Finite Group Databasebeta

← #37511011

Subgroups — #37511011

Order 22912.

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

normal subgroups only

This group has 226 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#26039192C211yesmaximal, characteristic, complemented
2#26038629C211yesmaximal, characteristic
3#26038617C211yesmaximal, characteristic
4#26038233C211yesmaximal, characteristic, complemented
5#26038229C211yesmaximal, characteristic, complemented
6#26038228C211yesmaximal, characteristic, complemented
7#458×C179C211yesmaximal, characteristic
8#19153143—21no—
9#19153112—21no—
10#19153106—21no—
11#19153106—21no—
12#19153103—21no—
13#19153100C2211yescharacteristic
14#19153094—21no—
15#19153094C2211yescharacteristic
16#19153092—21no—
17#19153090—21no—
18#19153040—21no—
19#19153030C2211yescharacteristic
20#19153028—21no—
21#19153028C2211yescharacteristic
22H(5,𝔽2)×C179C2211yescharacteristic
23#123×C179—21no—
24#123×C179C2211yescharacteristic
25#99×C179—21no—
26#99×C179C2211yescharacteristic
27C2×Dic358—21no—
28C2×Dic358D411yescharacteristic
29C22×Dic179D411yescharacteristic
30#15668249—41no—
31#15668247—41no—
32#15668243—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179—21no—
35Dic179⋊C4—21no—
36Dic179⋊C4—21no—
37Dic179⋊C4—21no—
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—41no—
41C22⋊Dic179—21no—
42C22⋊Dic179—21no—
43C22⋊Dic179—21no—
44C22⋊Dic179—21no—
45C22⋊Dic179—41no—
46C22⋊Dic179D411yescharacteristic
47C4×Dic179—21no—
48C4×Dic179D411yescharacteristic
49#41×C179—21no—
50#41×C179—41no—
51C2×D4×C179—21no—
52C2×D4×C179—21no—
53C2×D4×C179—41no—
54C2×D4×C179C2311yescharacteristic
55C22×C4×C179D411yescharacteristic
56C4⋊C4×C179—21no—
57C4⋊C4×C179—41no—
58C22⋊C4×C179—21no—
59C22⋊C4×C179—21no—
60C22⋊C4×C179—41no—
61C22⋊C4×C179D411yescharacteristic
62Dic358—41no—
63Dic358—81no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—21no—
68C2×Dic179—41no—
69C2×Dic179—41no—
70C179⋊C8—41no—
71C23×C179—21no—
72C23×C179—21no—
73C23×C179C2×D411yescharacteristic
74C23×C179C2×D411yescharacteristic
75Q8×C179—21no—
76D4×C179—21no—
77D4×C179—41no—
78D4×C179—41no—
79D4×C179—41no—
80D4×C179—41no—
81C2×C4×C179—21no—
82C2×C4×C179—21no—
83C2×C4×C179—21no—
84C2×C4×C179—21no—
85C2×C4×C179—41no—
86C2×C4×C179—41no—
87C2×C4×C179—41no—
88C2×C4×C179C2×D411yescharacteristic
89C22×C179—21no—
90C22×C179—21no—
91C22×C179—41no—
92C22×C179—41no—
93C22×C179—41no—
94C22×C179—81no—
95C22×C179—81no—
96C22×C179#12011yescharacteristic
97C4×C179—21no—
98C4×C179—41no—
99C4×C179—41no—
100C4×C179—41no—
101C4×C179—81no—
102Dic179—41no—
103Dic179—41no—
104Dic179—41no—
105Dic179—41no—
106Dic179—81no—
107C2×C179—21no—
108C2×C179—41no—
109C2×C179—41no—
110C2×C179—81no—
111C2×C179B(4,𝔽2)11yescharacteristic
112C179#220111yescharacteristic, complemented, Sylow
113#2201—1791nomaximal, Sylow
114#563—1791no—
115#458—1791no—
116#458D17911yescharacteristic
117#455—1791no—
118#354—1791no—
119#352—1791no—
120#352—1791no—
121H(5,𝔽2)C2×D17911yescharacteristic
122#137—3581no—
123C4⋊Q8—3581no—
124#126—3581no—
125#126—3581no—
126#125—3581no—
127#124—1791no—
128#123—21no—
129#123—3581no—
130#123—1791no—
131#123C2×D17911yescharacteristic
132C22⋊Q8—3581no—
133C22⋊Q8—3581no—
134#104—3581no—
135#100—1791no—
136#99—21no—
137#99—3581no—
138#99—1791no—
139#99C2×D17911yescharacteristic
140#41—21no—
141#41—41no—
142C2×Q8—3581no—
143C2×Q8—1791no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—41no—
147C2×D4C22×D17911yescharacteristic
148C22×C4—1791no—
149C22×C4C22⋊D17911yescharacteristic
150Q16—7161no—
151SD16—7161no—
152M16—3581no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—3581no—
156C4⋊C4—3581no—
157C4⋊C4—3581no—
158C4⋊C4—3581no—
159C4⋊C4—3581no—
160C4⋊C4—3581no—
161C4⋊C4—3581no—
162C4⋊C4—7161no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—41no—
166C22⋊C4—3581no—
167C22⋊C4—3581no—
168C22⋊C4—3581no—
169C22⋊C4—3581no—
170C22⋊C4—1791no—
171C22⋊C4—7161no—
172C22⋊C4C22⋊D17911yescharacteristic
173C42—3581no—
174C42—1791no—
175C23—21no—
176C23—21no—
177C23D4×D17911yescharacteristic
178C23C2×C22⋊D17911yescharacteristic
179Q8—21no—
180Q8—14321no—
181Q8—7161no—
182D4—21no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—41no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—3581no—
195C2×C4—3581no—
196C2×C4—3581no—
197C2×C4—3581no—
198C2×C4—7161no—
199C2×C4—7161no—
200C2×C4D4×D17911yescharacteristic
201C8—7161no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—81no—
209C22#1915308111yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—14321no—
216C4—7161no—
217C4—7161no—
218C4—7161no—
219C4—7161no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—81no—
224C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#458×C179C2
socleC2×C179B(4,𝔽2)
prime radicalH(5,𝔽2)×C179C22
hypercenter#458D179
hypocenterC179#2201
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #26039192C2semidirect
4. #26038233C2semidirect
5. #26038229C2semidirect
6. #26038228C2semidirect
112. C179#2201semidirect

Maximal subgroups

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

subgroupindex
#260391922
#260386292
#260386172
#260382332
#260382292
#260382282
#458×C1792
#2201179