Finite Group Databasebeta

← #37510687

Subgroups — #37510687

Order 22912.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#26039013C211yesmaximal, characteristic
2C2×#19153032C211yesmaximal, characteristic, complemented
3#26038237C212yesmaximal, complemented
4#26038235C212yesmaximal, complemented
5C2×#101×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C411yescharacteristic
7C2×#15668243—21no—
8#19153111C411yescharacteristic
9#19153111C2211yescharacteristic
10#19153102C411yescharacteristic
11#19153102C2211yescharacteristic
12#19153096—21no—
13#19153096—21no—
14#19153089—21no—
15#19153089—21no—
16#19153072C411yescharacteristic
17#19153032—21no—
18#19153032C2212yes—
19C22×Q8×C179C2211yescharacteristic
20C2×M16×C179—21no—
21#101×C179—21no—
22#101×C179C2212yes—
23#15668259—41no—
24#15668259—41no—
25C2×Dic358C2×C411yescharacteristic
26C2×D716C2×C411yescharacteristic
27C2×C22⋊D179—21no—
28C2×C4×D179—21no—
29#15668243—22no—
30#15668243—41no—
31C2×C179⋊C8—41no—
32C4⋊Dic179—21no—
33Dic179⋊C4—21no—
34Dic179⋊C4—21no—
35Dic179⋊C4—21no—
36Dic179⋊C4—21no—
37C22⋊Dic179—21no—
38(C2×D179)⋊C4—21no—
39(C2×D179)⋊C4—21no—
40(C2×D179)⋊C4—21no—
41(C2×D179)⋊C4—21no—
42C4×Dic179C2×C411yescharacteristic
43C4×Dic179C2×C411yescharacteristic
44C2×Q8×C179—41no—
45C2×Q8×C179—41no—
46C2×Q8×C179C2×C411yescharacteristic
47C2×Q8×C179D411yescharacteristic
48C2×Q8×C179D411yescharacteristic
49C2×Q8×C179C2311yescharacteristic
50C22×C4×C179C2×C411yescharacteristic
51C22×C4×C179D411yescharacteristic
52C22×C4×C179D411yescharacteristic
53M16×C179—22no—
54M16×C179—41no—
55C2×C8×C179—41no—
56C22×D179—21no—
57Dic358—41no—
58D716—41no—
59C22⋊D179—41no—
60C22⋊D179—41no—
61C2×Dic179—21no—
62C2×Dic179—21no—
63C2×Dic179—21no—
64C4×D179—42no—
65C179⋊C8—42no—
66C23×C179C22⋊C411yescharacteristic
67Q8×C179—41no—
68Q8×C179—41no—
69Q8×C179—41no—
70Q8×C179—41no—
71C2×C4×C179—21no—
72C2×C4×C179—21no—
73C2×C4×C179—21no—
74C2×C4×C179—21no—
75C2×C4×C179—21no—
76C2×C4×C179—21no—
77C2×C4×C179C22⋊C411yescharacteristic
78C2×C4×C179C22⋊C411yescharacteristic
79C2×C4×C179C22⋊C411yescharacteristic
80C2×C4×C179C22×C411yescharacteristic
81C2×C4×C179C2×D411yescharacteristic
82C2×C4×C179C2×D411yescharacteristic
83C8×C179—42no—
84C22×C179—41no—
85C22×C179#9911yescharacteristic
86C22×C179#9911yescharacteristic
87C22×C179C2×C22⋊C411yescharacteristic
88C2×D179—42no—
89C2×D179—41no—
90C4×C179—21no—
91C4×C179—21no—
92C4×C179—21no—
93C4×C179—21no—
94C4×C179—21no—
95C4×C179—21no—
96Dic179—41no—
97Dic179—41no—
98Dic179—41no—
99C2×C179—21no—
100C2×C179—21no—
101C2×C179—21no—
102C2×C179C2×#9911yescharacteristic
103D179—81no—
104C179#213311yescharacteristic, complemented, Sylow
105#2133—1791nomaximal, Sylow
106#536—1791no—
107C2×#101—1791no—
108C2×#101D17911yescharacteristic, complemented
109#356—1792no—
110#355—1792no—
111C2×#41—1791no—
112C22×Q8C2×D17911yescharacteristic
113C2×M16—21no—
114C2×M16—3581no—
115C4⋊Q8—1791no—
116C4⋊Q8—1791no—
117#124—1791no—
118#124—1791no—
119#123—3581no—
120#123—3581no—
121C22⋊Q8—3581no—
122C22⋊Q8—3581no—
123#117—1791no—
124#101—21no—
125#101—3581no—
126#101—1792no—
127#101C2×D17912yes—
128#41—7161no—
129#41—7161no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—1791no—
133C2×Q8C4×D17911yescharacteristic
134C2×Q8C22⋊D17911yescharacteristic
135C2×Q8D71611yescharacteristic
136C2×Q8C22×D17911yescharacteristic
137C2×D4—3581no—
138C2×D4—1791no—
139C22×C4—3581no—
140C22×C4C4×D17911yescharacteristic
141C22×C4C22⋊D17911yescharacteristic
142C22×C4D71611yescharacteristic
143M16—22no—
144M16—41no—
145M16—3582no—
146M16—7161no—
147C2×C8—41no—
148C2×C8—7161no—
149C4⋊C4—3581no—
150C4⋊C4—3581no—
151C4⋊C4—3581no—
152C4⋊C4—3581no—
153C4⋊C4—3581no—
154C22⋊C4—3581no—
155C22⋊C4—3581no—
156C22⋊C4—3581no—
157C22⋊C4—3581no—
158C22⋊C4—3581no—
159C42—1791no—
160C42—1791no—
161C23—3581no—
162C23(C2×D179)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—7161no—
168D4—7161no—
169D4—7161no—
170D4—7161no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—3581no—
178C2×C4—3581no—
179C2×C4—3581no—
180C2×C4—7162no—
181C2×C4(C2×D179)⋊C411yescharacteristic
182C2×C4(C2×D179)⋊C411yescharacteristic
183C2×C4(C2×D179)⋊C411yescharacteristic
184C2×C4C2×C4×D17911yescharacteristic
185C2×C4C2×C22⋊D17911yescharacteristic
186C2×C4C2×D71611yescharacteristic
187C8—42no—
188C8—7162no—
189C22—41no—
190C22—7162no—
191C22—7161no—
192C22#1915302711yescharacteristic
193C22#1915302711yescharacteristic
194C22C2×(C2×D179)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—7161no—
202C4—7161no—
203C4—7161no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—14321no—
208C2C2×#1915302711yescharacteristic
subgroupG/H
centerC2C2×#19153027
commutatorC2×C4×C179C22×C4
FrattiniC2×Q8C22×D179
FittingC2×#101×C179C2
socleC2×C179C2×#99
prime radicalC2×#15668259C4
cosocleC2×Q8×C179C23
hypercenterC2×#101D179
hypocenterC179#2133
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#19153032C2semidirect
3. #26038237C2semidirect
4. #26038235C2semidirect
5. C2×#101×C179C2semidirect
104. C179#2133semidirect
108. C2×#101D179semidirect

Maximal subgroups

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

subgroupindex
#260390132
C2×#191530322
#260382372
#260382352
C2×#101×C1792
#2133179