Finite Group Databasebeta

← #31909403

Subgroups — #31909403

Order 18176.

220 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 71 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#22590204C211yesmaximal, characteristic
2#22587250C211yesmaximal, characteristic, complemented
3#22586405C211yesmaximal, characteristic, complemented
4#22586309C211yesmaximal, characteristic, complemented
5#22586307C211yesmaximal, characteristic
6#22586261C211yesmaximal, characteristic, complemented
7C71×#1580C211yesmaximal, characteristic
8C2×#14556072C411yescharacteristic
9#17986129—21no—
10C4×Dic284C2211yescharacteristic
11#17985824C411yescharacteristic
12#17985802C411yescharacteristic
13#17985800C411yescharacteristic
14#17985775C2211yescharacteristic
15#17985775C2211yescharacteristic
16#17985750C2211yescharacteristic
17#17985731—21no—
18C4×D8×C71C2211yescharacteristic
19C8×D4×C71—21no—
20#334×C71C2211yescharacteristic
21D4⋊C8×C71—21no—
22#322×C71C2211yescharacteristic
23C2×Dic284C2×C411yescharacteristic
24C4×Dic142—21no—
25#14556072—22no—
26C2×C71⋊C16—21no—
27C2×C71⋊C16C2×C411yescharacteristic
28C2×C71⋊C16C2×C411yescharacteristic
29#14556062C2×C411yescharacteristic
30C4⋊(C71⋊C8)—21no—
31Dic71⋊C8—41no—
32#14556053—21no—
33#14556038—41no—
34C2×D8×C71C2×C411yescharacteristic
35C22×C8×C71—41no—
36C4×D4×C71—21no—
37#107×C71C2×C411yescharacteristic
38C4⋊C8×C71—21no—
39#102×C71—21no—
40C22⋊C8×C71—41no—
41C4×C8×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—21no—
45Dic284—21no—
46Dic284—21no—
47C2×C71⋊C8—41no—
48C4⋊Dic71—21no—
49Dic71⋊C4—41no—
50C4×Dic71—41no—
51C71⋊C16—22no—
52C71⋊C16—21no—
53C71⋊C16—21no—
54C2×D4×C71—21no—
55C22×C4×C71—41no—
56D8×C71—21no—
57D8×C71M1612yes—
58C2×C8×C71—21no—
59C2×C8×C71—21no—
60C2×C8×C71—41no—
61C2×C8×C71—82no—
62C2×C8×C71C22×C411yescharacteristic
63C2×C8×C71#4111yescharacteristic
64C4⋊C4×C71—21no—
65C22⋊C4×C71—41no—
66C42×C71C2×D411yescharacteristic
67C23×C71—41no—
68D4×C71—22no—
69D4×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71—81no—
72C2×C4×C71—81no—
73C2×C4×C71C4×D411yescharacteristic
74C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
75C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
76C2×Dic71—41no—
77C2×Dic71—41no—
78Dic142—41no—
79Dic142—41no—
80C8×C71—21no—
81C8×C71—41no—
82C8×C71—41no—
83C8×C71—81no—
84C8×C71C2×M1611yescharacteristic
85C8×C71#13111yescharacteristic
86C71⋊C8—81no—
87C22×C71—42no—
88C22×C71—81no—
89C22×C71—81no—
90C22×C71#44211yescharacteristic
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—81no—
94C4×C71#43511yescharacteristic
95C4×C71#44311yescharacteristic
96Dic71—81no—
97Dic71—81no—
98#12077—711nomaximal, Sylow
99C2×C71—82no—
100C2×C71#158611yescharacteristic
101C2×C71#218011yescharacteristic
102C2×C71AGL(1,ℤ/16)11yescharacteristic
103#2175—711no—
104#1593—711no—
105#1580D7111yescharacteristic
106#1368—711no—
107#1334—711no—
108#1333—711no—
109#1313—711no—
110C71#1207711yescharacteristic, complemented, Sylow
111C2×SD32—711no—
112#446—1421no—
113C4×Q16—711no—
114C4×D8C2×D7111yescharacteristic
115C8×D4—21no—
116#366—711no—
117#358—711no—
118#357—711no—
119#346—711no—
120#346—711no—
121#334—711no—
122#334C2×D7111yescharacteristic
123Q8⋊C8—1421no—
124D4⋊C8—21no—
125#322C2×D7111yescharacteristic
126C2×Q16—711no—
127C2×D8C4×D7111yescharacteristic
128C22×C8—41no—
129C4×Q8—1421no—
130C4×D4—21no—
131SD32—1422no—
132C2×C16—711no—
133C2×C16—711no—
134C2×C16—1421no—
135#107—711no—
136#107C4×D7111yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—2841no—
139C4⋊C8—1421no—
140#103—1421no—
141#102—21no—
142C22⋊C8—41no—
143#97—2841no—
144C4×C8C22⋊D7111yescharacteristic
145C4×C8C22⋊D7111yescharacteristic
146C4×C8C22×D7111yescharacteristic
147C2×Q8—1421no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—1421no—
151Q16—1421no—
152D8—21no—
153D8#1176584812yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8—82no—
158C2×C8—2841no—
159C2×C8C2×C4×D7111yescharacteristic
160C2×C8#1176586511yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—2841no—
163C4⋊C4—1421no—
164C22⋊C4—41no—
165C42—2841no—
166C42C2×C22⋊D7111yescharacteristic
167C16—1422no—
168C16—1421no—
169C16—1421no—
170C23—41no—
171Q8—2841no—
172Q8—2841no—
173D4—22no—
174D4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—2841no—
179C2×C4—2841no—
180C2×C4C4×C22⋊D7111yescharacteristic
181C2×C4#1455615311yescharacteristic
182C2×C4#1455615411yescharacteristic
183C8—21no—
184C8—41no—
185C8—41no—
186C8—81no—
187C8—5681no—
188C8C2×#1176584811yescharacteristic
189C8#1455613411yescharacteristic
190C22—42no—
191C22—81no—
192C22—81no—
193C22#1798610911yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—5681no—
198C4—5681no—
199C4#1798607011yescharacteristic
200C4#1798611411yescharacteristic
201C2—82no—
202C2#2258720411yescharacteristic
203C2#2259023111yescharacteristic
204C2#2259023611yescharacteristic
subgroupG/H
centerC22#17986109
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1580C2
socleC22×C71#442
prime radicalC2×D8×C71C2×C4
cosocleC4×C8×C71C23
hypercenter#1580D71
hypocenterC71#12077
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #22587250C2semidirect
3. #22586405C2semidirect
4. #22586309C2semidirect
6. #22586261C2semidirect
110. C71#12077semidirect

Maximal subgroups

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

subgroupindex
#225902042
#225872502
#225864052
#225863092
#225863072
#225862612
C71×#15802
#1207771