Finite Group Databasebeta

← #37509531

Subgroups — #37509531

Order 22912.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#26039008C211yesmaximal, characteristic
2#26038913C211yesmaximal, characteristic, complemented
3#26038457C211yesmaximal, characteristic, complemented
4#26038455C211yesmaximal, characteristic, complemented
5C2×#19153029C211yesmaximal, characteristic, complemented
6#26038195C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9C22×D716C2211yescharacteristic
10C2×#15668243C2211yescharacteristic
11#19153110C412yescomplemented
12#19153101C412yescomplemented
13#19153088—22no—
14#19153083—22no—
15C4×D716—22no—
16#19153075—22no—
17#19153072C2211yescharacteristic
18C22⋊C4×D179—22no—
19#19153041—22no—
20#19153039—22no—
21#19153029—22no—
22C2×M16×C179C2211yescharacteristic
23#117×C179C2211yescharacteristic
24#117×C179C2211yescharacteristic
25C23×D179D412yes—
26#15668259—21no—
27#15668259—21no—
28#15668259—21no—
29#15668259—21no—
30C2×Dic358C2×C411yescharacteristic, complemented
31C2×D716—24no—
32C2×D716—21no—
33C2×D716—21no—
34C2×D716—21no—
35C2×D716—21no—
36C2×D716C2×C412yes—
37C2×D716C2×C411yescharacteristic, complemented
38C2×D716D412yes—
39C2×C22⋊D179—21no—
40C2×C4×D179—24no—
41C2×C4×D179—21no—
42#15668243—21no—
43#15668243—21no—
44C2×C179⋊C8—21no—
45C4⋊Dic179—21no—
46C22⋊Dic179—21no—
47(C2×D179)⋊C4—24no—
48(C2×D179)⋊C4—24no—
49C4×Dic179D412yescomplemented
50C22×C4×C179C2311yescharacteristic
51M16×C179—21no—
52M16×C179—21no—
53C2×C8×C179—21no—
54C4⋊C4×C179—21no—
55C4⋊C4×C179—21no—
56C22⋊C4×C179—21no—
57C22⋊C4×C179—21no—
58C42×C179C2×C412yescomplemented
59C42×C179D412yescomplemented
60C22×D179—22no—
61C22×D179—22no—
62C22×D179—21no—
63C22×D179—82no—
64C22×D179C22⋊C414yes—
65Dic358—21no—
66Dic358—21no—
67D716—22no—
68D716—22no—
69D716—21no—
70D716—21no—
71D716—22no—
72D716—22no—
73D716—84no—
74C22⋊D179—41no—
75C22⋊D179—41no—
76C2×Dic179—21no—
77C2×Dic179—22no—
78C4×D179—41no—
79C4×D179—41no—
80C4×D179—84no—
81C179⋊C8—41no—
82C23×C179C2×D411yescharacteristic
83C2×C4×C179—22no—
84C2×C4×C179—22no—
85C2×C4×C179C22×C411yescharacteristic
86C2×C4×C179C2×D411yescharacteristic
87C2×C4×C179C2×D411yescharacteristic
88C2×C4×C179C2×D411yescharacteristic
89C2×C4×C179#4111yescharacteristic
90C2×C4×C179#4111yescharacteristic
91C8×C179—41no—
92C22×C179—41no—
93C22×C179C2×C22⋊C411yescharacteristic
94C22×C179#12011yescharacteristic
95C22×C179#12311yescharacteristic
96C2×D179—24no—
97C2×D179—44no—
98C2×D179—44no—
99C2×D179—42no—
100C2×D179—41no—
101C2×D179—82no—
102C4×C179—42no—
103C4×C179—42no—
104C4×C179C4×D411yescharacteristic
105C4×C179C4×D411yescharacteristic
106C4×C179#12111yescharacteristic
107C4×C179#12111yescharacteristic
108Dic179—42no—
109Dic179—41no—
110C2×C179—21no—
111C2×C179—21no—
112C2×C179—21no—
113C2×C179#38611yescharacteristic
114D179—44no—
115D179—81no—
116C179#191111yescharacteristic, complemented, Sylow
117#1911—1791nomaximal, Sylow
118#535—1791no—
119#518—1791no—
120#421—1791no—
121#421—1791no—
122C2×#100—1791no—
123H(3,ℤ/4)—1791no—
124H(3,ℤ/4)D17911yescharacteristic, complemented
125C2×#41—1791no—
126C22×D4—1791no—
127C2×M16—1791no—
128C2×M16C2×D17911yescharacteristic
129#127—1792no—
130#124—1792no—
131#121—3582no—
132#120—3582no—
133C4×D4—3582no—
134C4×D4—3582no—
135#117—1791no—
136#117C2×D17911yescharacteristic
137#117C2×D17911yescharacteristic
138C2×C22⋊C4—3582no—
139#104—3582no—
140#104—3582no—
141#100—3582no—
142C24—1792no—
143#41—3581no—
144#41—3581no—
145#41—3581no—
146#41—3581no—
147C2×Q8—1791no—
148C2×D4—3584no—
149C2×D4—3581no—
150C2×D4—3581no—
151C2×D4—3581no—
152C2×D4—3581no—
153C2×D4—3581no—
154C2×D4—1791no—
155C2×D4—1792no—
156C2×D4—1792no—
157C22×C4—3584no—
158C22×C4—3581no—
159C22×C4C22×D17911yescharacteristic
160M16—21no—
161M16—21no—
162M16—3581no—
163M16—3581no—
164C2×C8—21no—
165C2×C8—3581no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3581no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—3584no—
172C22⋊C4—3584no—
173C22⋊C4—3581no—
174C42—1792no—
175C42C4×D17912yescomplemented
176C42D71612yescomplemented
177C23—3582no—
178C23—3582no—
179C23—3581no—
180C23—14322no—
181C23—1794no—
182C23D4×D17911yescharacteristic
183Q8—3581no—
184Q8—3581no—
185D4—3582no—
186D4—3581no—
187D4—3582no—
188D4—3581no—
189D4—3582no—
190D4—3582no—
191D4—14324no—
192D4—7161no—
193D4—7161no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3581no—
197C2×C4—3582no—
198C2×C4—14324no—
199C2×C4—7161no—
200C2×C4—7161no—
201C2×C4C2×C4×D17911yescharacteristic
202C2×C4D4×D17911yescharacteristic
203C2×C4D4×D17911yescharacteristic
204C2×C4C2×D71611yescharacteristic
205C2×C4#1566825911yescharacteristic
206C2×C4#1566826011yescharacteristic
207C8—41no—
208C8—7161no—
209C22—41no—
210C22—3584no—
211C22—14322no—
212C22—7162no—
213C22—7164no—
214C22—7164no—
215C22—7161no—
216C22C22⋊C4×D17911yescharacteristic
217C22#1915308311yescharacteristic
218C22#1915309811yescharacteristic
219C4—42no—
220C4—42no—
221C4—7162no—
222C4—7161no—
223C4#1915307511yescharacteristic
224C4C4×D71611yescharacteristic
225C4#1915308611yescharacteristic
226C4#1915308811yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—14321no—
231C2—7164no—
232C2#2603831711yescharacteristic
subgroupG/H
centerC2#26038317
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
FittingH(3,ℤ/4)×C179C2
socleC2×C179#386
prime radical#26039008C2
cosocleC22×C4×C179C23
hypercenterH(3,ℤ/4)D179
hypocenterC179#1911
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #26038913C2semidirect
3. #26038457C2semidirect
4. #26038455C2semidirect
5. C2×#19153029C2semidirect
6. #26038195C2semidirect
7. H(3,ℤ/4)×C179C2semidirect
11. #19153110C4semidirect
12. #19153101C4semidirect
30. C2×Dic358C2×C4semidirect
37. C2×D716C2×C4semidirect
49. C4×Dic179D4semidirect
58. C42×C179C2×C4semidirect
59. C42×C179D4semidirect
116. C179#1911semidirect
124. H(3,ℤ/4)D179semidirect
175. C42C4×D179semidirect
176. C42D716semidirect

Maximal subgroups

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

subgroupindex
#260390082
#260389132
#260384572
#260384552
C2×#191530292
#260381952
H(3,ℤ/4)×C1792
#1911179