Finite Group Databasebeta

← #37508152

Subgroups — #37508152

Order 22912.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039092C211yesmaximal, characteristic
2#26038791C211yesmaximal, characteristic, complemented
3#26038730C211yesmaximal, characteristic, complemented
4#26038730C211yesmaximal, characteristic, complemented
5#26038166C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#332×C179C211yesmaximal, characteristic, complemented
8C2×#15668260—21no—
9#19153113C2211yescharacteristic
10#19153098—22no—
11#19153089—21no—
12#19153088—21no—
13C4×D716C2211yescharacteristic
14C4×D716C2211yescharacteristic
15C4⋊C4×D179—21no—
16#19153045—21no—
17#19153045—21no—
18#19153036—22no—
19#19153035—21no—
20#19153035—21no—
21C4×C179⋊C8C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25#15668260—42no—
26#15668260—42no—
27C2×Dic358—21no—
28C2×D716—22no—
29C2×D716D411yescharacteristic
30C2×D716D411yescharacteristic
31C2×C4×D179—22no—
32C2×C4×D179—21no—
33C2×C4×D179—21no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179D411yescharacteristic
38C4⋊Dic179D411yescharacteristic
39Dic179⋊C4—22no—
40(C2×D179)⋊C4—22no—
41(C2×D179)⋊C4—21no—
42(C2×D179)⋊C4—21no—
43C2×Q8×C179—21no—
44C2×C8×C179—21no—
45C2×C8×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D412yes—
48C42×C179C2311yescharacteristic
49C22×D179—21no—
50C22×D179—21no—
51Dic358—42no—
52D716—21no—
53D716—21no—
54D716—21no—
55D716—44no—
56D716SD1612yes—
57C2×Dic179—21no—
58C2×Dic179—21no—
59C2×Dic179—41no—
60C4×D179—44no—
61C4×D179—41no—
62C4×D179—41no—
63C4×D179—41no—
64C4×D179—41no—
65C179⋊C8—41no—
66Q8×C179—41no—
67Q8×C179—41no—
68C2×C4×C179—22no—
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D411yescharacteristic
71C2×C4×C179C2×D411yescharacteristic
72C8×C179—41no—
73C8×C179—41no—
74C22×C179#12011yescharacteristic
75C2×D179—22no—
76C2×D179—41no—
77C2×D179—41no—
78C2×D179—41no—
79C2×D179—41no—
80C2×D179—41no—
81C4×C179—21no—
82C4×C179—42no—
83C4×C179C2×SD1611yescharacteristic
84C4×C179#13511yescharacteristic
85C4×C179AGL(1,ℤ/8)11yescharacteristic
86C4×C179#13711yescharacteristic
87Dic179—41no—
88Dic179—41no—
89Dic179—81no—
90C2×C179#45011yescharacteristic
91C2×C179#45211yescharacteristic
92C2×C179#45411yescharacteristic
93D179—42no—
94D179—81no—
95C179#163511yescharacteristic, complemented, Sylow
96#1635—1791nomaximal, Sylow
97#547—1791no—
98#487—1791no—
99#476—1791no—
100#476—1791no—
101#332D17911yescharacteristic, complemented
102D4⋊C8—1791no—
103D4⋊C8—1791no—
104C2×#41—3581no—
105C4⋊Q8—1791no—
106C4⋊Q8C2×D17911yescharacteristic
107#123—3582no—
108C22⋊Q8—3581no—
109#121—3581no—
110C4×D4—1791no—
111C4×D4—1791no—
112C2×C4⋊C4—3581no—
113#106—3581no—
114#106—3581no—
115C4⋊C8C2×D17911yescharacteristic
116C4⋊C8C2×D17911yescharacteristic
117#103—3582no—
118#102—3581no—
119#102—3581no—
120C4×C8—1791no—
121#41—7162no—
122#41—7162no—
123C2×Q8—21no—
124C2×Q8—3581no—
125C2×D4—3582no—
126C2×D4—1791no—
127C2×D4—1791no—
128C22×C4—3582no—
129C22×C4—3581no—
130C22×C4—3581no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—3581no—
134C4⋊C4—21no—
135C4⋊C4—3582no—
136C4⋊C4—3581no—
137C4⋊C4—3581no—
138C4⋊C4—1791no—
139C4⋊C4—1791no—
140C4⋊C4D71612yes—
141C22⋊C4—3582no—
142C22⋊C4—3581no—
143C22⋊C4—3581no—
144C42C22×D17911yescharacteristic
145C23—3581no—
146C23—3581no—
147Q8—41no—
148Q8—41no—
149Q8—7162no—
150D4—3581no—
151D4—3581no—
152D4—3581no—
153D4—1792no—
154D4—7164no—
155C2×C4—22no—
156C2×C4—3581no—
157C2×C4—3581no—
158C2×C4—7164no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4—7161no—
162C2×C4—7161no—
163C2×C4—7161no—
164C2×C4D4×D17911yescharacteristic
165C2×C4D4×D17911yescharacteristic
166C2×C4C2×D71611yescharacteristic
167C8—41no—
168C8—41no—
169C8—7161no—
170C22—3582no—
171C22—7161no—
172C22—7161no—
173C22—7161no—
174C22—7161no—
175C22—7161no—
176C22#1915308311yescharacteristic
177C4—21no—
178C4—42no—
179C4—14321no—
180C4—7161no—
181C4—7161no—
182C4SD16×D17911yescharacteristic
183C4#1915313411yescharacteristic
184C4#1915313711yescharacteristic
185C4#1915314511yescharacteristic
186C2—14321no—
187C2—7162no—
188C2#2603859111yescharacteristic
189C2#2603860411yescharacteristic
190C2#2603861411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#332×C179C2
socleC22×C179#120
prime radical#26039092C2
hypercenter#332D179
hypocenterC179#1635
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #26038791C2semidirect
3. #26038730C2semidirect
4. #26038730C2semidirect
5. #26038166C2semidirect
6. #26038166C2semidirect
7. #332×C179C2semidirect
95. C179#1635semidirect
101. #332D179semidirect

Maximal subgroups

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

subgroupindex
#260390922
#260387912
#260387302
#260381662
#332×C1792
#1635179