Finite Group Databasebeta

← #37508071

Subgroups — #37508071

Order 22912.

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

normal subgroups only

This group has 236 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#26039149C211yesmaximal, characteristic, complemented
2#26038811C211yesmaximal, characteristic, complemented
3#26038631C211yesmaximal, characteristic
4#26038187C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#460×C179C211yesmaximal, characteristic, complemented
8C2×#15668247—22no—
9C2×#15668244—21no—
10#19153113C2211yescharacteristic
11#19153105—21no—
12#19153091—22no—
13#19153089—21no—
14C4×D716—21no—
15C4×D716C2211yescharacteristic
16C4⋊C4×D179—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18#19153034—21no—
19C4×C179⋊C8C2211yescharacteristic
20C2×SD16×C179—21no—
21#127×C179C2211yescharacteristic
22C4×Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24#102×C179—21no—
25C2×Dic358D412yes—
26C2×D716D411yescharacteristic
27C2×C4×D179—22no—
28C2×C4×D179—21no—
29#15668247—44no—
30#15668244—42no—
31C2×C179⋊C8—21no—
32C2×C179⋊C8—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179—21no—
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—22no—
37Dic179⋊C4—22no—
38(C2×D179)⋊C4—22no—
39(C2×D179)⋊C4—21no—
40C2×Q8×C179D411yescharacteristic
41C2×D4×C179—21no—
42C2×D4×C179—21no—
43C2×D4×C179—21no—
44SD16×C179—42no—
45C2×C8×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D411yescharacteristic
48C42×C179—21no—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51Dic358—42no—
52D716—21no—
53D716SD1612yescomplemented
54C2×Dic179—21no—
55C2×Dic179—22no—
56C4×D179—41no—
57C4×D179—41no—
58C4×D179—82no—
59C179⋊C8—22no—
60C179⋊C8—41no—
61C23×C179—41no—
62Q8×C179—21no—
63Q8×C179—21no—
64D4×C179—42no—
65D4×C179—41no—
66D4×C179—41no—
67D4×C179—41no—
68D4×C179—41no—
69C2×C4×C179—21no—
70C2×C4×C179—21no—
71C2×C4×C179C2×D411yescharacteristic
72C2×C4×C179C2×D411yescharacteristic
73C2×C4×C179C2×D411yescharacteristic
74C8×C179—41no—
75C22×C179—81no—
76C22×C179—81no—
77C22×C179—81no—
78C22×C179#12011yescharacteristic
79C2×D179—22no—
80C2×D179—41no—
81C2×D179—41no—
82C4×C179—21no—
83C4×C179—41no—
84C4×C179—41no—
85C4×C179C2×SD1611yescharacteristic
86C4×C179#13511yescharacteristic
87C4×C179AGL(1,ℤ/8)11yescharacteristic
88C4×C179#13711yescharacteristic
89Dic179—41no—
90Dic179—42no—
91C2×C179—161no—
92C2×C179#44811yescharacteristic
93C2×C179#44911yescharacteristic
94C2×C179#45311yescharacteristic
95D179—42no—
96C179#162211yescharacteristic, complemented, Sylow
97#1622—1791nomaximal, Sylow
98#554—1791no—
99#492—1791no—
100#460D17911yescharacteristic, complemented
101#459—1791no—
102#333—1791no—
103Q8⋊C8—1791no—
104D4⋊C8—1791no—
105C2×SD16—21no—
106C2×SD16—3582no—
107C2×D8—3581no—
108C4⋊Q8—1791no—
109#127C2×D17911yescharacteristic
110#125—3581no—
111C22⋊Q8—3582no—
112C22⋊Q8—3581no—
113C4×Q8C2×D17911yescharacteristic
114C4×D4—3581no—
115C4×D4—1791no—
116C2×C4⋊C4—3581no—
117C4⋊C8—1791no—
118C4⋊C8C2×D17911yescharacteristic
119#102—21no—
120#102—3581no—
121C4×C8—1791no—
122C2×Q8—1792no—
123C2×Q8C22⋊D17911yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—1791no—
128C22×C4—3582no—
129C22×C4—3581no—
130SD16—42no—
131SD16—7164no—
132D8—7162no—
133C2×C8—21no—
134C2×C8—3581no—
135C2×C8—3581no—
136C4⋊C4—21no—
137C4⋊C4—3582no—
138C4⋊C4—3582no—
139C4⋊C4—3581no—
140C4⋊C4—3581no—
141C4⋊C4—1791no—
142C4⋊C4C22⋊D17911yescharacteristic
143C22⋊C4—3582no—
144C22⋊C4—3581no—
145C42—21no—
146C42C22×D17911yescharacteristic
147C23—41no—
148C23—3581no—
149Q8—21no—
150Q8—21no—
151Q8—7162no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—3581no—
158D4—1792no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—3581no—
162C2×C4—3582no—
163C2×C4—14322no—
164C2×C4—7161no—
165C2×C4—7161no—
166C2×C4D4×D17911yescharacteristic
167C2×C4D4×D17911yescharacteristic
168C2×C4C2×C22⋊D17911yescharacteristic
169C8—41no—
170C8—3582no—
171C8—7161no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—3582no—
176C22—7161no—
177C22—7161no—
178C22#1915308111yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—7161no—
183C4—7162no—
184C4SD16×D17911yescharacteristic
185C4#1915313311yescharacteristic
186C4#1915313911yescharacteristic
187C4#1915314211yescharacteristic
188C2—161no—
189C2—7162no—
190C2#2603857611yescharacteristic
191C2#2603858211yescharacteristic
192C2#2603860511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#460×C179C2
socleC22×C179#120
prime radical#26038631C2
hypercenter#460D179
hypocenterC179#1622
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #26039149C2semidirect
2. #26038811C2semidirect
4. #26038187C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
6. #26038166C2semidirect
7. #460×C179C2semidirect
53. D716SD16semidirect
96. C179#1622semidirect
100. #460D179semidirect

Maximal subgroups

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

subgroupindex
#260391492
#260388112
#260386312
#260381872
Q8⋊(C179⋊C8)2
#260381662
#460×C1792
#1622179