Finite Group Databasebeta

← #37511019

Subgroups — #37511019

Order 22912.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#26039188C211yesmaximal, characteristic, complemented
2#26038622C211yesmaximal, characteristic, complemented
3#26038612C211yesmaximal, characteristic
4#26038238C211yesmaximal, characteristic, complemented
5#26038237C211yesmaximal, characteristic, complemented
6#26038235C211yesmaximal, characteristic, complemented
7#456×C179C211yesmaximal, characteristic, complemented
8#19153143—21no—
9#19153138—21no—
10#19153111C2211yescharacteristic
11#19153108—21no—
12#19153107—21no—
13#19153103—21no—
14#19153102C2211yescharacteristic
15#19153098—21no—
16#19153091—21no—
17#19153089—21no—
18#19153040—21no—
19#19153040—21no—
20#19153032C2211yescharacteristic
21#19153032C2211yescharacteristic
222−1+4×C179C2211yescharacteristic
23#137×C179—21no—
24C4⋊Q8×C179C2211yescharacteristic
25#104×C179—21no—
26#101×C179C2211yescharacteristic
27C2×Dic358D411yescharacteristic
28C2×D716D411yescharacteristic
29C2×C4×D179—21no—
30#15668249—41no—
31#15668247—41no—
32#15668246—41no—
33#15668244—41no—
34#15668243—21no—
35#15668243—21no—
36C4⋊Dic179—21no—
37Dic179⋊C4—21no—
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—21no—
41Dic179⋊C4—21no—
42(C2×D179)⋊C4—21no—
43(C2×D179)⋊C4—21no—
44(C2×D179)⋊C4—21no—
45(C2×D179)⋊C4—21no—
46(C2×D179)⋊C4—21no—
47C4×Dic179D411yescharacteristic
48C4×Dic179D411yescharacteristic
49#41×C179—21no—
50#41×C179—21no—
51#41×C179—21no—
52#41×C179—41no—
53C2×Q8×C179—41no—
54C2×Q8×C179D411yescharacteristic
55C2×Q8×C179C2311yescharacteristic
56Q16×C179—41no—
57SD16×C179—41no—
58M16×C179—21no—
59C4⋊C4×C179—21no—
60C4⋊C4×C179—21no—
61C42×C179D411yescharacteristic
62C22×D179—21no—
63Dic358—41no—
64D716—41no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—21no—
68C4×D179—81no—
69C179⋊C8—41no—
70C179⋊C8—41no—
71Q8×C179—21no—
72Q8×C179—21no—
73Q8×C179—21no—
74Q8×C179—41no—
75Q8×C179—41no—
76D4×C179—21no—
77D4×C179—21no—
78D4×C179—21no—
79D4×C179—41no—
80C2×C4×C179—21no—
81C2×C4×C179—21no—
82C2×C4×C179—41no—
83C2×C4×C179—41no—
84C2×C4×C179—41no—
85C2×C4×C179C2×D411yescharacteristic
86C2×C4×C179C2×D411yescharacteristic
87C2×C4×C179C2×D411yescharacteristic
88C8×C179—41no—
89C22×C179—41no—
90C22×C179#12011yescharacteristic
91C2×D179—42no—
92C2×D179—41no—
93C4×C179—21no—
94C4×C179—21no—
95C4×C179—21no—
96C4×C179—41no—
97C4×C179—41no—
98C4×C179—41no—
99Dic179—41no—
100Dic179—41no—
101Dic179—41no—
102C2×C179—21no—
103C2×C179—81no—
104C2×C179B(4,𝔽2)11yescharacteristic
105D179—81no—
106C179#220311yescharacteristic, complemented, Sylow
107#2203—1791nomaximal, Sylow
108#563—1791no—
109#456—1791no—
110#456D17911yescharacteristic, complemented
111#454—1791no—
112#356—1791no—
113#356—1791no—
114#355—1791no—
1152−1+4C2×D17911yescharacteristic
116#137—21no—
117#137—3581no—
118AGL(1,ℤ/8)—3581no—
119C4⋊Q8—1791no—
120C4⋊Q8C2×D17911yescharacteristic
121#126—3581no—
122#126—3581no—
123#125—3581no—
124#124—1791no—
125#123—3581no—
126C22⋊Q8—3581no—
127C22⋊Q8—3581no—
128#104—21no—
129#104—3581no—
130#104—3581no—
131#101—1791no—
132#101—1791no—
133#101C2×D17911yescharacteristic
134#41—21no—
135#41—21no—
136#41—21no—
137#41—41no—
138C2×Q8—41no—
139C2×Q8—1791no—
140C2×Q8C22⋊D17911yescharacteristic
141C2×Q8C22×D17911yescharacteristic
142C2×D4—1791no—
143C22×C4—3581no—
144Q16—41no—
145Q16—7161no—
146SD16—41no—
147SD16—7161no—
148SD16—7161no—
149D8—7161no—
150M16—21no—
151M16—3581no—
152M16—3581no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—3581no—
156C4⋊C4—3581no—
157C4⋊C4—3581no—
158C4⋊C4—3581no—
159C4⋊C4—3581no—
160C4⋊C4—3581no—
161C22⋊C4—3581no—
162C22⋊C4—3581no—
163C22⋊C4—3581no—
164C22⋊C4—3581no—
165C22⋊C4—3581no—
166C42—1791no—
167C42—1791no—
168C42C22⋊D17911yescharacteristic
169C23—3581no—
170Q8—21no—
171Q8—21no—
172Q8—21no—
173Q8—41no—
174Q8—41no—
175Q8—7161no—
176D4—21no—
177D4—21no—
178D4—21no—
179D4—41no—
180D4—7161no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3581no—
187C2×C4—3581no—
188C2×C4—3581no—
189C2×C4—14321no—
190C2×C4D4×D17911yescharacteristic
191C2×C4D4×D17911yescharacteristic
192C2×C4C2×C22⋊D17911yescharacteristic
193C8—41no—
194C8—7161no—
195C8—7161no—
196C22—41no—
197C22—7162no—
198C22—7161no—
199C22#1915308111yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—7161no—
207C4—7161no—
208C4—7161no—
209C2—21no—
210C2—81no—
211C2—14321no—
212C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#456×C179C2
socleC2×C179B(4,𝔽2)
prime radical#26038612C2
hypercenter#456D179
hypocenterC179#2203
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26039188C2semidirect
2. #26038622C2semidirect
4. #26038238C2semidirect
5. #26038237C2semidirect
6. #26038235C2semidirect
7. #456×C179C2semidirect
106. C179#2203semidirect
110. #456D179semidirect

Maximal subgroups

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

subgroupindex
#260391882
#260386222
#260386122
#260382382
#260382372
#260382352
#456×C1792
#2203179