Finite Group Databasebeta

← #37510064

Subgroups — #37510064

Order 22912.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#26038946C212yesmaximal, complemented
2#26038456C212yesmaximal, complemented
3#26038402C211yesmaximal, characteristic, complemented
4#26038196C211yesmaximal, characteristic, complemented
5#575×C179C211yesmaximal, characteristic, complemented
6C22×C4×D179C2211yescharacteristic
7C2×#15668243C2211yescharacteristic
8#19153094—22no—
9#19153089—22no—
10#19153084—22no—
11#19153081—22no—
12C4×C22⋊D179—24no—
13#19153072C2212yes—
14#19153040—24no—
15(C2×D179)⋊C8—22no—
16C2×#41×C179C2212yes—
17#137×C179—22no—
18AGL(1,ℤ/8)×C179—22no—
19#135×C179—22no—
20C2×M16×C179C2211yescharacteristic
21C23×D179D411yescharacteristic
22C2×C22⋊D179—24no—
23C22×Dic179D411yescharacteristic
24C2×C4×D179—21no—
25C2×C4×D179—22no—
26C2×C4×D179—22no—
27C2×C4×D179—21no—
28#15668243—22no—
29C2×C179⋊C8—21no—
30C4⋊Dic179—22no—
31Dic179⋊C4—24no—
32C22⋊Dic179—24no—
33C22⋊Dic179—22no—
34(C2×D179)⋊C4—24no—
35C4×Dic179D414yescomplemented
36#41×C179—22no—
37#41×C179—22no—
38#41×C179—22no—
39#41×C179—22no—
40C2×Q8×C179D412yescomplemented
41C2×D4×C179—22no—
42C2×D4×C179D412yescomplemented
43C22×C4×C179—22no—
44C22×C4×C179C2311yescharacteristic
45Q16×C179—22no—
46SD16×C179—24no—
47D8×C179—22no—
48M16×C179—22no—
49C2×C8×C179D412yescomplemented
50C22×D179—22no—
51C22×D179—21no—
52C22×D179—42no—
53C22⋊D179—48no—
54C2×Dic179—24no—
55C2×Dic179—22no—
56C2×Dic179—21no—
57C4×D179—41no—
58C4×D179—42no—
59C4×D179—41no—
60C179⋊C8—41no—
61C23×C179—22no—
62C23×C179C2×D411yescharacteristic
63Q8×C179—22no—
64Q8×C179—22no—
65D4×C179—22no—
66D4×C179—22no—
67D4×C179—42no—
68D4×C179—42no—
69C2×C4×C179—22no—
70C2×C4×C179—42no—
71C2×C4×C179—42no—
72C2×C4×C179C2×D412yes—
73C2×C4×C179C2×D412yes—
74C2×C4×C179C2×D411yescharacteristic
75C2×C4×C179#4111yescharacteristic
76C8×C179—22no—
77C22×C179—41no—
78C22×C179—44no—
79C22×C179—42no—
80C22×C179#12012yes—
81C22×C179#12111yescharacteristic
82C2×D179—24no—
83C2×D179—22no—
84C2×D179—42no—
85C2×D179—41no—
86C2×D179—41no—
87C4×C179—42no—
88C4×C179#12011yescharacteristic
89C4×C179#12112yes—
90C4×C179#12711yescharacteristic
91Dic179—44no—
92Dic179—41no—
93C2×C179—22no—
94C2×C179—21no—
95C2×C179—82no—
96C2×C179#39211yescharacteristic
97D179—42no—
98C179#201011yescharacteristic, complemented, Sylow
99#2010—1791nomaximal, Sylow
100#575D17911yescharacteristic, complemented
101#525—1792no—
102#421—1792no—
103#407—1791no—
104H(3,ℤ/4)—1791no—
105C2×#41C2×D17912yes—
106C23×C4—1791no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—1791no—
111C2×M16C2×D17911yescharacteristic
112#123—3582no—
113C22⋊Q8—3582no—
114#121—3582no—
115#120—3582no—
116C4×D4—3584no—
117#117—1792no—
118#104—3584no—
119C22⋊C8—3582no—
120C24—1791no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D17912yescomplemented
126C2×D4—22no—
127C2×D4—3584no—
128C2×D4C22⋊D17912yescomplemented
129C22×C4—22no—
130C22×C4—3582no—
131C22×C4—3581no—
132C22×C4—3582no—
133C22×C4—3581no—
134C22×C4—1791no—
135C22×C4C22×D17911yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—3582no—
141C2×C8—3581no—
142C2×C8C22⋊D17912yescomplemented
143C4⋊C4—3584no—
144C4⋊C4—3582no—
145C22⋊C4—3584no—
146C22⋊C4—3584no—
147C22⋊C4—3582no—
148C42—1794no—
149C23—22no—
150C23—3582no—
151C23—3581no—
152C23—7162no—
153C23D4×D17911yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—7168no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—3584no—
165C2×C4—3582no—
166C2×C4—3581no—
167C2×C4—7161no—
168C2×C4—7162no—
169C2×C4—7161no—
170C2×C4D4×D17912yes—
171C2×C4C2×C22⋊D17912yes—
172C2×C4C2×C22⋊D17911yescharacteristic
173C2×C4#1566825811yescharacteristic
174C8—22no—
175C8—7161no—
176C22—41no—
177C22—44no—
178C22—42no—
179C22—3584no—
180C22—3582no—
181C22—7162no—
182C22—7161no—
183C22—7161no—
184C22#1915308112yes—
185C22#1915308411yescharacteristic
186C4—42no—
187C4—7164no—
188C4—7161no—
189C4#1915308211yescharacteristic
190C4#1915308512yes—
191C4#1915310911yescharacteristic
192C2—22no—
193C2—21no—
194C2—82no—
195C2—7162no—
196C2#2603834711yescharacteristic
subgroupG/H
centerC4#19153109
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#575×C179C2
socleC2×C179#392
hypercenter#575D179
hypocenterC179#2010
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #26038946C2semidirect
2. #26038456C2semidirect
3. #26038402C2semidirect
4. #26038196C2semidirect
5. #575×C179C2semidirect
35. C4×Dic179D4semidirect
40. C2×Q8×C179D4semidirect
42. C2×D4×C179D4semidirect
49. C2×C8×C179D4semidirect
98. C179#2010semidirect
100. #575D179semidirect
125. C2×Q8C22⋊D179semidirect
128. C2×D4C22⋊D179semidirect
142. C2×C8C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260389462
#260384562
#260384022
#260381962
#575×C1792
#2010179