Finite Group Databasebeta

← #37508246

Subgroups — #37508246

Order 22912.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#26039104C211yesmaximal, characteristic
2#26038716C211yesmaximal, characteristic
3#26038646C211yesmaximal, characteristic, complemented
4#26038185C211yesmaximal, characteristic, complemented
5#26038171C211yesmaximal, characteristic, complemented
6#26038168C211yesmaximal, characteristic
7#500×C179C211yesmaximal, characteristic, complemented
8C2×Q8×D179—21no—
9C2×#15668249—21no—
10#19153112—21no—
11#19153091—22no—
12#19153089—21no—
13C4×Dic358C2211yescharacteristic
14C4×D716C2211yescharacteristic
15#19153075—21no—
16#19153046—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153037—21no—
20#19153033—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23#107×C179—22no—
24C4×C8×C179C2211yescharacteristic
25C2×Dic358—22no—
26C2×Dic358D411yescharacteristic
27Q8×D179—42no—
28Q8×D179—42no—
29C2×D716D411yescharacteristic
30C2×C4×D179—22no—
31C2×C4×D179—21no—
32#15668249—42no—
33C2×C179⋊C8—21no—
34C2×C179⋊C8—21no—
35C4⋊Dic179D411yescharacteristic
36C4⋊Dic179D411yescharacteristic
37Dic179⋊C4—22no—
38Dic179⋊C4—21no—
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41C4×Dic179—21no—
42C2×Q8×C179—21no—
43C2×Q8×C179—21no—
44C2×C8×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179—21no—
48C4⋊C4×C179D412yes—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51Dic358—21no—
52Dic358—44no—
53Dic358D812yes—
54D716—21no—
55D716Q1612yescomplemented
56C2×Dic179—21no—
57C2×Dic179—21no—
58C2×Dic179—21no—
59C4×D179—44no—
60C4×D179—41no—
61C4×D179—41no—
62C179⋊C8—41no—
63C179⋊C8—41no—
64Q8×C179—41no—
65Q8×C179—41no—
66Q8×C179—41no—
67Q8×C179—41no—
68C2×C4×C179—22no—
69C2×C4×C179—41no—
70C2×C4×C179C2×D411yescharacteristic
71C2×C4×C179C2×D411yescharacteristic
72C2×C4×C179C2×D411yescharacteristic
73C8×C179—22no—
74C22×C179#12011yescharacteristic
75C2×D179—22no—
76C2×D179—41no—
77C2×D179—41no—
78C4×C179—21no—
79C4×C179—42no—
80C4×C179—81no—
81C4×C179C2×D811yescharacteristic
82C4×C179C2×Q1611yescharacteristic
83C4×C179AGL(1,ℤ/8)11yescharacteristic
84C4×C179#13711yescharacteristic
85Dic179—22no—
86Dic179—41no—
87Dic179—41no—
88C2×C179#44711yescharacteristic
89C2×C179#45111yescharacteristic
90C2×C179#45611yescharacteristic
91D179—42no—
92C179#164911yescharacteristic, complemented, Sylow
93#1649—1791nomaximal, Sylow
94D4×Q8—1791no—
95#500D17911yescharacteristic, complemented
96#474—1791no—
97#462—1791no—
98#332—1791no—
99Q8⋊C8—1791no—
100D4⋊C8—1791no—
101C22×Q8—3581no—
102C2×Q16—3581no—
103C4⋊Q8—3581no—
104C4⋊Q8C2×D17911yescharacteristic
105C4⋊Q8C2×D17911yescharacteristic
106C22⋊Q8—3581no—
107C22⋊Q8—3582no—
108C4×Q8—1791no—
109C4×D4—3581no—
110C4×D4—1791no—
111#107—22no—
112#107—3581no—
113C4⋊C8—1791no—
114C4⋊C8—1791no—
115#103—3581no—
116#102—3581no—
117C4×C8C2×D17911yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—3582no—
121C2×Q8—1791no—
122C2×Q8—7162no—
123C2×Q8—7162no—
124C2×D4—1791no—
125C22×C4—3582no—
126C22×C4—3581no—
127Q16—7162no—
128C2×C8—21no—
129C2×C8—3581no—
130C2×C8—3581no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3582no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1791no—
138C4⋊C4C22⋊D17912yes—
139C22⋊C4—3582no—
140C22⋊C4—3581no—
141C42—3581no—
142C42C22×D17911yescharacteristic
143C23—3581no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—3581no—
149Q8—1792no—
150Q8—7164no—
151D4—3581no—
152D4—1792no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—3581no—
156C2×C4—3581no—
157C2×C4—3581no—
158C2×C4—7164no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4D4×D17911yescharacteristic
162C2×C4D4×D17911yescharacteristic
163C2×C4C2×C22⋊D17911yescharacteristic
164C8—22no—
165C8—7161no—
166C8—7161no—
167C22—3582no—
168C22—7161no—
169C22—7161no—
170C22#1915308111yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—3582no—
175C4—7161no—
176C4—7161no—
177C4D8×D17911yescharacteristic
178C4Q16×D17911yescharacteristic
179C4#1915313711yescharacteristic
180C4#1915314511yescharacteristic
181C2—7162no—
182C2#2603857111yescharacteristic
183C2#2603859311yescharacteristic
184C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#500×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#500D179
hypocenterC179#1649
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #26038646C2semidirect
4. #26038185C2semidirect
5. #26038171C2semidirect
7. #500×C179C2semidirect
55. D716Q16semidirect
92. C179#1649semidirect
95. #500D179semidirect

Maximal subgroups

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

subgroupindex
#260391042
#260387162
#260386462
#260381852
#260381712
#260381682
#500×C1792
#1649179