Finite Group Databasebeta

← #32864209

Subgroups — #32864209

Order 18176.

4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 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#22601988C213yesmaximal, complemented
2#22597746C2112yesmaximal, complemented
3#22597732C2112yesmaximal, complemented
4#22596804C213yesmaximal, complemented
5C71×2−1+6C211yesmaximal, characteristic, complemented
6C22×Q8×D71C2211yescharacteristic, complemented
7Q8×C22⋊D71—212no—
8#17986656—212no—
9#17986570C22118yescomplemented
10#17986506C22118yescomplemented
11#17986477C2213yescomplemented
12#17986189C22148yescomplemented
13#17986186C22116yescomplemented
14#17985973C22136yescomplemented
15#585×C71—212no—
16#585×C71C22112yes—
17C2×2−1+4×C71—212no—
18C2×2−1+4×C71C2213yes—
19C22×Dic142C2313yes—
20C2×Q8×D71—212no—
21C22×C4×D71C2313yes—
22#14556126—218no—
23#14556115—218no—
24#14556109C23136yes—
25#14556107—236no—
26#14556104—218no—
27#14556099—218no—
28#14556096C23112yes—
29Q8×Dic71—212no—
30C4×C22⋊D71—236no—
31#14556087C23118yes—
32#14556043C23148yes—
332−1+4×C71—248no—
342−1+4×C71—236no—
352−1+4×C71—224no—
36H(5,𝔽2)×C71—216no—
37H(5,𝔽2)×C71—236no—
38H(5,𝔽2)×C71C23116yes—
39C2×#41×C71—248no—
40C2×#41×C71—272no—
41C2×#41×C71D4112yescomplemented
42C2×#41×C71C23118yes—
43C22×Q8×C71—216no—
44C22×Q8×C71D4112yescomplemented
45C22×Q8×C71C2311yescharacteristic
46C23×D71C2411yescharacteristic
47C2×Dic142—218no—
48Q8×D71—416no—
49C2×C22⋊D71—212no—
50C22×Dic71C2413yes—
51C2×C4×D71—218no—
52C4⋊Dic71—218no—
53Dic71⋊C4—236no—
54C22⋊Dic71—236no—
55C22⋊Dic71C24112yes—
56(C2×D71)⋊C4—212no—
57C4×Dic71—218no—
58#41×C71—2144no—
59#41×C71—248no—
60#41×C71—248no—
61#41×C71—248no—
62#41×C71—272no—
63C2×Q8×C71—272no—
64C2×Q8×C71—248no—
65C2×Q8×C71C2×D4118yes—
66C2×Q8×C71C2×D4112yes—
67C2×D4×C71—272no—
68C2×D4×C71—248no—
69C2×D4×C71C2×D4118yes—
70C2×D4×C71C24112yes—
71C22×C4×C71—248no—
72C22×C4×C71C2×D4136yes—
73C22×C4×C71C2413yes—
74C23×C71—216no—
75C23×C71C22×D4112yes—
76C23×C71C2511yescharacteristic
77C22×D71—23no—
78C22×D71—42no—
79Q8×C71—216no—
80Q8×C71—236no—
81Q8×C71#120116yes—
82D4×C71—248no—
83D4×C71—224no—
84D4×C71—236no—
85C2×C4×C71—248no—
86C2×C4×C71—272no—
87C2×C4×C71C22×D4112yes—
88C2×C4×C71C22×D4118yes—
89C22⋊D71—424no—
90C2×Dic71—212no—
91C2×Dic71—29no—
92C4×D71—412no—
93Dic142—412no—
94C22×C71—212no—
95C22×C71—41no—
96C22×C71—412no—
97C22×C71—816no—
98C22×C71C23×D413yes—
99C2×D71—26no—
100C2×D71—43no—
101C2×D71—41no—
102C4×C71—212no—
103C4×C71C2×#120112yes—
104Dic71—412no—
105Dic71—43no—
106#60400—711nomaximal, Sylow
107C2×C71—23no—
108C2×C71—412no—
109C2×C71C22×#12011yescharacteristic
110D71—42no—
1112−1+6D7111yescharacteristic, complemented
112#3486—713no—
113#3028—7112no—
114#3025—7112no—
115#2884—713no—
116C71#6040011yescharacteristic, complemented, Sylow
117#585—212no—
118#585C2×D71112yes—
119C2×2−1+4—212no—
120C2×2−1+4C2×D7113yes—
121C23×Q8—711no—
122D4×Q8—14212no—
123#548—14212no—
124#536—7118no—
125#525—7118no—
126#519—713no—
127#458—7148no—
128B(4,𝔽2)—7116no—
129#410—7136no—
1302−1+4—248no—
1312−1+4—236no—
1322−1+4—224no—
133H(5,𝔽2)—216no—
134H(5,𝔽2)—236no—
135H(5,𝔽2)C22×D71116yes—
136C2×#41—248no—
137C2×#41—272no—
138C2×#41C22⋊D71112yescomplemented
139C2×#41C22×D71118yes—
140C22×Q8—216no—
141C22×Q8—713no—
142C22×Q8—14212no—
143C22×Q8C22⋊D71112yescomplemented
144C22×Q8C22×D7111yescharacteristic
145C23×C4—713no—
146C4⋊Q8—14218no—
147#124—14218no—
148#123—7136no—
149C22⋊Q8—14218no—
150C22⋊Q8—14236no—
151#121—14218no—
152#120—7112no—
153C4×Q8—14212no—
154C4×D4—14236no—
155#117—7118no—
156#99—7148no—
157C24—711no—
158#41—2144no—
159#41—248no—
160#41—248no—
161#41—248no—
162#41—272no—
163C2×Q8—272no—
164C2×Q8—248no—
165C2×Q8—28416no—
166C2×Q8—14218no—
167C2×Q8C2×C22⋊D71118yes—
168C2×Q8C2×C22⋊D71112yes—
169C2×D4—272no—
170C2×D4—248no—
171C2×D4—14212no—
172C2×D4C2×C22⋊D71118yes—
173C2×D4C23×D71112yes—
174C22×C4—248no—
175C22×C4—713no—
176C22×C4—14218no—
177C22×C4C2×C22⋊D71136yes—
178C22×C4C23×D7113yes—
179C4⋊C4—14236no—
180C4⋊C4—14218no—
181C22⋊C4—7112no—
182C22⋊C4—14236no—
183C22⋊C4—14212no—
184C42—14218no—
185C23—216no—
186C23—2842no—
187C23—1423no—
188C23C22×C22⋊D71112yes—
189C23C24×D7111yescharacteristic
190Q8—216no—
191Q8—236no—
192Q8—28412no—
193Q8#14556097116yes—
194D4—248no—
195D4—224no—
196D4—236no—
197D4—28424no—
198C2×C4—248no—
199C2×C4—272no—
200C2×C4—28412no—
201C2×C4—14212no—
202C2×C4—1429no—
203C2×C4C22×C22⋊D71112yes—
204C2×C4C22×C22⋊D71118yes—
205C22—212no—
206C22—41no—
207C22—412no—
208C22—816no—
209C22—2843no—
210C22—2841no—
211C22—1426no—
212C22C23×C22⋊D7113yes—
213C4—212no—
214C4—28412no—
215C4—2843no—
216C4C2×#14556097112yes—
217C2—23no—
218C2—412no—
219C2—2842no—
220C2C22×#1455609711yescharacteristic
subgroupG/H
centerC2C22×#14556097
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC71×2−1+6C2
socleC2×C71C22×#120
hypercenter2−1+6D71
hypocenterC71#60400
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #22601988C2semidirect
2. #22597746C2semidirect
3. #22597732C2semidirect
4. #22596804C2semidirect
5. C71×2−1+6C2semidirect
6. C22×Q8×D71C22semidirect
9. #17986570C22semidirect
10. #17986506C22semidirect
11. #17986477C22semidirect
12. #17986189C22semidirect
13. #17986186C22semidirect
14. #17985973C22semidirect
41. C2×#41×C71D4semidirect
44. C22×Q8×C71D4semidirect
111. 2−1+6D71semidirect
116. C71#60400semidirect
138. C2×#41C22⋊D71semidirect
143. C22×Q8C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
#226019882
#225977462
#225977322
#225968042
C71×2−1+62
#6040071