Finite Group Databasebeta

← #31911866

Subgroups — #31911866

Order 18176.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C16×D284C211yesmaximal, characteristic, complemented
2#22587726C211yesmaximal, characteristic, complemented
3#22587692C211yesmaximal, characteristic
4#22586412C211yesmaximal, characteristic, complemented
5#22586395C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic, complemented
7C71×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D71—21no—
9#17986201—21no—
10#17986191—21no—
11C8×D284C2211yescharacteristic
12C4⋊(C71⋊C16)C2211yescharacteristic
13(C2×D71)⋊C16—21no—
14#17985752C2211yescharacteristic
15#17985752C2211yescharacteristic
16#17985745—21no—
17#17985743—21no—
18#500×C71C2211yescharacteristic
19#493×C71C2211yescharacteristic
20#357×C71—21no—
21#357×C71—21no—
22C4×C16×C71C2211yescharacteristic
23C2×#11765852—41no—
24C2×#11765850—41no—
25C2×C8×D71—21no—
26C4×D284D411yescharacteristic
27C2×C71⋊C16—21no—
28C16×D71—42no—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)D411yescharacteristic
32#14556052—41no—
33#14556049—41no—
34(C2×D71)⋊C8—21no—
35C2×D8×C71—21no—
36C2×D8×C71—21no—
37C4⋊Q8×C71—21no—
38#127×C71—21no—
39C2×C16×C71D411yescharacteristic
40C2×C16×C71D411yescharacteristic
41#107×C71—21no—
42#107×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×D284D811yescharacteristic
45C2×C4×D71—21no—
46#11765852—82no—
47#11765850—82no—
48C2×C71⋊C8—21no—
49C2×C71⋊C8—41no—
50C2×C71⋊C8—41no—
51C8×D71—41no—
52C8×D71—41no—
53C4⋊Dic71D811yescharacteristic
54(C2×D71)⋊C4—21no—
55C71⋊C16—41no—
56C2×Q8×C71—41no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60D8×C71—41no—
61D8×C71—41no—
62D8×C71—81no—
63C2×C8×C71C2×D411yescharacteristic
64C2×C8×C71#4111yescharacteristic
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C42×C71C2×D411yescharacteristic
68C16×C71—21no—
69C16×C71—21no—
70C23×C71—81no—
71C22×D71—21no—
72Q8×C71—81no—
73Q8×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—81no—
77D4×C71—81no—
78D4×C71—161no—
79C2×C4×C71—81no—
80C2×C4×C71#12111yescharacteristic
81C2×C4×C71C2×D811yescharacteristic
82C2×C4×C71#13511yescharacteristic
83C2×Dic71—21no—
84D284—21no—
85D284D1612yes—
86C4×D71—41no—
87C4×D71—41no—
88C8×C71—21no—
89C8×C71#13511yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—41no—
92C71⋊C8—81no—
93C71⋊C8—81no—
94C22×C71—161no—
95C22×C71—161no—
96C22×C71—161no—
97C22×C71#46611yescharacteristic
98C2×D71—22no—
99C2×D71—41no—
100C2×D71—41no—
101C4×C71—21no—
102C4×C71—161no—
103C4×C71#46311yescharacteristic
104C4×C71#47111yescharacteristic
105C4×C71C2×D1611yescharacteristic
106C4×C71#50811yescharacteristic
107Dic71—41no—
108#12467—711nomaximal, Sylow
109C2×C71—321no—
110C2×C71#168111yescharacteristic
111C2×C71#221511yescharacteristic
112C2×C71#222311yescharacteristic
113D71—42no—
114#2240D7111yescharacteristic, complemented
115D4×C16—711no—
116#1673—711no—
117#1667—711no—
118#1371—711no—
119#1364—711no—
120#1361—711no—
121C71#1246711yescharacteristic, complemented, Sylow
122C22×C16—1421no—
123#500C2×D7111yescharacteristic
124#493C2×D7111yescharacteristic
125#461—1421no—
126#459—1421no—
127C8×D4—711no—
128C4⋊C16—711no—
129#357—21no—
130#357—21no—
131C22⋊C16—1421no—
132C4×C16C2×D7111yescharacteristic
133#335—711no—
134#335—711no—
135#332—1421no—
136#331—1421no—
137C2×SD16—2841no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—2841no—
141C22×C8—1421no—
142C4⋊Q8—21no—
143#127—21no—
144C4×D4—711no—
145C2×C16—2842no—
146C2×C16—1421no—
147C2×C16C22⋊D7111yescharacteristic
148C2×C16C22⋊D7111yescharacteristic
149#107—21no—
150#107—21no—
151C4⋊C8—711no—
152C4⋊C8—1421no—
153C4⋊C8—1421no—
154#103—2841no—
155#102—2841no—
156C22⋊C8—1421no—
157C4×C8C22×D7111yescharacteristic
158C2×Q8—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—711no—
163C22×C4—1421no—
164SD16—5682no—
165D8—41no—
166D8—41no—
167D8—81no—
168D8—5682no—
169C2×C8—2841no—
170C2×C8—2841no—
171C2×C8—2841no—
172C2×C8—2841no—
173C2×C8—1421no—
174C2×C8C2×C22⋊D7111yescharacteristic
175C2×C8#1176586411yescharacteristic
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—711no—
179C22⋊C4—1421no—
180C42D4×D7111yescharacteristic
181C16—21no—
182C16—21no—
183C16—2841no—
184C23—81no—
185C23—1421no—
186Q8—81no—
187Q8—81no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192D4—161no—
193D4—712no—
194D4—1421no—
195C2×C4—81no—
196C2×C4—2841no—
197C2×C4—2841no—
198C2×C4—1421no—
199C2×C4#1455609911yescharacteristic
200C2×C4D8×D7111yescharacteristic
201C2×C4#1455614711yescharacteristic
202C8—21no—
203C8—2841no—
204C8—5681no—
205C8—5681no—
206C8#1455614811yescharacteristic
207C8#1455615311yescharacteristic
208C22—161no—
209C22—161no—
210C22—161no—
211C22—2841no—
212C22—2841no—
213C22—1422no—
214C22#1798623211yescharacteristic
215C4—21no—
216C4—161no—
217C4—2841no—
218C4#1798621111yescharacteristic
219C4#1798625911yescharacteristic
220C4D16×D7111yescharacteristic
221C4#1798642811yescharacteristic
222C2—2842no—
223C2—321no—
224C2#2258777411yescharacteristic
225C2#2259042311yescharacteristic
226C2#2259046711yescharacteristic
subgroupG/H
centerC22#17986232
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2240C2
socleC22×C71#466
prime radical#22587692C2
hypercenter#2240D71
hypocenterC71#12467
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D284C2semidirect
2. #22587726C2semidirect
4. #22586412C2semidirect
5. #22586395C2semidirect
6. #22586386C2semidirect
7. C71×#2240C2semidirect
114. #2240D71semidirect
121. C71#12467semidirect

Maximal subgroups

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

subgroupindex
C16×D2842
#225877262
#225876922
#225864122
#225863952
#225863862
C71×#22402
#1246771