Finite Group Databasebeta

← #31911737

Subgroups — #31911737

Order 18176.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#22590596C211yesmaximal, characteristic
2#22590248C211yesmaximal, characteristic, complemented
3#22587711C211yesmaximal, characteristic, complemented
4#22587701C211yesmaximal, characteristic
5#22586388C211yesmaximal, characteristic, complemented
6#22586385C211yesmaximal, characteristic
7C71×#1371C211yesmaximal, characteristic, complemented
8C2×#14556073—21no—
9C2×#14556073—21no—
10#17986379C2211yescharacteristic
11#17986376C2211yescharacteristic
12#17986285—21no—
13#17986200—21no—
14C8×Dic142C2211yescharacteristic
15C4⋊(C71⋊C16)C2211yescharacteristic
16Dic71⋊C16—21no—
17#17985774—21no—
18#17985744—21no—
19#17985742—21no—
20C4×C16×C71C2211yescharacteristic
21#335×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284—21no—
24C2×Dic284—21no—
25C2×#11765854—41no—
26C2×D568—21no—
27C2×D568—21no—
28#14556128—21no—
29#14556125—21no—
30C4×Dic142D411yescharacteristic
31#14556073—42no—
32#14556073—42no—
33C2×C71⋊C16—21no—
34#14556060—41no—
35C4⋊(C71⋊C8)D411yescharacteristic
36Dic71⋊C8—21no—
37#14556053—41no—
38#14556050—41no—
39C8×Dic71—21no—
40C2×C16×C71D411yescharacteristic
41C2×C16×C71D411yescharacteristic
42C4⋊C8×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×Dic142D811yescharacteristic
47C2×D284—41no—
48C2×D284—41no—
49C2×D284—41no—
50Dic284—41no—
51Dic284—41no—
52Dic284—81no—
53#11765854—82no—
54D568—41no—
55D568—41no—
56D568—81no—
57C2×C71⋊C8—21no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71D811yescharacteristic
61Dic71⋊C4—21no—
62C4×Dic71—21no—
63C71⋊C16—41no—
64C2×C8×C71—41no—
65C2×C8×C71—41no—
66C2×C8×C71C2×D411yescharacteristic
67C2×C8×C71#4111yescharacteristic
68C42×C71C2×D411yescharacteristic
69C16×C71—21no—
70C16×C71SD1612yescomplemented
71C22×D71—81no—
72C2×C4×C71#12111yescharacteristic
73C2×C4×C71C2×D811yescharacteristic
74C2×C4×C71#13511yescharacteristic
75C2×Dic71—21no—
76C2×Dic71—21no—
77C2×Dic71—81no—
78D284—81no—
79D284—81no—
80D284—81no—
81D284—81no—
82D284—161no—
83Dic142—21no—
84Dic142—21no—
85Dic142—81no—
86Dic142—81no—
87C8×C71—21no—
88C8×C71—81no—
89C8×C71—81no—
90C8×C71C2×SD1611yescharacteristic
91C8×C71AGL(1,ℤ/8)11yescharacteristic
92C71⋊C8—41no—
93C22×C71#46611yescharacteristic
94C2×D71—161no—
95C2×D71—161no—
96C2×D71—161no—
97C4×C71—21no—
98C4×C71#46011yescharacteristic
99C4×C71#47211yescharacteristic
100C4×C71#50911yescharacteristic
101C4×C71#51011yescharacteristic
102Dic71—41no—
103Dic71—41no—
104Dic71—161no—
105#12444—711nomaximal, Sylow
106C2×C71#166811yescharacteristic
107C2×C71#222011yescharacteristic
108C2×C71#222311yescharacteristic
109D71—321no—
110#2248—711no—
111#2183—711no—
112#1670—711no—
113#1668—711no—
114#1371D7111yescharacteristic, complemented
115#1362—711no—
116#1361—711no—
117C71#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1421no—
119C2×SD32—1421no—
120#494—711no—
121#493—711no—
122#475—1421no—
123#460—1421no—
124C8×Q8—711no—
125C4⋊C16—711no—
126C4⋊C16—1421no—
127#346—1421no—
128C4×C16C2×D7111yescharacteristic
129#335C2×D7111yescharacteristic
130#335C2×D7111yescharacteristic
131#332—1421no—
132#331—1421no—
133C2×Q16—1421no—
134C2×Q16—1421no—
135C2×SD16—2841no—
136C2×D8—1421no—
137C2×D8—1421no—
138C4⋊Q8—1421no—
139#127—1421no—
140C4×Q8—711no—
141SD32—2842no—
142SD32—2842no—
143C2×C16—1421no—
144C2×C16D28411yescharacteristic
145C2×C16D28411yescharacteristic
146#106—2841no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—711no—
150C4⋊C8—1421no—
151#103—2841no—
152#102—2841no—
153C4×C8—1421no—
154C4×C8C22×D7111yescharacteristic
155C2×Q8—2841no—
156C2×Q8—711no—
157C2×D4—2841no—
158C2×D4—2841no—
159C2×D4—2841no—
160Q16—2841no—
161Q16—2841no—
162Q16—5681no—
163SD16—5682no—
164D8—2841no—
165D8—2841no—
166D8—5681no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—1421no—
170C2×C8C2×D28411yescharacteristic
171C2×C8#1176586611yescharacteristic
172C4⋊C4—2841no—
173C4⋊C4—2841no—
174C4⋊C4—711no—
175C4⋊C4—1421no—
176C42—1421no—
177C42D4×D7111yescharacteristic
178C16—21no—
179C16—2841no—
180C16#1176585412yescomplemented
181C23—5681no—
182Q8—5681no—
183Q8—5681no—
184Q8—1421no—
185Q8—1421no—
186D4—5681no—
187D4—5681no—
188D4—5681no—
189D4—5681no—
190D4—11361no—
191C2×C4—5681no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4#1455610311yescharacteristic
195C2×C4D8×D7111yescharacteristic
196C2×C4#1455615111yescharacteristic
197C8—21no—
198C8—81no—
199C8—81no—
200C8—2841no—
201C8C2×#1176585411yescharacteristic
202C8#1455615611yescharacteristic
203C22—11361no—
204C22—11361no—
205C22—11361no—
206C22#1798623611yescharacteristic
207C4—21no—
208C4—2841no—
209C4—2841no—
210C4—11361no—
211C4#1798620011yescharacteristic
212C4#1798627011yescharacteristic
213C4#1798643311yescharacteristic
214C4#1798644111yescharacteristic
215C2—22721no—
216C2#2258770111yescharacteristic
217C2#2259045411yescharacteristic
218C2#2259047111yescharacteristic
subgroupG/H
centerC22#17986236
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1371C2
socleC22×C71#466
prime radical#17986376C22
hypercenter#1371D71
hypocenterC71#12444
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1371

Complements

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

NH ≅ G/Ntype
2. #22590248C2semidirect
3. #22587711C2semidirect
5. #22586388C2semidirect
7. C71×#1371C2semidirect
70. C16×C71SD16semidirect
114. #1371D71semidirect
117. C71#12444semidirect
180. C16#11765854semidirect

Maximal subgroups

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

subgroupindex
#225905962
#225902482
#225877112
#225877012
#225863882
#225863852
C71×#13712
#1244471