Finite Group Databasebeta

← #37507337

Subgroups — #37507337

Order 22912.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×#19153075C211yesmaximal, characteristic
2#26038401C212yesmaximal, complemented
3#26038165C211yesmaximal, characteristic, complemented
4#26038162C212yesmaximal, complemented
5#324×C179C211yesmaximal, characteristic, complemented
6C22×D716C411yescharacteristic
7C22×C4×D179C411yescharacteristic
8C22×C4×D179C2211yescharacteristic
9C2×#15668242—22no—
10#19153075—22no—
11#19153075—22no—
12C2×(C2×D179)⋊C4C411yescharacteristic
13C2×(C2×D179)⋊C4C2211yescharacteristic
14C2×C4×Dic179C411yescharacteristic
15C22⋊(C179⋊C8)C2212yes—
16(C2×D179)⋊C8—22no—
17C2×C4⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19C23×D179C2×C411yescharacteristic
20C23×D179C2×C411yescharacteristic
21C2×D716—22no—
22C2×D716—22no—
23C2×D716—22no—
24C22×Dic179C2×C411yescharacteristic
25C22×Dic179C2×C411yescharacteristic
26C2×C4×D179—21no—
27C2×C4×D179—21no—
28C2×C4×D179—42no—
29C2×C4×D179—42no—
30C2×C4×D179D412yes—
31C2×C4×D179D412yes—
32#15668242—44no—
33C2×C179⋊C8—22no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—22no—
36C4×Dic179—21no—
37C4×Dic179—21no—
38C22×C4×C179C2×C411yescharacteristic
39C22×C4×C179C2×C411yescharacteristic
40C22×C4×C179C2311yescharacteristic
41C2×C8×C179—22no—
42C4⋊C4×C179—22no—
43C22×D179—21no—
44C22×D179—21no—
45C22×D179—21no—
46C22×D179—21no—
47C22×D179—41no—
48C22×D179—41no—
49C22×D179—41no—
50C22×D179—41no—
51C22×D179C22⋊C412yes—
52C22×D179M1612yescomplemented
53D716—28no—
54C2×Dic179—22no—
55C2×Dic179—21no—
56C2×Dic179—41no—
57C2×Dic179C22⋊C411yescharacteristic
58C2×Dic179C22⋊C411yescharacteristic
59C4×D179—24no—
60C4×D179—42no—
61C4×D179—42no—
62C4×D179—42no—
63C179⋊C8—42no—
64C23×C179C22×C411yescharacteristic
65C2×C4×C179—21no—
66C2×C4×C179—21no—
67C2×C4×C179—21no—
68C2×C4×C179—41no—
69C2×C4×C179—41no—
70C2×C4×C179M1612yescomplemented
71C2×C4×C179C2×D412yes—
72C8×C179—42no—
73C22×C179—21no—
74C22×C179—21no—
75C22×C179C2×C22⋊C411yescharacteristic
76C22×C179C2×M1611yescharacteristic
77C22×C179C2×M1611yescharacteristic
78C2×D179—22no—
79C2×D179—22no—
80C2×D179—22no—
81C2×D179—22no—
82C2×D179—42no—
83C2×D179—42no—
84C2×D179—42no—
85C2×D179—41no—
86C2×D179—41no—
87C2×D179—42no—
88C4×C179—22no—
89C4×C179—22no—
90C4×C179—41no—
91Dic179—21no—
92Dic179—41no—
93Dic179#9912yes—
94C2×C179—21no—
95C2×C179—21no—
96C2×C179#40711yescharacteristic
97C2×C179C2×#9911yescharacteristic
98C2×C179#41311yescharacteristic
99D179—42no—
100D179—42no—
101C179#146511yescharacteristic, complemented, Sylow
102#1465—1791nomaximal, Sylow
103C2×C4×D4—1791no—
104#407—1792no—
105#324—1791no—
106#324D17911yescharacteristic, complemented
107#323—1792no—
108C22×D4—1791no—
109C23×C4—1791no—
110C23×C4—1791no—
111C2×M16—3582no—
112C4×D4—3582no—
113C4×D4—3582no—
114C2×C4⋊C4C2×D17911yescharacteristic
115C2×C22⋊C4—1791no—
116C2×C22⋊C4—1791no—
117C2×C42—1791no—
118C22⋊C8—3582no—
119C22⋊C8—1792no—
120C22⋊C8C2×D17912yes—
121C24—1791no—
122C24—1791no—
123C2×D4—3582no—
124C2×D4—3582no—
125C2×D4—3582no—
126C22×C4—3581no—
127C22×C4—3581no—
128C22×C4—1791no—
129C22×C4—1792no—
130C22×C4—1791no—
131C22×C4—1792no—
132C22×C4—7162no—
133C22×C4—7162no—
134C22×C4C4×D17911yescharacteristic
135C22×C4C4×D17911yescharacteristic
136C22×C4C22×D17911yescharacteristic
137M16—7164no—
138C2×C8—22no—
139C2×C8—3582no—
140C4⋊C4—22no—
141C22⋊C4—3582no—
142C22⋊C4—3582no—
143C42—3581no—
144C42—3581no—
145C23—3581no—
146C23—3581no—
147C23—3581no—
148C23—3581no—
149C23—1792no—
150C23—1792no—
151C23—7161no—
152C23—7161no—
153C23—7161no—
154C23—7161no—
155C23C2×C4×D17911yescharacteristic
156D4—3588no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3584no—
163C2×C4—3582no—
164C2×C4—3581no—
165C2×C4—1791no—
166C2×C4—1791no—
167C2×C4—7162no—
168C2×C4—7162no—
169C2×C4—7162no—
170C2×C4—7161no—
171C2×C4#1566824212yescomplemented
172C2×C4D4×D17912yes—
173C8—42no—
174C8—7162no—
175C22—21no—
176C22—21no—
177C22—3582no—
178C22—3582no—
179C22—3582no—
180C22—3582no—
181C22—7162no—
182C22—7162no—
183C22—7162no—
184C22—7162no—
185C22—7161no—
186C22—7161no—
187C22C22⋊C4×D17911yescharacteristic
188C22M16×D17911yescharacteristic
189C22C2×#1566824211yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—3581no—
194C4—1792no—
195C4—7161no—
196C2—21no—
197C2—21no—
198C2—7162no—
199C2—7162no—
200C2#2603840111yescharacteristic
201C2#99×D17911yescharacteristic
202C2#2603842211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D179
commutatorC23×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179C2×C22⋊C4
prime radicalC22×D716C4
cosocleC22×C4×C179C23
hypercenter#324D179
hypocenterC179#1465
LCSG ▷ C23×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26038401C2semidirect
3. #26038165C2semidirect
4. #26038162C2semidirect
5. #324×C179C2semidirect
52. C22×D179M16semidirect
70. C2×C4×C179M16semidirect
101. C179#1465semidirect
106. #324D179semidirect
171. C2×C4#15668242semidirect

Maximal subgroups

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

subgroupindex
C2×#191530752
#260384012
#260381652
#260381622
#324×C1792
#1465179