Finite Group Databasebeta

← #37507955

Subgroups — #37507955

Order 22912.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#26039064C211yesmaximal, characteristic
2#26038595C211yesmaximal, characteristic, complemented
3#26038350C211yesmaximal, characteristic
4#26038177C211yesmaximal, characteristic, complemented
5#26038176C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic
7#451×C179C211yesmaximal, characteristic, complemented
8C2×#15668249—21no—
9#19153093—21no—
10#19153092C2211yescharacteristic
11#19153091—21no—
12#19153091—21no—
13#19153090—21no—
14#19153089—21no—
15C2×(C2×D179)⋊C4—21no—
16C2×(C2×D179)⋊C4C2211yescharacteristic
17C22⋊C4×D179—21no—
18#19153037—21no—
19C22⋊(C179⋊C8)C2211yescharacteristic
20C179⋊#95—21no—
21C179⋊#95C2211yescharacteristic
22C22×Q8×C179C2211yescharacteristic
23C2×Q16×C179—21no—
24C22⋊Q8×C179C2211yescharacteristic
25#103×C179—21no—
26C22⋊C8×C179C2211yescharacteristic
27C23×D179D411yescharacteristic
28C2×Dic358—21no—
29C2×Dic358D411yescharacteristic
30C22×Dic179—21no—
31C22×Dic179D411yescharacteristic
32C2×C4×D179—21no—
33C2×C4×D179—21no—
34#15668249—42no—
35C2×C179⋊C8—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179D411yescharacteristic
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—21no—
41C22⋊Dic179—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—42no—
47C2×Q8×C179—21no—
48C2×Q8×C179—21no—
49C2×Q8×C179—42no—
50C2×Q8×C179D411yescharacteristic
51C22×C4×C179—21no—
52C22×C4×C179C2311yescharacteristic
53Q16×C179—42no—
54C2×C8×C179—21no—
55C4⋊C4×C179—21no—
56C4⋊C4×C179D411yescharacteristic
57C22⋊C4×C179—21no—
58C22×D179—21no—
59C22×D179—21no—
60C22×D179—41no—
61C22×D179—41no—
62C22×D179Q1612yescomplemented
63Dic358—41no—
64Dic358—81no—
65C2×Dic179—21no—
66C2×Dic179—21no—
67C2×Dic179—21no—
68C2×Dic179—41no—
69C2×Dic179—41no—
70C2×Dic179—41no—
71C2×Dic179—41no—
72C4×D179—81no—
73C4×D179—81no—
74C179⋊C8—41no—
75C23×C179C2×D411yescharacteristic
76Q8×C179—22no—
77Q8×C179—22no—
78Q8×C179—41no—
79Q8×C179—41no—
80Q8×C179—41no—
81C2×C4×C179—22no—
82C2×C4×C179—21no—
83C2×C4×C179—21no—
84C2×C4×C179—41no—
85C2×C4×C179—41no—
86C2×C4×C179—41no—
87C2×C4×C179C2×D411yescharacteristic
88C2×C4×C179C2×D411yescharacteristic
89C8×C179—41no—
90C22×C179—21no—
91C22×C179—21no—
92C22×C179#12011yescharacteristic
93C22×C179C2×Q1611yescharacteristic
94C22×C179#13711yescharacteristic
95C2×D179—22no—
96C2×D179—41no—
97C2×D179—41no—
98C2×D179—42no—
99C2×D179—81no—
100C4×C179—21no—
101C4×C179—21no—
102C4×C179—41no—
103C4×C179—42no—
104C4×C179—41no—
105Dic179—41no—
106Dic179—41no—
107Dic179—41no—
108Dic179—81no—
109C2×C179—21no—
110C2×C179—21no—
111C2×C179#45111yescharacteristic
112C2×C179#45511yescharacteristic
113C2×C179B(4,𝔽2)11yescharacteristic
114D179—42no—
115C179#160211yescharacteristic, complemented, Sylow
116#1602—1791nomaximal, Sylow
117#543—1791no—
118#451—1791no—
119#451D17911yescharacteristic, complemented
120#393—1791no—
121#328—1791no—
122#328—1791no—
123#323—1791no—
124C22×Q8C2×D17911yescharacteristic
125C2×Q16—21no—
126C2×Q16—3581no—
127C22⋊Q8—3581no—
128C22⋊Q8—3581no—
129C22⋊Q8—3581no—
130C22⋊Q8—3581no—
131C22⋊Q8—3581no—
132C22⋊Q8—1791no—
133C22⋊Q8C2×D17911yescharacteristic
134C2×C22⋊C4—3581no—
135C2×C22⋊C4—3581no—
136C2×C22⋊C4—1791no—
137#103—21no—
138#103—3581no—
139C22⋊C8—1791no—
140C22⋊C8C2×D17911yescharacteristic
141#95—3581no—
142#95—1791no—
143C24—1791no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3581no—
148C2×Q8—1791no—
149C2×Q8C22⋊D17911yescharacteristic
150C22×C4—21no—
151C22×C4—3581no—
152C22×C4—3581no—
153C22×C4—3581no—
154C22×C4—1791no—
155C22×C4C22×D17911yescharacteristic
156Q16—42no—
157Q16—7162no—
158C2×C8—21no—
159C2×C8—3581no—
160C4⋊C4—21no—
161C4⋊C4—3581no—
162C4⋊C4—3581no—
163C4⋊C4—3581no—
164C4⋊C4—3581no—
165C4⋊C4—1791no—
166C4⋊C4C22⋊D17911yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3581no—
169C22⋊C4—3581no—
170C22⋊C4—3581no—
171C22⋊C4—3581no—
172C22⋊C4—3581no—
173C22⋊C4—7162no—
174C23—3581no—
175C23—3581no—
176C23—1792no—
177C23—7161no—
178C23—7161no—
179C23D4×D17911yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—14321no—
186Q8—7161no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3581no—
194C2×C4—3581no—
195C2×C4—3581no—
196C2×C4—14321no—
197C2×C4—14321no—
198C2×C4—7161no—
199C2×C4—7161no—
200C2×C4—7161no—
201C2×C4—7161no—
202C2×C4D4×D17911yescharacteristic
203C2×C4C2×C22⋊D17911yescharacteristic
204C8—41no—
205C8—7161no—
206C22—21no—
207C22—21no—
208C22—3582no—
209C22—14321no—
210C22—7162no—
211C22—7161no—
212C22—7161no—
213C22#1915308111yescharacteristic
214C22Q16×D17911yescharacteristic
215C22#1915314511yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—14321no—
222C4—7161no—
223C4—7161no—
224C4—7161no—
225C2—21no—
226C2—21no—
227C2—7162no—
228C2#2603859311yescharacteristic
229C2#2603861511yescharacteristic
230C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#451×C179C2
socleC22×C179#120
prime radicalC23×D179D4
hypercenter#451D179
hypocenterC179#1602
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #26038595C2semidirect
4. #26038177C2semidirect
5. #26038176C2semidirect
7. #451×C179C2semidirect
62. C22×D179Q16semidirect
115. C179#1602semidirect
119. #451D179semidirect

Maximal subgroups

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

subgroupindex
#260390642
#260385952
#260383502
#260381772
#260381762
#260381622
#451×C1792
#1602179