Finite Group Databasebeta

← #31911873

Subgroups — #31911873

Order 18176.

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

normal subgroups only

This group has 234 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#22587736C211yesmaximal, characteristic, complemented
3#22587692C211yesmaximal, characteristic
4#22586414C211yesmaximal, characteristic, complemented
5#22586398C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic, complemented
7C71×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D71—21no—
9#17986286—21no—
10#17986191—21no—
11C8×D284C2211yescharacteristic
12C4⋊(C71⋊C16)C2211yescharacteristic
13(C2×D71)⋊C16—21no—
14#17985752C2211yescharacteristic
15#17985752C2211yescharacteristic
16#17985748—21no—
17#17985743—21no—
18#500×C71C2211yescharacteristic
19#493×C71C2211yescharacteristic
20#357×C71—21no—
21#357×C71—21no—
22C4×C16×C71C2211yescharacteristic
23C2×#11765850—41no—
24C2×C8×D71—21no—
25C4×D284D411yescharacteristic
26C2×C71⋊C16—21no—
27C16×D71—42no—
28#14556059—41no—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)D411yescharacteristic
32#14556049—41no—
33#14556048—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#11765850—82no—
47C2×C71⋊C8—21no—
48C2×C71⋊C8—41no—
49C2×C71⋊C8—41no—
50C8×D71—41no—
51C8×D71—41no—
52C4⋊Dic71D811yescharacteristic
53(C2×D71)⋊C4—21no—
54C71⋊C16—41no—
55C2×Q8×C71—41no—
56C2×D4×C71—41no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59D8×C71—41no—
60D8×C71—41no—
61D8×C71—81no—
62C2×C8×C71C2×D411yescharacteristic
63C2×C8×C71#4111yescharacteristic
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C42×C71C2×D411yescharacteristic
67C16×C71—21no—
68C16×C71—21no—
69C23×C71—81no—
70C22×D71—21no—
71Q8×C71—161no—
72D4×C71—81no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—161no—
77C2×C4×C71—81no—
78C2×C4×C71#12111yescharacteristic
79C2×C4×C71C2×D811yescharacteristic
80C2×C4×C71#13511yescharacteristic
81C2×Dic71—21no—
82D284—21no—
83D284SD3212yes—
84C4×D71—41no—
85C4×D71—41no—
86C8×C71—21no—
87C8×C71#13511yescharacteristic
88C8×C71AGL(1,ℤ/8)11yescharacteristic
89C71⋊C8—41no—
90C71⋊C8—81no—
91C71⋊C8—81no—
92C22×C71—161no—
93C22×C71—161no—
94C22×C71—161no—
95C22×C71#46611yescharacteristic
96C2×D71—22no—
97C2×D71—41no—
98C2×D71—41no—
99C4×C71—21no—
100C4×C71—161no—
101C4×C71#46311yescharacteristic
102C4×C71#47111yescharacteristic
103C4×C71C2×SD3211yescharacteristic
104C4×C71#50811yescharacteristic
105Dic71—41no—
106#12468—711nomaximal, Sylow
107C2×C71—321no—
108C2×C71#168111yescharacteristic
109C2×C71#221711yescharacteristic
110C2×C71#222111yescharacteristic
111D71—42no—
112#2240D7111yescharacteristic, complemented
113D4×C16—711no—
114#1675—711no—
115#1667—711no—
116#1372—711no—
117#1365—711no—
118#1361—711no—
119C71#1246811yescharacteristic, complemented, Sylow
120C22×C16—1421no—
121#500C2×D7111yescharacteristic
122#493C2×D7111yescharacteristic
123#476—1421no—
124#459—1421no—
125C8×D4—711no—
126C4⋊C16—711no—
127#357—21no—
128#357—21no—
129C22⋊C16—1421no—
130C4×C16C2×D7111yescharacteristic
131#335—711no—
132#335—711no—
133#333—1421no—
134#331—1421no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—2841no—
138C22×C8—1421no—
139C4⋊Q8—21no—
140#127—21no—
141C4×D4—711no—
142C2×C16—2842no—
143C2×C16—1421no—
144C2×C16C22⋊D7111yescharacteristic
145C2×C16C22⋊D7111yescharacteristic
146#107—21no—
147#107—21no—
148#106—2841no—
149C4⋊C8—711no—
150C4⋊C8—1421no—
151C4⋊C8—1421no—
152#102—2841no—
153#102—2841no—
154C22⋊C8—1421no—
155C4×C8C22×D7111yescharacteristic
156C2×Q8—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—711no—
161C22×C4—1421no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—5682no—
166C2×C8—2841no—
167C2×C8—2841no—
168C2×C8—2841no—
169C2×C8—2841no—
170C2×C8—1421no—
171C2×C8C2×C22⋊D7111yescharacteristic
172C2×C8#1176586411yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—711no—
176C22⋊C4—1421no—
177C42D4×D7111yescharacteristic
178C16—21no—
179C16—21no—
180C16—2841no—
181C23—81no—
182C23—1421no—
183Q8—161no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—712no—
190D4—1421no—
191C2×C4—81no—
192C2×C4—2841no—
193C2×C4—2841no—
194C2×C4—1421no—
195C2×C4#1455609911yescharacteristic
196C2×C4D8×D7111yescharacteristic
197C2×C4#1455614711yescharacteristic
198C8—21no—
199C8—2841no—
200C8—5681no—
201C8—5681no—
202C8#1455614811yescharacteristic
203C8#1455615311yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—2841no—
208C22—2841no—
209C22—1422no—
210C22#1798623211yescharacteristic
211C4—21no—
212C4—161no—
213C4—2841no—
214C4#1798621111yescharacteristic
215C4#1798625911yescharacteristic
216C4SD32×D7111yescharacteristic
217C4#1798643011yescharacteristic
218C2—2842no—
219C2—321no—
220C2#2258777411yescharacteristic
221C2#2259043311yescharacteristic
222C2#2259045711yescharacteristic
subgroupG/H
centerC22#17986232
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2240C2
socleC22×C71#466
prime radical#22587692C2
hypercenter#2240D71
hypocenterC71#12468
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. #22587736C2semidirect
4. #22586414C2semidirect
5. #22586398C2semidirect
6. #22586386C2semidirect
7. C71×#2240C2semidirect
112. #2240D71semidirect
119. C71#12468semidirect

Maximal subgroups

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

subgroupindex
C16×D2842
#225877362
#225876922
#225864142
#225863982
#225863862
C71×#22402
#1246871