Finite Group Databasebeta

← #31911581

Subgroups — #31911581

Order 18176.

256 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 63 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#22590488C211yesmaximal, characteristic, complemented
2#22590372C211yesmaximal, characteristic
3#22587929C211yesmaximal, characteristic, complemented
4#22587134C211yesmaximal, characteristic, complemented
5#22586403C211yesmaximal, characteristic, complemented
6#22586399C211yesmaximal, characteristic, complemented
7C71×#1361C211yesmaximal, characteristic, complemented
8C2×#14556069—21no—
9#17986404C2211yescharacteristic
10#17986371—21no—
11#17986345—21no—
12C4×D568C2211yescharacteristic
13#17986064—21no—
14#17985827—21no—
15C4⋊(C71⋊C16)C2211yescharacteristic
16#17985801—21no—
17#17985800—21no—
18#17985726—21no—
19C8×C71⋊C8C2211yescharacteristic
20#493×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×#11765853—42no—
24C2×D568D411yescharacteristic
25C2×C8×D71—41no—
26#14556128—21no—
27C4×D284—21no—
28#14556069—42no—
29C2×C71⋊C16—21no—
30#14556062—21no—
31#14556062—21no—
32#14556062D411yescharacteristic
33#14556050—21no—
34#14556049—41no—
35#14556049—41no—
36(C2×D71)⋊C8—41no—
37C4×C71⋊C8D411yescharacteristic
38C4×C71⋊C8D411yescharacteristic
39C2×D8×C71—21no—
40C2×D8×C71—21no—
41#127×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×D284—21no—
47C2×C4×D71—41no—
48#11765853—84no—
49D568—21no—
50D568SD1612yescomplemented
51C2×C71⋊C8—21no—
52C2×C71⋊C8D811yescharacteristic
53C2×C71⋊C8D811yescharacteristic
54C8×D71—82no—
55C4⋊Dic71—21no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58(C2×D71)⋊C4—41no—
59C71⋊C16—41no—
60C2×D4×C71—41no—
61C2×D4×C71—41no—
62C2×D4×C71—41no—
63D8×C71—41no—
64D8×C71—41no—
65D8×C71—81no—
66C2×C8×C71—41no—
67C2×C8×C71C2×D411yescharacteristic
68C2×C8×C71#4111yescharacteristic
69C42×C71C2×D411yescharacteristic
70C16×C71—41no—
71C23×C71—81no—
72C22×D71—41no—
73D4×C71—82no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—81no—
77D4×C71—81no—
78C2×C4×C71#12111yescharacteristic
79C2×C4×C71C2×D811yescharacteristic
80C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
81C2×Dic71—41no—
82C2×Dic71—81no—
83D284—22no—
84D284—41no—
85C4×D71—81no—
86C4×D71—81no—
87Dic142—82no—
88C8×C71—21no—
89C8×C71—81no—
90C8×C71C2×SD1611yescharacteristic
91C8×C71#13511yescharacteristic
92C71⋊C8—22no—
93C71⋊C8—21no—
94C71⋊C8—21no—
95C22×C71—161no—
96C22×C71—161no—
97C22×C71—161no—
98C22×C71#45911yescharacteristic
99C2×D71—42no—
100C2×D71—81no—
101C2×D71—81no—
102C4×C71—21no—
103C4×C71#46511yescharacteristic
104C4×C71#47111yescharacteristic
105C4×C71#50811yescharacteristic
106C4×C71#50911yescharacteristic
107Dic71—81no—
108Dic71—161no—
109#12418—711nomaximal, Sylow
110C2×C71—321no—
111C2×C71#166511yescharacteristic
112C2×C71#221011yescharacteristic
113C2×C71#221311yescharacteristic
114D71—82no—
115#2226—711no—
116#2206—711no—
117#1706—711no—
118C8×D8—711no—
119#1367—711no—
120#1365—711no—
121#1361D7111yescharacteristic, complemented
122C71#1241811yescharacteristic, complemented, Sylow
123C2×D16—1421no—
124#500—711no—
125#493C2×D7111yescharacteristic
126#492—1421no—
127#486—1421no—
128C4×D8—711no—
129C8×D4—1421no—
130#367—1421no—
131C4⋊C16—711no—
132C4⋊C16C2×D7111yescharacteristic
133#357—1421no—
134#357—1421no—
135#335C2×D7111yescharacteristic
136D4⋊C8—1421no—
137C82—711no—
138C2×SD16—2842no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—711no—
142C22×C8—2841no—
143C4⋊Q8—1421no—
144#127—21no—
145C4×D4—1421no—
146D16—2842no—
147C2×C16—21no—
148C2×C16—1421no—
149#107—711no—
150#107—1421no—
151#107—1421no—
152C4⋊C8—21no—
153#102—2841no—
154#102—2841no—
155#102—1421no—
156C22⋊C8—2841no—
157C4×C8—711no—
158C4×C8—711no—
159C4×C8C22×D7111yescharacteristic
160C2×Q8—2841no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—1421no—
165C22×C4—2841no—
166SD16—5684no—
167D8—41no—
168D8—41no—
169D8—81no—
170D8—712no—
171D8—1421no—
172C2×C8—41no—
173C2×C8—5682no—
174C2×C8—711no—
175C2×C8—711no—
176C2×C8—1421no—
177C2×C8D4×D7111yescharacteristic
178C2×C8#1176586511yescharacteristic
179C4⋊C4—2841no—
180C4⋊C4—2841no—
181C4⋊C4—1421no—
182C22⋊C4—2841no—
183C42D4×D7111yescharacteristic
184C16—41no—
185C16—2841no—
186C23—81no—
187C23—2841no—
188Q8—5682no—
189D4—82no—
190D4—81no—
191D4—81no—
192D4—81no—
193D4—81no—
194D4—2841no—
195D4—1422no—
196C2×C4—2841no—
197C2×C4—5681no—
198C2×C4—5681no—
199C2×C4—5681no—
200C2×C4#1455610111yescharacteristic
201C2×C4D8×D7111yescharacteristic
202C2×C4#1455615511yescharacteristic
203C8—21no—
204C8—81no—
205C8—1422no—
206C8—1421no—
207C8—1421no—
208C8SD16×D7111yescharacteristic
209C8#1455614711yescharacteristic
210C22—161no—
211C22—161no—
212C22—161no—
213C22—2842no—
214C22—5681no—
215C22—5681no—
216C22#1798619311yescharacteristic
217C4—21no—
218C4—5681no—
219C4—11361no—
220C4#1798622711yescharacteristic
221C4#1798626111yescharacteristic
222C4#1798643011yescharacteristic
223C4#1798643311yescharacteristic
224C2—321no—
225C2—5682no—
226C2#2258768011yescharacteristic
227C2#2259039411yescharacteristic
228C2#2259041511yescharacteristic
subgroupG/H
centerC22#17986193
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1361C2
socleC22×C71#459
prime radical#22590372C2
hypercenter#1361D71
hypocenterC71#12418
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
1. #22590488C2semidirect
3. #22587929C2semidirect
4. #22587134C2semidirect
5. #22586403C2semidirect
6. #22586399C2semidirect
7. C71×#1361C2semidirect
50. D568SD16semidirect
121. #1361D71semidirect
122. C71#12418semidirect

Maximal subgroups

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

subgroupindex
#225904882
#225903722
#225879292
#225871342
#225864032
#225863992
C71×#13612
#1241871