Finite Group Databasebeta

← #31908972

Subgroups — #31908972

Order 18176.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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#22598999C211yesmaximal, characteristic
2#22590156C211yesmaximal, characteristic
3#22586402C212yesmaximal, complemented
4#22586378C212yesmaximal, complemented
5C71×#2153C211yesmaximal, characteristic
6#17986611C412yes—
7#17986401C2211yescharacteristic
8#17986379C2211yescharacteristic
9#17986368C412yes—
10#17986257—21no—
11#17986240—21no—
12#17986107—21no—
13C4×Dic284—21no—
14#17985820C2212yes—
15#17985781—21no—
16#17985777—21no—
17C2×M32×C71—21no—
18#405×C71C2211yescharacteristic
19#364×C71—21no—
20#364×C71C2212yes—
21C2×Dic284—21no—
22C2×Dic284—21no—
23#14556128C2×C412yes—
24#14556121—42no—
25#14556120C2×C412yes—
26#14556118—42no—
27#14556107—22no—
28C4×Dic142—22no—
29#14556068—22no—
30#14556062—21no—
31#14556060—21no—
32#14556053—22no—
33#14556053—22no—
34C2×M16×C71D411yescharacteristic
35C22×C8×C71D411yescharacteristic
36#117×C71C2×C411yescharacteristic
37M32×C71—22no—
38M32×C71—41no—
39C2×C16×C71—41no—
40#97×C71C2×C411yescharacteristic
41#97×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—22no—
45Dic284—42no—
46Dic284—41no—
47Dic284—41no—
48C4⋊Dic71—22no—
49C4⋊Dic71—42no—
50Dic71⋊C4—42no—
51Dic71⋊C4—42no—
52Dic71⋊C4—42no—
53C22⋊Dic71—42no—
54C4×Dic71—42no—
55C71⋊C16—42no—
56C22×C4×C71C22⋊C411yescharacteristic
57M16×C71—22no—
58C2×C8×C71—21no—
59C2×C8×C71—21no—
60C2×C8×C71C22⋊C411yescharacteristic
61C2×C8×C71C22⋊C411yescharacteristic
62C2×C8×C71C2×D411yescharacteristic
63C2×C8×C71C2×D411yescharacteristic
64C4⋊C4×C71—21no—
65C22⋊C4×C71—21no—
66C42×C71C22⋊C411yescharacteristic
67C42×C71C22×C411yescharacteristic
68C16×C71—42no—
69C23×C71#10011yescharacteristic
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71#10011yescharacteristic
75C2×C4×C71C2×C22⋊C411yescharacteristic
76C2×Dic71—42no—
77C2×Dic71—42no—
78Dic142—42no—
79Dic142—42no—
80C8×C71—21no—
81C8×C71—21no—
82C8×C71#13512yes—
83C8×C71AGL(1,ℤ/8)11yescharacteristic
84C8×C71#13711yescharacteristic
85C22×C71—21no—
86C22×C71—41no—
87C22×C71C2×#10011yescharacteristic
88C4×C71—21no—
89C4×C71—41no—
90C4×C71—41no—
91C4×C71#41611yescharacteristic
92C4×C71#41711yescharacteristic
93Dic71—82no—
94Dic71—82no—
95#11990—711nomaximal, Sylow
96C2×C71—21no—
97C2×C71—41no—
98C2×C71#155111yescharacteristic
99#3177—711no—
100#2166—711no—
101#2153D7111yescharacteristic
102#1366—712no—
103#1358—712no—
104C71#1199011yescharacteristic, complemented, Sylow
105#541—712no—
106C2×M32—21no—
107#499—711no—
108#494—711no—
109#491—712no—
110#470—1421no—
111#467—1421no—
112#441—1421no—
113C4×Q16—1421no—
114#405C2×D7111yescharacteristic
115#364—21no—
116#364—712no—
117#364C2×D7112yes—
118#349—1421no—
119#347—1421no—
120C2×Q16—1421no—
121C2×Q16—1421no—
122C2×M16D28411yescharacteristic
123C22×C8C22⋊D7111yescharacteristic
124C4⋊Q8—712no—
125#126—2842no—
126#125—2842no—
127#125—712no—
128C22⋊Q8—1422no—
129C4×Q8—1422no—
130#117C4×D7111yescharacteristic
131M32—22no—
132M32—41no—
133M32—1422no—
134C2×C16—41no—
135#107—1421no—
136#106—1421no—
137#103—1422no—
138#103—1422no—
139#97C4×D7111yescharacteristic
140#97D28411yescharacteristic
141C4×C8C22⋊D7111yescharacteristic
142C4×C8C22×D7111yescharacteristic
143C2×Q8—1422no—
144C22×C4(C2×D71)⋊C411yescharacteristic
145Q16—2842no—
146Q16—2841no—
147Q16—2841no—
148M16—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8(C2×D71)⋊C411yescharacteristic
152C2×C8(C2×D71)⋊C411yescharacteristic
153C2×C8C2×C22⋊D7111yescharacteristic
154C2×C8C2×D28411yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—2842no—
157C4⋊C4—2842no—
158C4⋊C4—2842no—
159C4⋊C4—2842no—
160C4⋊C4—1422no—
161C22⋊C4—21no—
162C22⋊C4—2842no—
163C42—2842no—
164C42(C2×D71)⋊C411yescharacteristic
165C42C2×C4×D7111yescharacteristic
166C16—42no—
167C16—2842no—
168C23#1455604411yescharacteristic
169Q8—2842no—
170Q8—2842no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—2842no—
176C2×C4—2842no—
177C2×C4#1455604411yescharacteristic
178C2×C4C2×(C2×D71)⋊C411yescharacteristic
179C8—21no—
180C8—21no—
181C8#1455615012yes—
182C8#1455615411yescharacteristic
183C8#1455615911yescharacteristic
184C22—21no—
185C22—41no—
186C22C2×#1455604411yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—5682no—
191C4—5682no—
192C4#1798599611yescharacteristic
193C4#1798599811yescharacteristic
194C2—21no—
195C2—41no—
196C2#2258703311yescharacteristic
subgroupG/H
centerC2#22587033
commutatorC42×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#2153C2
socleC2×C71#1551
prime radicalC23×C71#100
cosocleC4×C8×C71C23
hypercenter#2153D71
hypocenterC71#11990
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
3. #22586402C2semidirect
4. #22586378C2semidirect
104. C71#11990semidirect

Maximal subgroups

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

subgroupindex
#225989992
#225901562
#225864022
#225863782
C71×#21532
#1199071