—41no—
21#14556087—41no—
22#14556047—22no—
23#14556045—41no—
24#14556043—22no—
25#14556043—41no—
26C2×#41×C71D411yescharacteristic
27C2×#41×C71D411yescharacteristic
28C22×C8×C71—21no—
29C4⋊Q8×C71—21no—
30C4⋊Q8×C71D411yescharacteristic
31#127×C71D411yescharacteristic
32#125×C71C2×C412yes—
33#124×C71C2×C411yescharacteristic
34#124×C71C2311yescharacteristic
35C22⋊Q8×C71—41no—
36#121×C71—41no—
37C4×Q8×C71—41no—
38C4×D4×C71—41no—
39#117×C71—21no—
40C2×C42×C71C2×C411yescharacteristic
41#103×C71—22no—
42#103×C71—22no—
43#102×C71—22no—
44#102×C71—22no—
45C4×C8×C71—21no—
46C4×C8×C71C2×C412yes—
47#11765849—42no—
48#11765849—81no—
49C2×C71⋊C8—81no—
50C4⋊Dic71—81no—
51C22⋊Dic71—42no—
52C4×Dic71—81no—
53#41×C71—41no—
54#41×C71—41no—
55#41×C71—41no—
56#41×C71—41no—
57C2×Q8×C71—41no—
58C2×Q8×C71C22⋊C411yescharacteristic
59C2×Q8×C71C2×D411yescharacteristic
60C2×D4×C71—21no—
61C2×D4×C71—21no—
62C2×D4×C71—41no—
63C2×D4×C71C22⋊C411yescharacteristic
64C2×D4×C71C2×D411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71C22⋊C411yescharacteristic
69C2×C8×C71—22no—
70C2×C8×C71—41no—
71C2×C8×C71—41no—
72C4⋊C4×C71—22no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—42no—
75C4⋊C4×C71—41no—
76C22⋊C4×C71—41no—
77C22⋊C4×C71—41no—
78C22⋊C4×C71—41no—
79C42×C71—21no—
80C42×C71—41no—
81C42×C71C22⋊C411yescharacteristic
82C42×C71C22×C411yescharacteristic
83C23×C71—21no—
84C23×C71—21no—
85C23×C71#9911yescharacteristic
86Q8×C71—41no—
87Q8×C71—41no—
88Q8×C71—82no—
89D4×C71—41no—
90D4×C71—41no—
91D4×C71—82no—
92D4×C71—81no—
93D4×C71—81no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—42no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—42no—
106C2×C4×C71#9911yescharacteristic
107C2×C4×C71C2×C22⋊C411yescharacteristic
108C2×Dic71—82no—
109C8×C71—42no—
110C71⋊C8—82no—
111C22×C71—21no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71—81no—
116C22×C71—81no—
117C22×C71C2×#9911yescharacteristic
118C4×C71—22no—
119C4×C71—21no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71—82no—
124C4×C71#35411yescharacteristic
125C4×C71#35411yescharacteristic
126Dic71—162no—
127#13602—711nomaximal, Sylow
128C2×C71—21no—
129C2×C71—41no—
130C2×C71—81no—
131C2×C71—81no—
132C2×C71C2×#35411yescharacteristic
133#3131D7111yescharacteristic
134#2132—711no—
135#2126—711no—
136#1414—714no—
137C71#1360211yescharacteristic, complemented, Sylow
138#533Dic7111yescharacteristic, complemented
139#532C2×D7111yescharacteristic
140#488Dic7112yescomplemented
141#488C2×D7112yes—
142#487—21no—
143#486—21no—
144#416—21no—
145#416—21no—
146#413—1421no—
147#410—1421no—
148C2×C4×C8Dic7111yescharacteristic, complemented
149#356—1421no—
150#355—712no—
151#354—712no—
152#353—1421no—
153C2×#41C22⋊D7111yescharacteristic
154C2×#41C22⋊D7111yescharacteristic
155C2×M16—2841no—
156C22×C8—21no—
157C4⋊Q8—21no—
158C4⋊Q8C22⋊D7111yescharacteristic
159#127C22⋊D7111yescharacteristic
160#125C2×Dic7112yes—
161#124C2×Dic7111yescharacteristic
162#124C22×D7111yescharacteristic
163C22⋊Q8—41no—
164#121—41no—
165C4×Q8—41no—
166C4×D4—41no—
167#117—21no—
168#117—2841no—
169C2×C42C2×Dic7111yescharacteristic
170#103—22no—
171#103—22no—
172#102—22no—
173#102—22no—
174#101—1422no—
175#100—2841no—
176#99—2841no—
177#99—1422no—
178C4×C8—21no—
179C4×C8C2×Dic7112yes—
180#41—41no—
181#41—41no—
182#41—41no—
183#41—41no—
184C2×Q8—41no—
185C2×Q8C22⋊Dic7111yescharacteristic
186C2×Q8C2×C22⋊D7111yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4C22⋊Dic7111yescharacteristic
191C2×D4C2×C22⋊D7111yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4C22⋊Dic7111yescharacteristic
196M16—2842no—
197M16—5681no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—5681no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—5681no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—2842no—
211C42—21no—
212C42—41no—
213C42—5681no—
214C42C22⋊Dic7111yescharacteristic
215C42C22×Dic7111yescharacteristic
216C23—21no—
217C23—21no—
218C23#1455604311yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—82no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—5682no—
240C2×C4#1455604311yescharacteristic
241C2×C4C2×C22⋊Dic7111yescharacteristic
242C8—42no—
243C8—5682no—
244C22—21no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—81no—
249C22—81no—
250C22C2×#1455604311yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4—11362no—
258C4#1798579411yescharacteristic
259C4#1798579411yescharacteristic
260C2—21no—
261C2—41no—
262C2—81no—
263C2—81no—
264C2C2×#1798579411yescharacteristic
subgroupG/H
centerC4#17985794
commutatorC42×C71C22×C4
Frattini#124C22×D71
FittingC71×#3131C2
socleC2×C71C2×#354
prime radical#532×C71C22
cosocle#124×C71C23
hypercenter#3131D71
hypocenterC71#13602
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3131

Complements

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

NH ≅ G/Ntype
3. #22586515C2semidirect
11. #533×C71C4semidirect
13. #488×C71C4semidirect
19. C2×C4×C8×C71C4semidirect
137. C71#13602semidirect
138. #533Dic71semidirect
140. #488Dic71semidirect
148. C2×C4×C8Dic71semidirect

Maximal subgroups

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

subgroupindex
#225900252
#225900052
#225865152
C71×#31312
#1360271