Finite Group Databasebeta

← #31911149

Subgroups — #31911149

Order 18176.

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

normal subgroups only

This group has 312 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#22590565C211yesmaximal, characteristic, complemented
2#22590375C211yesmaximal, characteristic
3#22587700C211yesmaximal, characteristic, complemented
4#22586398C211yesmaximal, characteristic, complemented
5Q8⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
6#22586301C211yesmaximal, characteristic, complemented
7C71×#3380C211yesmaximal, characteristic, complemented
8C2×#14556069—21no—
9#17986376C2211yescharacteristic
10#17986199—21no—
11C4⋊(C71⋊C16)C2211yescharacteristic
12#17985799—22no—
13#17985799—21no—
14C4×C71⋊C16C2211yescharacteristic
15#17985752C2211yescharacteristic
16#17985742—21no—
17#554×C71—21no—
18#500×C71C2211yescharacteristic
19#499×C71—21no—
20#478×C71—22no—
21#476×C71—21no—
22C8×Q8×C71C2211yescharacteristic
23C4×D8×C71—21no—
24C4×D8×C71C2211yescharacteristic
25C2×#11765852—41no—
26C2×D568—21no—
27C2×D568—21no—
28#14556125—21no—
29#14556069—42no—
30C2×C71⋊C16—21no—
31C2×C71⋊C16—21no—
32C4⋊(C71⋊C8)—21no—
33#14556048—41no—
34C2×D8×C71D411yescharacteristic
35C4⋊Q8×C71—21no—
36#125×C71—22no—
37C22⋊Q8×C71—42no—
38C22⋊Q8×C71—41no—
39C4×Q8×C71D411yescharacteristic
40C4×D4×C71—22no—
41C4×D4×C71—21no—
42C2×C4⋊C4×C71—41no—
43#107×C71—21no—
44#107×C71D411yescharacteristic
45#107×C71D412yes—
46#106×C71—22no—
47C4⋊C8×C71—21no—
48C4⋊C8×C71D411yescharacteristic
49#102×C71—22no—
50#102×C71—21no—
51C4×C8×C71—21no—
52C4×C8×C71C2311yescharacteristic
53C2×D284—41no—
54C2×D284—41no—
55C2×D284—41no—
56#11765852—82no—
57D568—42no—
58D568—41no—
59D568—41no—
60C2×C71⋊C8—41no—
61C71⋊C16—41no—
62C71⋊C16—41no—
63C2×Q8×C71—41no—
64C2×Q8×C71D811yescharacteristic
65C2×D4×C71—21no—
66C22×C4×C71—42no—
67C22×C4×C71—41no—
68D8×C71—21no—
69D8×C71SD1612yescomplemented
70C2×C8×C71—21no—
71C2×C8×C71C2×D411yescharacteristic
72C2×C8×C71C2×D411yescharacteristic
73C4⋊C4×C71—21no—
74C4⋊C4×C71—21no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71D811yescharacteristic
80C22⋊C4×C71—42no—
81C22⋊C4×C71—41no—
82C42×C71—21no—
83C42×C71C2×D411yescharacteristic
84C23×C71—41no—
85C22×D71—81no—
86Q8×C71—21no—
87Q8×C71—21no—
88Q8×C71—161no—
89D4×C71—22no—
90D4×C71—41no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—41no—
94C2×C4×C71—42no—
95C2×C4×C71—81no—
96C2×C4×C71—81no—
97C2×C4×C71—162no—
98C2×C4×C71#12011yescharacteristic
99C2×C4×C71C2×D811yescharacteristic
100C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
101D284—81no—
102D284—81no—
103D284—81no—
104D284—81no—
105D284—161no—
106C8×C71—21no—
107C8×C71—41no—
108C8×C71C2×SD1611yescharacteristic
109C8×C71#13711yescharacteristic
110C71⋊C8—81no—
111C22×C71—42no—
112C22×C71—81no—
113C22×C71—81no—
114C22×C71#44711yescharacteristic
115C2×D71—161no—
116C2×D71—161no—
117C2×D71—161no—
118C4×C71—21no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71—81no—
122C4×C71—82no—
123C4×C71#44811yescharacteristic
124C4×C71#45411yescharacteristic
125C4×C71#50811yescharacteristic
126C4×C71#50911yescharacteristic
127#12355—711nomaximal, Sylow
128C2×C71—82no—
129C2×C71#162711yescharacteristic
130C2×C71#218611yescharacteristic
131C2×C71#219211yescharacteristic
132D71—321no—
133#3380D7111yescharacteristic, complemented
134#2240—711no—
135#2206—711no—
136#1668—711no—
137#1365—711no—
138Q8⋊C16—711no—
139#1331—711no—
140C71#1235511yescharacteristic, complemented, Sylow
141#554—21no—
142C2×D16—1421no—
143#500C2×D7111yescharacteristic
144#499—21no—
145#493—711no—
146#478—22no—
147#476—21no—
148#460—1421no—
149C8×Q8C2×D7111yescharacteristic
150C4×D8—21no—
151C4×D8C2×D7111yescharacteristic
152C4⋊C16—711no—
153#357—1421no—
154#357—1422no—
155C4×C16—711no—
156#335—711no—
157#331—1421no—
158C2×SD16—2841no—
159C2×D8—1421no—
160C2×D8—1421no—
161C2×D8C22⋊D7111yescharacteristic
162C4⋊Q8—21no—
163#127—1421no—
164#125—22no—
165C22⋊Q8—42no—
166C22⋊Q8—41no—
167C4×Q8C22⋊D7111yescharacteristic
168C4×D4—22no—
169C4×D4—21no—
170C2×C4⋊C4—41no—
171D16—2842no—
172C2×C16—1421no—
173C2×C16—1421no—
174#107—21no—
175#107C22⋊D7111yescharacteristic
176#107C22⋊D7112yes—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—1421no—
180C4⋊C8C22⋊D7111yescharacteristic
181#102—22no—
182#102—21no—
183#102—2841no—
184C4×C8—21no—
185C4×C8C22×D7111yescharacteristic
186C2×Q8—41no—
187C2×Q8#1176585011yescharacteristic
188C2×D4—21no—
189C2×D4—2841no—
190C2×D4—2841no—
191C2×D4—2841no—
192C22×C4—42no—
193C22×C4—41no—
194SD16—5682no—
195D8—21no—
196D8—2842no—
197D8—2841no—
198D8—2841no—
199D8#1176585212yescomplemented
200C2×C8—21no—
201C2×C8—2841no—
202C2×C8C2×C22⋊D7111yescharacteristic
203C2×C8C2×C22⋊D7111yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4#1176585011yescharacteristic
211C22⋊C4—42no—
212C22⋊C4—41no—
213C42—21no—
214C42C2×C22⋊D7111yescharacteristic
215C16—2841no—
216C16—2841no—
217C23—41no—
218C23—5681no—
219Q8—21no—
220Q8—21no—
221Q8—161no—
222D4—22no—
223D4—41no—
224D4—5681no—
225D4—5681no—
226D4—5681no—
227D4—5681no—
228D4—11361no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—162no—
236C2×C4#1455609711yescharacteristic
237C2×C4C2×#1176585011yescharacteristic
238C2×C4#1455615411yescharacteristic
239C8—21no—
240C8—41no—
241C8—5681no—
242C8C2×#1176585211yescharacteristic
243C8#1455615911yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—11361no—
248C22—11361no—
249C22—11361no—
250C22#1798613311yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4#1798613811yescharacteristic
257C4#1798617211yescharacteristic
258C4#1798642911yescharacteristic
259C4#1798643411yescharacteristic
260C2—82no—
261C2—22721no—
262C2#2258744311yescharacteristic
263C2#2259026311yescharacteristic
264C2#2259029711yescharacteristic
subgroupG/H
centerC22#17986133
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#3380C2
socleC22×C71#447
prime radical#22590375C2
hypercenter#3380D71
hypocenterC71#12355
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #22590565C2semidirect
3. #22587700C2semidirect
4. #22586398C2semidirect
5. Q8⋊(C71⋊C16)C2semidirect
6. #22586301C2semidirect
7. C71×#3380C2semidirect
69. D8×C71SD16semidirect
133. #3380D71semidirect
140. C71#12355semidirect
199. D8#11765852semidirect

Maximal subgroups

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

subgroupindex
#225905652
#225903752
#225877002
#225863982
Q8⋊(C71⋊C16)2
#225863012
C71×#33802
#1235571