Finite Group Databasebeta

← #31911662

Subgroups — #31911662

Order 18176.

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

normal subgroups only

This group has 236 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
1#22590592C211yesmaximal, characteristic, complemented
2#22590159C211yesmaximal, characteristic
3#22587706C211yesmaximal, characteristic
4#22587702C211yesmaximal, characteristic
5#22586411C211yesmaximal, characteristic, complemented
6#22586390C211yesmaximal, characteristic, complemented
7C71×#1362C211yesmaximal, characteristic, complemented
8C2×#14556074—21no—
9C2×#14556074—21no—
10C2×C16×D71—21no—
11#17986379C2211yescharacteristic
12#17986280—21no—
13#17986276—21no—
14C8×D284C2211yescharacteristic
15(C2×D71)⋊C16—21no—
16C4×C71⋊C16C2211yescharacteristic
17#17985752C2211yescharacteristic
18#17985745—21no—
19#17985744—21no—
20#494×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284—21no—
24C2×Dic284—21no—
25C2×C8×D71—21no—
26#14556128—21no—
27C4×D284D411yescharacteristic
28#14556074—42no—
29#14556074—42no—
30C2×C71⋊C16D411yescharacteristic
31C2×C71⋊C16D411yescharacteristic
32C16×D71—42no—
33#14556062—41no—
34#14556061—41no—
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)D411yescharacteristic
37#14556050—41no—
38#14556048—41no—
39(C2×D71)⋊C8—21no—
40C2×Q16×C71—21no—
41C2×Q16×C71—21no—
42C4⋊Q8×C71—21no—
43C2×C16×C71—21no—
44C4⋊C8×C71—21no—
45C4×C8×C71C2311yescharacteristic
46C2×Dic142—41no—
47C2×D284D811yescharacteristic
48C2×C4×D71—21no—
49Dic284—41no—
50Dic284—41no—
51Dic284—81no—
52C2×C71⋊C8—21no—
53C2×C71⋊C8—41no—
54C8×D71—41no—
55C8×D71—41no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71D811yescharacteristic
59(C2×D71)⋊C4—21no—
60C71⋊C16—21no—
61C71⋊C16D812yes—
62C2×Q8×C71—41no—
63Q16×C71—41no—
64Q16×C71—41no—
65Q16×C71—81no—
66C2×C8×C71—41no—
67C2×C8×C71C2×D411yescharacteristic
68C2×C8×C71#4111yescharacteristic
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C42×C71C2×D411yescharacteristic
72C16×C71—41no—
73C22×D71—21no—
74Q8×C71—81no—
75Q8×C71—81no—
76C2×C4×C71—81no—
77C2×C4×C71#12111yescharacteristic
78C2×C4×C71C2×D811yescharacteristic
79C2×C4×C71#13511yescharacteristic
80C2×Dic71—21no—
81C2×Dic71—81no—
82D284—21no—
83D284Q3212yescomplemented
84C4×D71—41no—
85C4×D71—41no—
86Dic142—81no—
87Dic142—81no—
88C8×C71—21no—
89C8×C71—81no—
90C8×C71C2×D811yescharacteristic
91C8×C71AGL(1,ℤ/8)11yescharacteristic
92C71⋊C8—41no—
93C71⋊C8—81no—
94C22×C71#46611yescharacteristic
95C2×D71—22no—
96C2×D71—41no—
97C2×D71—41no—
98C4×C71—21no—
99C4×C71—161no—
100C4×C71#45911yescharacteristic
101C4×C71#47211yescharacteristic
102C4×C71C2×Q3211yescharacteristic
103C4×C71#50811yescharacteristic
104Dic71—41no—
105Dic71—161no—
106#12431—711nomaximal, Sylow
107C2×C71#166711yescharacteristic
108C2×C71#221611yescharacteristic
109C2×C71#222211yescharacteristic
110D71—42no—
111#2247—711no—
112D4×C16—711no—
113#1669—711no—
114#1669—711no—
115#1371—711no—
116#1362—711no—
117#1362D7111yescharacteristic, complemented
118C71#1243111yescharacteristic, complemented, Sylow
119C2×Q32—1421no—
120C2×Q32—1421no—
121C22×C16—1421no—
122#494—711no—
123#494C2×D7111yescharacteristic
124#474—1421no—
125#474—1421no—
126C8×D4—711no—
127C4⋊C16C2×D7111yescharacteristic
128C22⋊C16—1421no—
129C4×C16—711no—
130#335—711no—
131#335C2×D7111yescharacteristic
132#332—1421no—
133#332—1421no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—1421no—
137C2×Q16—1421no—
138C22×C8—1421no—
139C4⋊Q8—21no—
140C4⋊Q8—1421no—
141C4×D4—711no—
142Q32—2842no—
143Q32—2842no—
144C2×C16—21no—
145C2×C16—2842no—
146C2×C16—711no—
147C2×C16—711no—
148#107—2841no—
149#107—2841no—
150C4⋊C8—21no—
151C4⋊C8—711no—
152C4⋊C8—1421no—
153#102—2841no—
154#102—2841no—
155C22⋊C8—1421no—
156C4×C8C22×D7111yescharacteristic
157C2×Q8—41no—
158C2×Q8—2841no—
159C2×D4—711no—
160C22×C4—1421no—
161Q16—41no—
162Q16—41no—
163Q16—81no—
164Q16—2841no—
165Q16—2841no—
166Q16—5681no—
167C2×C8—41no—
168C2×C8—2841no—
169C2×C8—2841no—
170C2×C8—2841no—
171C2×C8—1421no—
172C2×C8D4×D7111yescharacteristic
173C2×C8#1176586511yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2841no—
177C4⋊C4—2841no—
178C4⋊C4—711no—
179C22⋊C4—1421no—
180C42D4×D7111yescharacteristic
181C16—41no—
182C16—712no—
183C16—1421no—
184C23—1421no—
185Q8—81no—
186Q8—81no—
187Q8—5681no—
188Q8—5681no—
189D4—712no—
190D4—1421no—
191C2×C4—81no—
192C2×C4—2841no—
193C2×C4—2841no—
194C2×C4—5681no—
195C2×C4—1421no—
196C2×C4#1455610111yescharacteristic
197C2×C4D8×D7111yescharacteristic
198C2×C4#1455614911yescharacteristic
199C8—21no—
200C8—81no—
201C8—2841no—
202C8—5681no—
203C8D8×D7111yescharacteristic
204C8#1455615511yescharacteristic
205C22—2841no—
206C22—2841no—
207C22—1422no—
208C22#1798623411yescharacteristic
209C4—21no—
210C4—161no—
211C4—2841no—
212C4—11361no—
213C4#1798619311yescharacteristic
214C4#1798626811yescharacteristic
215C4Q32×D7111yescharacteristic
216C4#1798643011yescharacteristic
217C2—2842no—
218C2#2258769411yescharacteristic
219C2#2259043011yescharacteristic
220C2#2259046411yescharacteristic
subgroupG/H
centerC22#17986234
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1362C2
socleC22×C71#466
prime radicalC2×D284D8
hypercenter#1362D71
hypocenterC71#12431
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #22590592C2semidirect
5. #22586411C2semidirect
6. #22586390C2semidirect
7. C71×#1362C2semidirect
83. D284Q32semidirect
117. #1362D71semidirect
118. C71#12431semidirect

Maximal subgroups

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

subgroupindex
#225905922
#225901592
#225877062
#225877022
#225864112
#225863902
C71×#13622
#1243171