Finite Group Databasebeta

← #31911874

Subgroups — #31911874

Order 18176.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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#22590563C211yesmaximal, characteristic, complemented
2D4×C71⋊C16C211yesmaximal, characteristic, complemented
3#22587739C211yesmaximal, characteristic, complemented
4#22586413C211yesmaximal, characteristic, complemented
5#22586399C211yesmaximal, characteristic, complemented
6#22586386C211yesmaximal, characteristic, complemented
7C71×#1667C211yesmaximal, characteristic
8C22×C71⋊C16—21no—
9#17986404C2211yescharacteristic
10#17986289—21no—
11C4⋊(C71⋊C16)C2211yescharacteristic
12#17985800—21no—
13#17985800—21no—
14C22⋊(C71⋊C16)—21no—
15C4×C71⋊C16C2211yescharacteristic
16#17985752C2211yescharacteristic
17#17985747—21no—
18#493×C71C2211yescharacteristic
19#459×C71—21no—
20C8×D4×C71C2211yescharacteristic
21#335×C71C2211yescharacteristic
22#331×C71—21no—
23#14556128—21no—
24C2×C71⋊C16—21no—
25C2×C71⋊C16—42no—
26C2×C71⋊C16D411yescharacteristic
27C2×C71⋊C16D411yescharacteristic
28#14556062—21no—
29#14556062—21no—
30#14556059—41no—
31C4⋊(C71⋊C8)—21no—
32#14556049—41no—
33C2×D8×C71—21no—
34C2×D8×C71—21no—
35C2×D8×C71—41no—
36C22×C8×C71—21no—
37#127×C71—21no—
38C4×D4×C71D411yescharacteristic
39C4⋊C8×C71—21no—
40C4⋊C8×C71D411yescharacteristic
41#102×C71—41no—
42C22⋊C8×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×Dic142—41no—
45C2×C71⋊C8—41no—
46C4⋊Dic71—41no—
47C4⋊Dic71—41no—
48C71⋊C16—21no—
49C71⋊C16—21no—
50C71⋊C16—41no—
51C2×D4×C71—41no—
52C2×D4×C71—41no—
53C2×D4×C71—41no—
54C2×D4×C71D811yescharacteristic
55C22×C4×C71—21no—
56D8×C71—41no—
57D8×C71—41no—
58D8×C71—81no—
59D8×C71—82no—
60C2×C8×C71—21no—
61C2×C8×C71—41no—
62C2×C8×C71—41no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C4⋊C4×C71D811yescharacteristic
67C22⋊C4×C71—21no—
68C42×C71C2×D411yescharacteristic
69C23×C71—21no—
70C23×C71—81no—
71D4×C71—21no—
72D4×C71—81no—
73D4×C71—81no—
74D4×C71—81no—
75D4×C71—81no—
76D4×C71—161no—
77D4×C71SD3212yes—
78C2×C4×C71—21no—
79C2×C4×C71—41no—
80C2×C4×C71—41no—
81C2×C4×C71#12111yescharacteristic
82C2×C4×C71C2×D811yescharacteristic
83C2×C4×C71#13511yescharacteristic
84C2×Dic71—81no—
85Dic142—161no—
86C8×C71—21no—
87C8×C71—41no—
88C8×C71—81no—
89C8×C71#13511yescharacteristic
90C8×C71AGL(1,ℤ/8)11yescharacteristic
91C71⋊C8—81no—
92C22×C71—22no—
93C22×C71—41no—
94C22×C71—41no—
95C22×C71—161no—
96C22×C71—161no—
97C22×C71—161no—
98C22×C71#46611yescharacteristic
99C4×C71—21no—
100C4×C71—41no—
101C4×C71#46311yescharacteristic
102C4×C71#47111yescharacteristic
103C4×C71C2×SD3211yescharacteristic
104C4×C71#50811yescharacteristic
105Dic71—161no—
106#12468—711nomaximal, Sylow
107C2×C71—42no—
108C2×C71—321no—
109C2×C71#168111yescharacteristic
110C2×C71#221711yescharacteristic
111C2×C71#222111yescharacteristic
112#2240—711no—
113D4×C16—711no—
114#1675—711no—
115#1667D7111yescharacteristic
116#1372—711no—
117#1365—711no—
118#1361—711no—
119C71#1246811yescharacteristic, complemented, Sylow
120C22×C16—1421no—
121#500—711no—
122#493C2×D7111yescharacteristic
123#476—1421no—
124#459—21no—
125C8×D4C2×D7111yescharacteristic
126C4⋊C16—711no—
127#357—1421no—
128#357—1421no—
129C22⋊C16—1421no—
130C4×C16—711no—
131#335—711no—
132#335C2×D7111yescharacteristic
133#333—1421no—
134#331—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—41no—
138C22×C8—21no—
139C4⋊Q8—1421no—
140#127—21no—
141C4×D4C22⋊D7111yescharacteristic
142C2×C16—2842no—
143C2×C16—711no—
144C2×C16—711no—
145C2×C16—1421no—
146#107—1421no—
147#107—1421no—
148#106—2841no—
149C4⋊C8—21no—
150C4⋊C8—1421no—
151C4⋊C8C22⋊D7111yescharacteristic
152#102—41no—
153#102—2841no—
154C22⋊C8—21no—
155C4×C8C22×D7111yescharacteristic
156C2×Q8—2841no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4#1176585011yescharacteristic
161C22×C4—21no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—82no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—2841no—
171C2×C8D4×D7111yescharacteristic
172C2×C8#1176586411yescharacteristic
173C4⋊C4—2841no—
174C4⋊C4—2841no—
175C4⋊C4#1176585011yescharacteristic
176C22⋊C4—21no—
177C42C2×C22⋊D7111yescharacteristic
178C16—2841no—
179C16—1421no—
180C16—1421no—
181C23—21no—
182C23—81no—
183Q8—11361no—
184D4—21no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—161no—
190D4#1455607212yes—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—5681no—
195C2×C4#1455610011yescharacteristic
196C2×C4C2×#1176585011yescharacteristic
197C2×C4#1455614811yescharacteristic
198C8—21no—
199C8—41no—
200C8—81no—
201C8—5681no—
202C8#1455614711yescharacteristic
203C8#1455615211yescharacteristic
204C22—22no—
205C22—41no—
206C22—41no—
207C22—161no—
208C22—161no—
209C22—161no—
210C22#1798623311yescharacteristic
211C4—21no—
212C4—41no—
213C4—11361no—
214C4#1798621211yescharacteristic
215C4#1798626011yescharacteristic
216C4C2×#1455607211yescharacteristic
217C4#1798642911yescharacteristic
218C2—42no—
219C2—321no—
220C2#2258777511yescharacteristic
221C2#2259043411yescharacteristic
222C2#2259045811yescharacteristic
subgroupG/H
centerC22#17986233
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
Fitting, prime radicalC71×#1667C2
socleC22×C71#466
hypercenter#1667D71
hypocenterC71#12468
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1667

Complements

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

NH ≅ G/Ntype
1. #22590563C2semidirect
2. D4×C71⋊C16C2semidirect
3. #22587739C2semidirect
4. #22586413C2semidirect
5. #22586399C2semidirect
6. #22586386C2semidirect
119. C71#12468semidirect

Maximal subgroups

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

subgroupindex
#225905632
D4×C71⋊C162
#225877392
#225864132
#225863992
#225863862
C71×#16672
#1246871