Finite Group Databasebeta

← #31912136

Subgroups — #31912136

Order 18176.

306 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 83 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#22590564C211yesmaximal, characteristic, complemented
3#22590563C211yesmaximal, characteristic, complemented
4#22587957C211yesmaximal, characteristic, complemented
5#22587954C211yesmaximal, characteristic
6C16×C71⋊C8C211yesmaximal, characteristic, complemented
7C71×#2240C211yesmaximal, characteristic, complemented
8#17986404C2211yescharacteristic
9#17986402—21no—
10#17986378—21no—
11#17986376C2211yescharacteristic
12#17986375—21no—
13#17986375—21no—
14#17985801—22no—
15#17985800—22no—
16#17985799—22no—
17C4×C71⋊C16C2211yescharacteristic
18C8×C71⋊C8C2211yescharacteristic
19#500×C71C2211yescharacteristic
20#493×C71C2211yescharacteristic
21#357×C71—22no—
22C4×C16×C71C2211yescharacteristic
23C2×#11765855—42no—
24C2×D568—21no—
25C2×D568—21no—
26C2×#11765850—42no—
27C2×#11765850—42no—
28#14556128—21no—
29#14556125—21no—
30C2×C71⋊C16D412yes—
31#14556062—21no—
32#14556062—21no—
33#14556061—42no—
34C4×C71⋊C8D411yescharacteristic
35C4×C71⋊C8D411yescharacteristic
36C2×D8×C71—21no—
37C2×D8×C71—21no—
38C4⋊Q8×C71—21no—
39#127×C71—21no—
40C2×C16×C71D412yes—
41#107×C71—21no—
42#107×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—41no—
45C2×D284—41no—
46C2×D284—41no—
47C2×D284—41no—
48#11765855—84no—
49D568—42no—
50D568—42no—
51#11765850—84no—
52#11765850—84no—
53C2×C71⋊C8D812yes—
54C2×C71⋊C8D812yes—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57C71⋊C16—22no—
58C2×Q8×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71—41no—
62D8×C71—42no—
63D8×C71—42no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71C2×D411yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C42×C71C2×D411yescharacteristic
69C16×C71—22no—
70C23×C71—81no—
71C22×D71—81no—
72Q8×C71—82no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—82no—
76D4×C71—82no—
77C2×C4×C71—81no—
78C2×C4×C71#12711yescharacteristic
79C2×C4×C71C2×D811yescharacteristic
80C2×C4×C71C2×D811yescharacteristic
81C2×Dic71—81no—
82D284—81no—
83D284—81no—
84D284—82no—
85D284—82no—
86Dic142—82no—
87C8×C71#13512yes—
88C8×C71#13512yes—
89C71⋊C8D1614yes—
90C71⋊C8SD3214yes—
91C22×C71—161no—
92C22×C71—161no—
93C22×C71—161no—
94C22×C71#49311yescharacteristic
95C2×D71—161no—
96C2×D71—161no—
97C2×D71—161no—
98C4×C71—161no—
99C4×C71#49511yescharacteristic
100C4×C71#49511yescharacteristic
101C4×C71C2×D1612yes—
102C4×C71C2×SD3212yes—
103Dic71—161no—
104#12509—711nomaximal, Sylow
105C2×C71—321no—
106C2×C71#170911yescharacteristic
107C2×C71#224611yescharacteristic
108C2×C71#224811yescharacteristic
109D71—321no—
110#2240—711no—
111#2240—711no—
112#2240—711no—
113#2240D7111yescharacteristic, complemented
114#1713—711no—
115#1712—711no—
116C8×C16—711no—
117C71#1250911yescharacteristic, complemented, Sylow
118#500—711no—
119#500—1421no—
120#500C2×D7111yescharacteristic
121#494—1421no—
122#493—711no—
123#493—1421no—
124#493—1421no—
125#493C2×D7111yescharacteristic
126#357—22no—
127#357—1422no—
128#357—1422no—
129#357—1422no—
130C4×C16—711no—
131C4×C16C2×D7111yescharacteristic
132C82—711no—
133C2×Q16—2842no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—2842no—
137C2×D8—2842no—
138C2×D8—1421no—
139C2×D8—1421no—
140C4⋊Q8—21no—
141C4⋊Q8—1421no—
142#127—21no—
143#127—1421no—
144C2×C16—712no—
145C2×C16C22⋊D7112yes—
146#107—21no—
147#107—21no—
148#107—2842no—
149#107—1421no—
150#107—1421no—
151C4×C8—711no—
152C4×C8—711no—
153C4×C8C22×D7111yescharacteristic
154C2×Q8—41no—
155C2×Q8—2841no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—2841no—
160C2×D4—2841no—
161C2×D4—2841no—
162Q16—5684no—
163D8—42no—
164D8—42no—
165D8—2842no—
166D8—2842no—
167D8—5684no—
168D8—5684no—
169C2×C8—712no—
170C2×C8—712no—
171C2×C8D4×D7111yescharacteristic
172C2×C8C2×C22⋊D7111yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—2841no—
176C4⋊C4—2841no—
177C42D4×D7111yescharacteristic
178C16—22no—
179C16—1422no—
180C23—81no—
181C23—5681no—
182Q8—82no—
183Q8—5682no—
184D4—81no—
185D4—81no—
186D4—82no—
187D4—82no—
188D4—5681no—
189D4—5681no—
190D4—5682no—
191D4—5682no—
192C2×C4—81no—
193C2×C4—5681no—
194C2×C4#1455612411yescharacteristic
195C2×C4D8×D7111yescharacteristic
196C2×C4D8×D7111yescharacteristic
197C8—714no—
198C8—714no—
199C8#1455614812yes—
200C8#1455614912yes—
201C22—161no—
202C22—161no—
203C22—161no—
204C22—11361no—
205C22—11361no—
206C22—11361no—
207C22#1798637411yescharacteristic
208C4—161no—
209C4—11361no—
210C4#1798638011yescharacteristic
211C4#1798638111yescharacteristic
212C4D16×D7112yes—
213C4SD32×D7112yes—
214C2—321no—
215C2—22721no—
216C2#2258794411yescharacteristic
217C2#2259058811yescharacteristic
218C2#2259059411yescharacteristic
subgroupG/H
centerC22#17986374
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2240C2
socleC22×C71#493
prime radical#22587954C2
hypercenter#2240D71
hypocenterC71#12509
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. #22590565C2semidirect
2. #22590564C2semidirect
3. #22590563C2semidirect
4. #22587957C2semidirect
6. C16×C71⋊C8C2semidirect
7. C71×#2240C2semidirect
113. #2240D71semidirect
117. C71#12509semidirect

Maximal subgroups

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

subgroupindex
#225905652
#225905642
#225905632
#225879572
#225879542
C16×C71⋊C82
C71×#22402
#1250971