Finite Group Databasebeta

← #37510124

Subgroups — #37510124

Order 22912.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19153081C212yesmaximal, complemented
2C2×#19153028C212yesmaximal, complemented
3#26038294C211yesmaximal, characteristic, complemented
4#26038206C211yesmaximal, characteristic, complemented
5#534×C179C211yesmaximal, characteristic, complemented
6C24×D179C2211yescharacteristic
7C22×C22⋊D179—22no—
8#19153081—24no—
9#19153081—24no—
10C2×C22⋊Dic179C2211yescharacteristic
11C2×C22⋊Dic179C2212yes—
12C2×(C2×D179)⋊C4—22no—
13C2×(C2×D179)⋊C4—22no—
14#19153028—24no—
15C22×D4×C179C2212yes—
16#123×C179—22no—
17#121×C179—22no—
18#120×C179—22no—
19C2×C22⋊C4×C179C2211yescharacteristic
20C23×D179—21no—
21C23×D179—22no—
22C23×D179—22no—
23C23×D179—21no—
24C23×D179—44no—
25C23×D179D412yescomplemented
26C2×C22⋊D179—28no—
27C2×C22⋊D179—48no—
28C22×Dic179—22no—
29C22×Dic179—21no—
30C22⋊Dic179—22no—
31C22⋊Dic179—22no—
32C22⋊Dic179D414yescomplemented
33(C2×D179)⋊C4—28no—
34(C2×D179)⋊C4—44no—
35C24×C179D412yescomplemented
36C24×C179C2311yescharacteristic
37C2×D4×C179—22no—
38C2×D4×C179—22no—
39C2×D4×C179—22no—
40C2×D4×C179—22no—
41C2×D4×C179—22no—
42C2×D4×C179—22no—
43C2×D4×C179—22no—
44C22×C4×C179D412yescomplemented
45C22×C4×C179D412yescomplemented
46C4⋊C4×C179—22no—
47C22⋊C4×C179—24no—
48C22⋊C4×C179—22no—
49C22×D179—24no—
50C22×D179—28no—
51C22×D179—24no—
52C22×D179—22no—
53C22×D179—44no—
54C22×D179—48no—
55C22×D179—44no—
56C22×D179—42no—
57C22×D179—44no—
58C22×D179—41no—
59C22×D179—42no—
60C22×D179—41no—
61C22⋊D179—416no—
62C2×Dic179—24no—
63C2×Dic179—44no—
64C2×Dic179—42no—
65C2×Dic179—41no—
66C23×C179—22no—
67C23×C179—22no—
68C23×C179—22no—
69C23×C179—42no—
70C23×C179—44no—
71C23×C179C2×D411yescharacteristic
72C23×C179C2×D412yes—
73C23×C179C2×D412yes—
74C23×C179C2×D411yescharacteristic
75C23×C179#4111yescharacteristic
76D4×C179—42no—
77D4×C179—42no—
78D4×C179—42no—
79D4×C179—42no—
80D4×C179—82no—
81C2×C4×C179—22no—
82C2×C4×C179—22no—
83C2×C4×C179—22no—
84C2×C4×C179—22no—
85C2×C4×C179—42no—
86C22×C179—22no—
87C22×C179—24no—
88C22×C179—41no—
89C22×C179—42no—
90C22×C179—41no—
91C22×C179—44no—
92C22×C179—42no—
93C22×C179—44no—
94C22×C179—82no—
95C22×C179#12012yes—
96C22×C179#12011yescharacteristic
97C22×C179#12111yescharacteristic
98C22×C179#12112yes—
99C22×C179#12711yescharacteristic
100C2×D179—28no—
101C2×D179—24no—
102C2×D179—44no—
103C2×D179—44no—
104C2×D179—42no—
105C2×D179—48no—
106C2×D179—42no—
107C2×D179—44no—
108C4×C179—42no—
109C4×C179—42no—
110Dic179—44no—
111Dic179—81no—
112C2×C179—22no—
113C2×C179—21no—
114C2×C179—21no—
115C2×C179—22no—
116C2×C179—82no—
117C2×C179#39211yescharacteristic
118C2×C179B(4,𝔽2)12yes—
119D179—44no—
120C179#202111yescharacteristic, complemented, Sylow
121#2021—1791nomaximal, Sylow
122#534D17911yescharacteristic, complemented
123C2×#120—1792no—
124C2×#99—1792no—
125#379—1791no—
126#342—1791no—
127C25—1791no—
128C22×D4—3582no—
129C22×D4C2×D17912yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—3584no—
134#120—3584no—
135C2×C22⋊C4—3582no—
136C2×C22⋊C4—3582no—
137C2×C22⋊C4—1791no—
138C2×C22⋊C4—1792no—
139C2×C22⋊C4C2×D17911yescharacteristic
140#99—3584no—
141C24—3581no—
142C24—3582no—
143C24—3582no—
144C24—3581no—
145C24—1792no—
146C24—7164no—
147C24C22⋊D17912yescomplemented
148C24C22×D17911yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—3588no—
157C2×D4—7168no—
158C22×C4—3582no—
159C22×C4—3581no—
160C22×C4C22⋊D17912yescomplemented
161C22×C4C22⋊D17912yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—3588no—
166C22⋊C4—3582no—
167C22⋊C4—3582no—
168C22⋊C4—1794no—
169C22⋊C4—7164no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—3584no—
176C23—3588no—
177C23—3584no—
178C23—3582no—
179C23—7162no—
180C23—7164no—
181C23—7164no—
182C23—7168no—
183C23—7164no—
184C23—7161no—
185C23—7162no—
186C23—7161no—
187C23D4×D17911yescharacteristic
188C23D4×D17912yes—
189C23C2×C22⋊D17912yes—
190C23C2×C22⋊D17911yescharacteristic
191C23#1566825811yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—71616no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3584no—
204C2×C4—7162no—
205C2×C4—7161no—
206C2×C4—7164no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—3588no—
217C22—3584no—
218C22—7164no—
219C22—7164no—
220C22—7168no—
221C22—7162no—
222C22—7162no—
223C22—7164no—
224C22#1915308112yes—
225C22#1915308211yescharacteristic
226C22#1915308411yescharacteristic
227C22#1915308512yes—
228C22#1915310911yescharacteristic
229C4—42no—
230C4—42no—
231C4—14321no—
232C4—7164no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—7164no—
239C2#2603834711yescharacteristic
240C2#2603862612yes—
subgroupG/H
centerC22#19153082
commutator, cosocleC24×C179C23
FrattiniC24C22×D179
Fitting#534×C179C2
socleC22×C179#120
hypercenter#534D179
hypocenterC179#2021
LCSG ▷ C24×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#19153081C2semidirect
2. C2×#19153028C2semidirect
3. #26038294C2semidirect
4. #26038206C2semidirect
5. #534×C179C2semidirect
25. C23×D179D4semidirect
32. C22⋊Dic179D4semidirect
35. C24×C179D4semidirect
44. C22×C4×C179D4semidirect
45. C22×C4×C179D4semidirect
120. C179#2021semidirect
122. #534D179semidirect
147. C24C22⋊D179semidirect
160. C22×C4C22⋊D179semidirect
161. C22×C4C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191530812
C2×#191530282
#260382942
#260382062
#534×C1792
#2021179