Finite Group Databasebeta

← #37507515

Subgroups — #37507515

Order 22912.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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×#19153087C211yesmaximal, characteristic
2#26038403C211yesmaximal, characteristic, complemented
3#26038172C214yesmaximal, complemented
4C4×C22⋊C4×C179C211yesmaximal, characteristic, complemented
5C22×D716C411yescharacteristic, complemented
6C22×C22⋊D179—21no—
7C2×#15668243—21no—
8#19153087—21no—
9#19153087—21no—
10#19153087C412yescomplemented
11#19153087C2212yes—
12C2×C4⋊Dic179C411yescharacteristic, complemented
13C2×(C2×D179)⋊C4—21no—
14C22⋊(C179⋊C8)—21no—
15C22⋊(C179⋊C8)C2212yescomplemented
16C23×C4×C179C2211yescharacteristic
17C2×C22⋊C4×C179—21no—
18C2×C42×C179—21no—
19#95×C179C2212yescomplemented
20C23×D179—21no—
21C2×D716—21no—
22C2×D716—42no—
23C2×D716C2×C412yes—
24C2×C22⋊D179—22no—
25C2×C22⋊D179—22no—
26C2×C22⋊D179—44no—
27C22×Dic179—21no—
28#15668243—42no—
29C2×C179⋊C8—22no—
30C4⋊Dic179—21no—
31C4⋊Dic179C2×C412yes—
32(C2×D179)⋊C4—22no—
33(C2×D179)⋊C4—22no—
34C24×C179D411yescharacteristic
35C22×C4×C179—22no—
36C22×C4×C179—42no—
37C22×C4×C179C2×C411yescharacteristic
38C22×C4×C179C2×C411yescharacteristic
39C22×C4×C179D411yescharacteristic
40C22×C4×C179D411yescharacteristic
41C22×C4×C179D411yescharacteristic
42C22×C4×C179C2311yescharacteristic
43C22⋊C4×C179—24no—
44C42×C179—42no—
45C22×D179—22no—
46C22×D179—42no—
47C22×D179—42no—
48C22×D179—42no—
49D716—42no—
50D716—42no—
51C22⋊D179—44no—
52C22⋊D179—44no—
53C2×Dic179—22no—
54C2×Dic179—42no—
55C179⋊C8—42no—
56C23×C179—22no—
57C23×C179—22no—
58C23×C179—41no—
59C23×C179C22⋊C411yescharacteristic
60C23×C179C22⋊C411yescharacteristic
61C23×C179C2×D411yescharacteristic
62C2×C4×C179—22no—
63C2×C4×C179—22no—
64C2×C4×C179—22no—
65C2×C4×C179—24no—
66C2×C4×C179—41no—
67C2×C4×C179—41no—
68C2×C4×C179—41no—
69C2×C4×C179—42no—
70C2×C4×C179—42no—
71C2×C4×C179C22⋊C411yescharacteristic
72C2×C4×C179C22⋊C411yescharacteristic
73C2×C4×C179C22×C411yescharacteristic
74C2×C4×C179C2×D411yescharacteristic
75C22×C179—21no—
76C22×C179—21no—
77C22×C179—21no—
78C22×C179—21no—
79C22×C179—21no—
80C22×C179—21no—
81C22×C179—21no—
82C22×C179—44no—
83C22×C179#10412yes—
84C22×C179C2×C22⋊C411yescharacteristic
85C22×C179AGL(1,ℤ/8)12yes—
86C2×D179—42no—
87C2×D179—42no—
88C2×D179—42no—
89C2×D179—84no—
90C4×C179—21no—
91C4×C179—21no—
92C4×C179—21no—
93C4×C179—44no—
94C4×C179#9912yes—
95Dic179—42no—
96C2×C179—22no—
97C2×C179—22no—
98C2×C179—41no—
99C2×C179C2×#9911yescharacteristic
100C2×C179#41811yescharacteristic
101C2×C179C2×#10411yescharacteristic
102D179—82no—
103C179#150711yescharacteristic, complemented, Sylow
104#1507—1791nomaximal, Sylow
105C2×#121—1791no—
106#407—1791no—
107C4×C22⋊C4D17911yescharacteristic, complemented
108#327—1794no—
109C22×D4—3581no—
110C22×D4—1791no—
111C23×C4C2×D17911yescharacteristic
112C2×M16—3581no—
113#121—3581no—
114#121—3581no—
115#121—1792no—
116#121—1792no—
117C2×C4⋊C4—1791no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—3581no—
120C2×C42—21no—
121C22⋊C8—3581no—
122C22⋊C8—1792no—
123#95C2×D17912yescomplemented
124C24—3581no—
125C24C22⋊D17911yescharacteristic
126C2×D4—3582no—
127C2×D4—3582no—
128C2×D4—3581no—
129C2×D4—1792no—
130C2×D4—7162no—
131C2×D4—7164no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—3581no—
135C22×C4C4×D17911yescharacteristic
136C22×C4C4×D17911yescharacteristic
137C22×C4C22⋊D17911yescharacteristic
138C22×C4D71611yescharacteristic
139C22×C4D71611yescharacteristic
140C22×C4C22×D17911yescharacteristic
141M16—7162no—
142C2×C8—3582no—
143C4⋊C4—3581no—
144C4⋊C4—1792no—
145C22⋊C4—24no—
146C22⋊C4—3582no—
147C22⋊C4—3582no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—3582no—
153C23—7162no—
154C23—7162no—
155C23—7162no—
156C23(C2×D179)⋊C411yescharacteristic
157C23(C2×D179)⋊C411yescharacteristic
158C23C2×C22⋊D17911yescharacteristic
159D4—7162no—
160D4—7164no—
161D4—7164no—
162D4—7162no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—3582no—
173C2×C4—7162no—
174C2×C4(C2×D179)⋊C411yescharacteristic
175C2×C4(C2×D179)⋊C411yescharacteristic
176C2×C4C2×C4×D17911yescharacteristic
177C2×C4C2×D71611yescharacteristic
178C8—7162no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—14324no—
188C22—7162no—
189C22—7162no—
190C22—7162no—
191C22#1915304112yes—
192C22C2×(C2×D179)⋊C411yescharacteristic
193C22#1915313812yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—7162no—
199C4#1915302712yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—14322no—
204C2C2×#1915302711yescharacteristic
205C2#2603844311yescharacteristic
206C2C2×#1915304111yescharacteristic
subgroupG/H
centerC22C2×(C2×D179)⋊C4
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
FittingC4×C22⋊C4×C179C2
socleC22×C179C2×C22⋊C4
prime radicalC2×#19153087C2
cosocleC22×C4×C179C23
hypercenterC4×C22⋊C4D179
hypocenterC179#1507
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #26038403C2semidirect
3. #26038172C2semidirect
4. C4×C22⋊C4×C179C2semidirect
5. C22×D716C4semidirect
10. #19153087C4semidirect
12. C2×C4⋊Dic179C4semidirect
15. C22⋊(C179⋊C8)C22semidirect
19. #95×C179C22semidirect
103. C179#1507semidirect
107. C4×C22⋊C4D179semidirect
123. #95C2×D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191530872
#260384032
#260381722
C4×C22⋊C4×C1792
#1507179