Finite Group Databasebeta

← #31908645

Subgroups — #31908645

Order 18176.

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

normal subgroups only

This group has 326 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#22598715C211yesmaximal, characteristic
2#22590152C211yesmaximal, characteristic, complemented
3#22586372C214yesmaximal, complemented
4C71×#2165C211yesmaximal, characteristic, complemented
5#17986556C411yescharacteristic
6C2×#14556125C411yescharacteristic
7#17986364C412yes—
8#17986364C2212yes—
9#17986346—21no—
10#17986346—21no—
11#17986005—21no—
12#17986005—21no—
13#17985820—21no—
14#17985781—21no—
15#17985777C2212yescomplemented
16C4×M16×C71C2211yescharacteristic
17#364×C71—21no—
18#349×C71—21no—
19#347×C71C2212yescomplemented
20C22×D284—21no—
21C22×D284—41no—
22#14556128—21no—
23#14556125—21no—
24#14556125—41no—
25#14556125C2×C412yes—
26#14556120C2×C412yes—
27#14556107—41no—
28C4×Dic142—41no—
29#14556087—21no—
30#14556068—22no—
31#14556050—22no—
32#14556050—22no—
33#14556050—22no—
34#14556050—22no—
35C2×M16×C71D411yescharacteristic
36C2×M16×C71D411yescharacteristic
37C2×C42×C71C2×C411yescharacteristic
38M32×C71—22no—
39#97×C71C2×C411yescharacteristic
40#97×C71C2311yescharacteristic
41C4×C8×C71D411yescharacteristic
42C4×C8×C71D411yescharacteristic
43C23×D71—41no—
44C2×Dic142—41no—
45C2×D284—22no—
46C2×D284—42no—
47C2×D284—42no—
48C2×D284—42no—
49C2×D284—42no—
50C2×D284—41no—
51C2×D284—84no—
52C4⋊Dic71—22no—
53C4⋊Dic71—41no—
54Dic71⋊C4—42no—
55C22⋊Dic71—41no—
56C4×Dic71—41no—
57C71⋊C16—42no—
58C22×C4×C71—21no—
59C22×C4×C71C22⋊C411yescharacteristic
60M16×C71—41no—
61M16×C71—41no—
62C2×C8×C71C22⋊C411yescharacteristic
63C2×C8×C71C22⋊C411yescharacteristic
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71C2×D411yescharacteristic
66C42×C71—21no—
67C42×C71C22⋊C411yescharacteristic
68C42×C71C22×C411yescharacteristic
69C16×C71—42no—
70C23×C71#10011yescharacteristic
71C22×D71—42no—
72C22×D71—82no—
73C22×D71—84no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—42no—
79C2×C4×C71#10011yescharacteristic
80C2×C4×C71C2×C22⋊C411yescharacteristic
81C2×Dic71—42no—
82D284—42no—
83D284—42no—
84D284—82no—
85D284—84no—
86D284—84no—
87Dic142—82no—
88C8×C71—21no—
89C8×C71—21no—
90C8×C71—21no—
91C8×C71—21no—
92C22×C71—21no—
93C22×C71—41no—
94C22×C71C2×#10011yescharacteristic
95C2×D71—84no—
96C2×D71—82no—
97C2×D71—164no—
98C4×C71—22no—
99C4×C71—21no—
100C4×C71—41no—
101C4×C71#42111yescharacteristic
102C4×C71#42111yescharacteristic
103Dic71—82no—
104#11915—711nomaximal, Sylow
105C2×C71—21no—
106C2×C71—41no—
107C2×C71#153311yescharacteristic
108D71—162no—
109#3138—711no—
110#2165—711no—
111#2165D7111yescharacteristic, complemented
112#1355—714no—
113C71#1191511yescharacteristic, complemented, Sylow
114#533—711no—
115C2×#127—711no—
116#490—712no—
117#490—712no—
118#486—1421no—
119#486—1421no—
120#418—1421no—
121#418—1421no—
122C4×M16C2×D7111yescharacteristic
123#364—21no—
124#364—1421no—
125#349—21no—
126#349—1421no—
127#347—712no—
128#347C2×D7112yescomplemented
129C22×D4—2841no—
130C22×D4—1421no—
131C2×M16D28411yescharacteristic
132C2×M16C22⋊D7111yescharacteristic
133C4⋊Q8—1421no—
134#127—2841no—
135#127—712no—
136#127—1421no—
137#125—712no—
138C22⋊Q8—2841no—
139C4×Q8—2841no—
140#117—1421no—
141C2×C42C4×D7111yescharacteristic
142M32—22no—
143M32—1422no—
144#102—1422no—
145#102—1422no—
146#102—1422no—
147#102—1422no—
148#97C4×D7111yescharacteristic
149#97C22×D7111yescharacteristic
150C4×C8D28411yescharacteristic
151C4×C8C22⋊D7111yescharacteristic
152C24—2841no—
153C2×Q8—2841no—
154C2×D4—2842no—
155C2×D4—2842no—
156C2×D4—2842no—
157C2×D4—2842no—
158C2×D4—2841no—
159C2×D4—5684no—
160C2×D4—1422no—
161C22×C4—21no—
162C22×C4(C2×D71)⋊C411yescharacteristic
163M16—41no—
164M16—41no—
165C2×C8(C2×D71)⋊C411yescharacteristic
166C2×C8(C2×D71)⋊C411yescharacteristic
167C2×C8C2×C22⋊D7111yescharacteristic
168C2×C8C2×D28411yescharacteristic
169C4⋊C4—2842no—
170C4⋊C4—2841no—
171C4⋊C4—1422no—
172C22⋊C4—2841no—
173C42—21no—
174C42—2841no—
175C42(C2×D71)⋊C411yescharacteristic
176C42C2×C4×D7111yescharacteristic
177C16—42no—
178C16—2842no—
179C23—2842no—
180C23—5682no—
181C23—5684no—
182C23#1455604411yescharacteristic
183Q8—5682no—
184D4—2842no—
185D4—2842no—
186D4—5682no—
187D4—5684no—
188D4—5684no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—2842no—
195C2×C4#1455604411yescharacteristic
196C2×C4C2×(C2×D71)⋊C411yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C22—21no—
202C22—41no—
203C22—5684no—
204C22—5682no—
205C22—11364no—
206C22C2×#1455604411yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—5682no—
211C4#1798601511yescharacteristic
212C4#1798601711yescharacteristic
213C2—21no—
214C2—41no—
215C2—11362no—
216C2#2258696311yescharacteristic
subgroupG/H
centerC2#22586963
commutatorC42×C71C22×C4
Frattini#97C22×D71
FittingC71×#2165C2
socleC2×C71#1533
prime radicalC2×#14556125C4
cosocle#97×C71C23
hypercenter#2165D71
hypocenterC71#11915
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #22590152C2semidirect
3. #22586372C2semidirect
4. C71×#2165C2semidirect
15. #17985777C22semidirect
19. #347×C71C22semidirect
111. #2165D71semidirect
113. C71#11915semidirect
128. #347C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225987152
#225901522
#225863722
C71×#21652
#1191571