Finite Group Databasebeta

← #37510127

Subgroups — #37510127

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
1#26039001C211yesmaximal, characteristic, complemented
2C2×#19153083C212yesmaximal, complemented
3C2×#19153027C212yesmaximal, complemented
4#26038294C211yesmaximal, characteristic, complemented
5#342×C179C211yesmaximal, characteristic, complemented
6C24×D179C2211yescharacteristic
7C22×D716—22no—
8C22×C22⋊D179C2212yes—
9#19153097—22no—
10#19153087—22no—
11#19153083—24no—
12#19153083—24no—
13#19153083—22no—
14C2×C22⋊Dic179C2211yescharacteristic
15C2×(C2×D179)⋊C4—22no—
16C2×(C2×D179)⋊C4—22no—
17#19153027—24no—
18C2×C22⋊C4×C179C2212yes—
19C2×C22⋊C4×C179C2211yescharacteristic
20C23×D179—21no—
21C23×D179—22no—
22C23×D179—22no—
23C23×D179—21no—
24C23×D179—44no—
25C23×D179D412yescomplemented
26C23×D179D412yescomplemented
27C2×D716—28no—
28C2×D716—22no—
29C2×D716—48no—
30C2×C22⋊D179—22no—
31C2×C22⋊D179—22no—
32C2×C22⋊D179—22no—
33C2×C22⋊D179—22no—
34C2×C22⋊D179—22no—
35C2×C22⋊D179—22no—
36C22×Dic179—21no—
37C22×Dic179D412yescomplemented
38C4⋊Dic179—22no—
39C22⋊Dic179—22no—
40(C2×D179)⋊C4—28no—
41(C2×D179)⋊C4—24no—
42(C2×D179)⋊C4—44no—
43C24×C179C2311yescharacteristic
44C22×C4×C179—22no—
45C22×C4×C179D412yescomplemented
46C22⋊C4×C179—22no—
47C22⋊C4×C179—22no—
48C22⋊C4×C179D414yescomplemented
49C22×D179—24no—
50C22×D179—28no—
51C22×D179—24no—
52C22×D179—22no—
53C22×D179—22no—
54C22×D179—22no—
55C22×D179—22no—
56C22×D179—44no—
57C22×D179—48no—
58C22×D179—44no—
59C22×D179—42no—
60C22×D179—44no—
61C22×D179—41no—
62C22×D179—42no—
63C22×D179—41no—
64C22×D179—44no—
65D716—416no—
66D716—82no—
67C22⋊D179—42no—
68C22⋊D179—42no—
69C22⋊D179—42no—
70C22⋊D179—42no—
71C2×Dic179—22no—
72C2×Dic179—22no—
73C2×Dic179—22no—
74C2×Dic179—42no—
75C2×Dic179—41no—
76C23×C179—42no—
77C23×C179C2×D412yes—
78C23×C179C2×D411yescharacteristic
79C23×C179C2×D411yescharacteristic
80C23×C179C2×D412yes—
81C23×C179#4111yescharacteristic
82C2×C4×C179—24no—
83C2×C4×C179—22no—
84C2×C4×C179—44no—
85C2×C4×C179—42no—
86C22×C179—22no—
87C22×C179—24no—
88C22×C179—41no—
89C22×C179—42no—
90C22×C179—41no—
91C22×C179#12012yes—
92C22×C179#12011yescharacteristic
93C22×C179#12112yes—
94C22×C179#12111yescharacteristic
95C22×C179#12711yescharacteristic
96C2×D179—28no—
97C2×D179—24no—
98C2×D179—44no—
99C2×D179—44no—
100C2×D179—42no—
101C2×D179—48no—
102C2×D179—42no—
103C2×D179—44no—
104C2×D179—44no—
105C2×D179—42no—
106C2×D179—44no—
107C2×D179—82no—
108C4×C179—44no—
109C4×C179—42no—
110Dic179—42no—
111Dic179—81no—
112C2×C179—21no—
113C2×C179—22no—
114C2×C179—21no—
115C2×C179—22no—
116C2×C179#39211yescharacteristic
117C2×C179B(4,𝔽2)12yes—
118D179—44no—
119D179—82no—
120C179#202111yescharacteristic, complemented, Sylow
121#2021—1791nomaximal, Sylow
122#534—1791no—
123C2×#120—1792no—
124C2×#99—1792no—
125#379—1791no—
126#342D17911yescharacteristic, complemented
127C25—1791no—
128C22×D4—3582no—
129C22×D4—1792no—
130#123—3582no—
131#121—3582no—
132#120—3584no—
133#120—3582no—
134#120—3584no—
135C2×C22⋊C4—3582no—
136C2×C22⋊C4—3582no—
137C2×C22⋊C4—1791no—
138C2×C22⋊C4C2×D17911yescharacteristic
139C2×C22⋊C4C2×D17912yes—
140#99—3584no—
141C24—3582no—
142C24—3581no—
143C24—3582no—
144C24—3581no—
145C24—1792no—
146C24—1792no—
147C24—7164no—
148C24C22×D17911yescharacteristic
149C2×D4—3582no—
150C2×D4—3588no—
151C2×D4—3582no—
152C2×D4—3582no—
153C2×D4—3582no—
154C2×D4—3582no—
155C2×D4—3582no—
156C2×D4—3582no—
157C2×D4—7168no—
158C22×C4—22no—
159C22×C4—3581no—
160C22×C4—1792no—
161C22×C4D71612yescomplemented
162C4⋊C4—3582no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3588no—
166C22⋊C4—3584no—
167C22⋊C4—3582no—
168C22⋊C4—7164no—
169C22⋊C4D71614yescomplemented
170C23—42no—
171C23—3584no—
172C23—3588no—
173C23—3584no—
174C23—3582no—
175C23—3582no—
176C23—3582no—
177C23—3582no—
178C23—7164no—
179C23—7162no—
180C23—7164no—
181C23—7164no—
182C23—7168no—
183C23—7164no—
184C23—7161no—
185C23—7162no—
186C23—7161no—
187C23D4×D17912yes—
188C23D4×D17911yescharacteristic
189C23C2×D71611yescharacteristic
190C23C2×D71612yes—
191C23#1566826011yescharacteristic
192D4—14322no—
193D4—71616no—
194D4—7162no—
195D4—7162no—
196D4—7162no—
197D4—7162no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—3582no—
203C2×C4—3582no—
204C2×C4—3582no—
205C2×C4—7162no—
206C2×C4—7161no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3588no—
213C22—3584no—
214C22—14322no—
215C22—7164no—
216C22—7164no—
217C22—7164no—
218C22—7162no—
219C22—7168no—
220C22—7162no—
221C22—7164no—
222C22—7162no—
223C22—7164no—
224C22#1915308312yes—
225C22#1915308311yescharacteristic
226C22#1915308812yes—
227C22#1915308811yescharacteristic
228C22#1915311011yescharacteristic
229C4—44no—
230C4—42no—
231C4—14321no—
232C4—7162no—
233C2—22no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—14322no—
238C2—7164no—
239C2#2603834911yescharacteristic
240C2#2603862712yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC24×C179C23
FrattiniC24C22×D179
Fitting#342×C179C2
socleC22×C179#120
hypercenter#342D179
hypocenterC179#2021
LCSG ▷ C24×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26039001C2semidirect
2. C2×#19153083C2semidirect
3. C2×#19153027C2semidirect
4. #26038294C2semidirect
5. #342×C179C2semidirect
25. C23×D179D4semidirect
26. C23×D179D4semidirect
37. C22×Dic179D4semidirect
45. C22×C4×C179D4semidirect
48. C22⋊C4×C179D4semidirect
120. C179#2021semidirect
126. #342D179semidirect
161. C22×C4D716semidirect
169. C22⋊C4D716semidirect

Maximal subgroups

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

subgroupindex
#260390012
C2×#191530832
C2×#191530272
#260382942
#342×C1792
#2021179