Finite Group Databasebeta

← #31901732

Subgroups — #31901732

Order 18176.

364 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598864C211yesmaximal, characteristic
2#22586998C212yesmaximal
3#22586838C212yesmaximal
4#22586171C211yesmaximal, characteristic
5C71×#1940C211yesmaximal, characteristic
6#17986556C2212yes—
7#17986408—22no—
8#17986408—22no—
9#17986052C2212yes—
10#17986037—21no—
11#17986037—21no—
12#17985747—22no—
13#17985744—22no—
14#17985735—22no—
15#17985735—22no—
16C2×C4⋊C8×C71—21no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C2×#97×C71C2211yescharacteristic
19C4×C4⋊C4×C71C2211yescharacteristic
20#336×C71—21no—
21#14556128D412yes—
22#14556128D412yes—
23#14556120—22no—
24#14556107—22no—
25#14556107—22no—
26C4×Dic142—22no—
27C4×Dic142—22no—
28#14556087—22no—
29#14556062—21no—
30#14556062—21no—
31#14556062—21no—
32#14556062—21no—
33#14556060—22no—
34#14556060—22no—
35C4⋊(C71⋊C8)—22no—
36C22⋊(C71⋊C8)—22no—
37C4×C71⋊C8—22no—
38C22×C8×C71—21no—
39C22×C8×C71D411yescharacteristic
40C22×C8×C71D411yescharacteristic
41C2×C4⋊C4×C71—21no—
42C2×C42×C71—21no—
43C2×C42×C71C2311yescharacteristic
44C4⋊C8×C71—22no—
45C4⋊C8×C71—42no—
46#97×C71—22no—
47#95×C71D411yescharacteristic
48#95×C71D411yescharacteristic
49C2×Dic142—22no—
50C2×Dic142—22no—
51C2×C71⋊C8—42no—
52C4⋊Dic71—22no—
53C4⋊Dic71—22no—
54C4⋊Dic71—22no—
55C4⋊Dic71—22no—
56Dic71⋊C4—42no—
57Dic71⋊C4—42no—
58C22⋊Dic71—42no—
59C4×Dic71—42no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71#4111yescharacteristic
65C2×C8×C71—21no—
66C2×C8×C71—21no—
67C2×C8×C71—21no—
68C2×C8×C71—21no—
69C2×C8×C71—41no—
70C2×C8×C71—41no—
71C2×C8×C71—41no—
72C2×C8×C71SD1612yes—
73C2×C8×C71Q1612yes—
74C4⋊C4×C71—42no—
75C42×C71—41no—
76C42×C71—41no—
77C42×C71C2×D412yes—
78C42×C71#4112yes—
79C23×C71#12111yescharacteristic
80C2×C4×C71—22no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71#12011yescharacteristic
89C2×C4×C71#12112yes—
90C2×C4×C71#12312yes—
91C2×C4×C71#12411yescharacteristic
92C2×C4×C71C2×SD1611yescharacteristic
93C2×C4×C71C2×Q1611yescharacteristic
94C2×C4×C71#13711yescharacteristic
95C2×C4×C71#13711yescharacteristic
96C2×Dic71—42no—
97C2×Dic71—42no—
98Dic142—82no—
99Dic142—82no—
100C8×C71—22no—
101C8×C71—22no—
102C8×C71—81no—
103C71⋊C8—82no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71#39411yescharacteristic
107C22×C71#46111yescharacteristic
108C22×C71#46211yescharacteristic
109C4×C71—22no—
110C4×C71—22no—
111C4×C71—42no—
112C4×C71—81no—
113C4×C71#44811yescharacteristic
114C4×C71#45111yescharacteristic
115C4×C71#48711yescharacteristic
116C4×C71#49111yescharacteristic
117Dic71—82no—
118Dic71—82no—
119#10548—711nomaximal, Sylow
120C2×C71—21no—
121C2×C71—21no—
122C2×C71#204311yescharacteristic
123C2×C71#205011yescharacteristic
124C2×C71#205311yescharacteristic
125#3162—711no—
126#1940D7111yescharacteristic
127#1542—712no—
128#1503—712no—
129#1275—711no—
130C71#1054811yescharacteristic, complemented, Sylow
131#533—712no—
132#501—1422no—
133#501—1422no—
134#431—712no—
135#427—1421no—
136#427—1421no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×D7111yescharacteristic
139C2×#97C2×D7111yescharacteristic
140C4×C4⋊C4C2×D7111yescharacteristic
141#336—21no—
142#333—1422no—
143#332—1422no—
144#328—1422no—
145#328—1422no—
146C22×C8—21no—
147C22×C8C22⋊D7111yescharacteristic
148C22×C8C22⋊D7111yescharacteristic
149C4⋊Q8—712no—
150C4⋊Q8—712no—
151#125—1422no—
152C22⋊Q8—1422no—
153C22⋊Q8—1422no—
154C4×Q8—1422no—
155C4×Q8—1422no—
156#117—1422no—
157C2×C4⋊C4—21no—
158C2×C42—21no—
159C2×C42C22×D7111yescharacteristic
160#107—1421no—
161#107—1421no—
162#107—1421no—
163#107—1421no—
164#106—1422no—
165#106—1422no—
166C4⋊C8—22no—
167C4⋊C8—42no—
168C4⋊C8—1422no—
169C22⋊C8—1422no—
170#97—22no—
171C4×C8—1422no—
172#95D28411yescharacteristic
173#95D28411yescharacteristic
174C2×Q8—1422no—
175C2×Q8—1422no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4C2×C22⋊D7111yescharacteristic
179C22×C4C2×D28411yescharacteristic
180C22×C4#1176586511yescharacteristic
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—2842no—
189C2×C8#1176585312yes—
190C2×C8#1176585512yes—
191C4⋊C4—42no—
192C4⋊C4—2842no—
193C4⋊C4—2842no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C4⋊C4—1422no—
197C4⋊C4—1422no—
198C22⋊C4—2842no—
199C42—41no—
200C42—41no—
201C42—2842no—
202C42D4×D7112yes—
203C42#1176586612yes—
204C23#1455610211yescharacteristic
205Q8—5682no—
206Q8—5682no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—2842no—
216C2×C4—2842no—
217C2×C4#1455609611yescharacteristic
218C2×C4#1455610312yes—
219C2×C4#1455611312yes—
220C2×C4#1455611711yescharacteristic
221C2×C4C2×#1176585311yescharacteristic
222C2×C4C2×#1176585511yescharacteristic
223C2×C4#1455615811yescharacteristic
224C2×C4#1455615811yescharacteristic
225C8—22no—
226C8—22no—
227C8—81no—
228C8—5682no—
229C22—21no—
230C22—21no—
231C22#1798591611yescharacteristic
232C22#1798620411yescharacteristic
233C22#1798620911yescharacteristic
234C4—22no—
235C4—22no—
236C4—42no—
237C4—81no—
238C4—5682no—
239C4—5682no—
240C4#1798613711yescharacteristic
241C4#1798615411yescharacteristic
242C4#1798635111yescharacteristic
243C4#1798636911yescharacteristic
244C2—21no—
245C2—21no—
246C2#2258958611yescharacteristic
247C2#2258962711yescharacteristic
248C2#2258964611yescharacteristic
subgroupG/H
centerC2×C4#14556117
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1940C2
socleC22×C71#394
prime radicalC23×C71#121
hypercenter#1940D71
hypocenterC71#10548
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1940

Complements

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

NH ≅ G/Ntype
130. C71#10548semidirect

Maximal subgroups

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

subgroupindex
#225988642
#225869982
#225868382
#225861712
C71×#19402
#1054871