Finite Group Databasebeta

← #31908643

Subgroups — #31908643

Order 18176.

272 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 75 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#22589945C211yesmaximal, characteristic
3#22586374C214yesmaximal, complemented
4C71×#2166C211yesmaximal, characteristic
5#17986611C412yes—
6C2×#14556068—21no—
7#17986401C411yescharacteristic
8#17986379C411yescharacteristic
9#17986368C2212yes—
10#17986257—21no—
11#17986240—21no—
12#17986107—21no—
13C4×Dic284—21no—
14#17985777—21no—
15#17985777C2212yes—
16#405×C71C2211yescharacteristic
17#364×C71—21no—
18#349×C71—21no—
19#347×C71C2212yes—
20C2×Dic284—21no—
21C2×Dic284—21no—
22#14556128C2×C412yes—
23#14556121—42no—
24#14556120C2×C412yes—
25#14556118—42no—
26#14556107—22no—
27C4×Dic142—22no—
28#14556068—22no—
29#14556068—22no—
30C2×C71⋊C16—22no—
31#14556062—21no—
32#14556060—21no—
33#14556053—22no—
34#14556053—22no—
35C2×M16×C71D411yescharacteristic
36C22×C8×C71D411yescharacteristic
37#117×C71C2×C411yescharacteristic
38M32×C71—22no—
39#97×C71D411yescharacteristic
40#97×C71C2311yescharacteristic
41C4×C8×C71C2×C411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C2×Dic142—22no—
44Dic284—42no—
45Dic284—41no—
46Dic284—41no—
47C4⋊Dic71—22no—
48C4⋊Dic71—42no—
49Dic71⋊C4—42no—
50Dic71⋊C4—42no—
51Dic71⋊C4—42no—
52C22⋊Dic71—42no—
53C4×Dic71—42no—
54C71⋊C16—24no—
55C22×C4×C71C22⋊C411yescharacteristic
56M16×C71—41no—
57C2×C8×C71—21no—
58C2×C8×C71—21no—
59C2×C8×C71C22⋊C411yescharacteristic
60C2×C8×C71C22⋊C411yescharacteristic
61C2×C8×C71C2×D411yescharacteristic
62C2×C8×C71C2×D411yescharacteristic
63C4⋊C4×C71—21no—
64C22⋊C4×C71—21no—
65C42×C71C22⋊C411yescharacteristic
66C42×C71C22×C411yescharacteristic
67C16×C71—42no—
68C23×C71#10011yescharacteristic
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71#10011yescharacteristic
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C2×Dic71—42no—
76C2×Dic71—42no—
77Dic142—42no—
78Dic142—42no—
79C8×C71—21no—
80C8×C71—21no—
81C8×C71—21no—
82C8×C71#10411yescharacteristic
83C8×C71#10411yescharacteristic
84C22×C71—21no—
85C22×C71—41no—
86C22×C71C2×#10011yescharacteristic
87C4×C71—21no—
88C4×C71—41no—
89C4×C71—41no—
90C4×C71C2×#10411yescharacteristic
91C4×C71#42111yescharacteristic
92Dic71—82no—
93Dic71—82no—
94#11914—711nomaximal, Sylow
95C2×C71—21no—
96C2×C71—41no—
97C2×C71#153011yescharacteristic
98#3177—711no—
99#2166D7111yescharacteristic
100#2110—711no—
101#1356—714no—
102C71#1191411yescharacteristic, complemented, Sylow
103#541—712no—
104C2×M32—1421no—
105#499—711no—
106#494—711no—
107#491—712no—
108#470—1421no—
109#467—1421no—
110#441—1421no—
111C4×Q16—1421no—
112#405C2×D7111yescharacteristic
113#364—21no—
114#349—21no—
115#347—712no—
116#347—1421no—
117#347C2×D7112yes—
118C2×Q16—1421no—
119C2×Q16—1421no—
120C2×M16C22⋊D7111yescharacteristic
121C22×C8D28411yescharacteristic
122C4⋊Q8—712no—
123#126—2842no—
124#125—2842no—
125#125—712no—
126C22⋊Q8—1422no—
127C4×Q8—1422no—
128#117C4×D7111yescharacteristic
129M32—22no—
130M32—1422no—
131M32—1422no—
132C2×C16—1422no—
133#107—1421no—
134#106—1421no—
135#103—1422no—
136#103—1422no—
137#97D28411yescharacteristic
138#97C22×D7111yescharacteristic
139C4×C8C4×D7111yescharacteristic
140C4×C8C22⋊D7111yescharacteristic
141C2×Q8—1422no—
142C22×C4(C2×D71)⋊C411yescharacteristic
143Q16—2842no—
144Q16—2841no—
145Q16—2841no—
146M16—41no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8(C2×D71)⋊C411yescharacteristic
150C2×C8(C2×D71)⋊C411yescharacteristic
151C2×C8C2×C22⋊D7111yescharacteristic
152C2×C8C2×D28411yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—2842no—
155C4⋊C4—2842no—
156C4⋊C4—2842no—
157C4⋊C4—2842no—
158C4⋊C4—1422no—
159C22⋊C4—21no—
160C22⋊C4—2842no—
161C42—2842no—
162C42(C2×D71)⋊C411yescharacteristic
163C42C2×C4×D7111yescharacteristic
164C16—42no—
165C16—1424no—
166C23#1455604411yescharacteristic
167Q8—2842no—
168Q8—2842no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—2842no—
174C2×C4—2842no—
175C2×C4#1455604411yescharacteristic
176C2×C4C2×(C2×D71)⋊C411yescharacteristic
177C8—21no—
178C8—21no—
179C8—21no—
180C8#1455605411yescharacteristic
181C8#1455605411yescharacteristic
182C22—21no—
183C22—41no—
184C22C2×#1455604411yescharacteristic
185C4—21no—
186C4—41no—
187C4—41no—
188C4—5682no—
189C4—5682no—
190C4C2×#1455605411yescharacteristic
191C4#1798601711yescharacteristic
192C2—21no—
193C2—41no—
194C2#2258695211yescharacteristic
subgroupG/H
centerC2#22586952
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#2166C2
socleC2×C71#1530
prime radicalC23×C71#100
cosocle#97×C71C23
hypercenter#2166D71
hypocenterC71#11914
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
3. #22586374C2semidirect
102. C71#11914semidirect

Maximal subgroups

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

subgroupindex
#225989992
#225899452
#225863742
C71×#21662
#1191471