Finite Group Databasebeta

← #37508250

Subgroups — #37508250

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#26038842C211yesmaximal, characteristic, complemented
3#26038720C211yesmaximal, characteristic
4#26038650C211yesmaximal, characteristic, complemented
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#332×C179C211yesmaximal, characteristic, complemented
8C2×Q8×D179—21no—
9C2×Dic716—21no—
10#19153113C2211yescharacteristic
11#19153112—21no—
12#19153091—22no—
13#19153089—21no—
14C4×Dic358C2211yescharacteristic
15C4×D716C2211yescharacteristic
16#19153075—21no—
17#19153047—21no—
18#19153046—22no—
19#19153038—21no—
20#19153035—21no—
21C4×C179⋊C8C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25C2×Dic358—22no—
26C2×Dic358—21no—
27C2×Dic358D411yescharacteristic
28Q8×D179—42no—
29Q8×D179—42no—
30C2×D716D411yescharacteristic
31C2×C4×D179—22no—
32C2×C4×D179—21no—
33Dic716—42no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179—21no—
37C4⋊Dic179D411yescharacteristic
38C4⋊Dic179D411yescharacteristic
39Dic179⋊C4—22no—
40Dic179⋊C4—21no—
41(C2×D179)⋊C4—22no—
42(C2×D179)⋊C4—21no—
43C4×Dic179—21no—
44C2×Q8×C179—21no—
45C2×C8×C179—21no—
46C2×C8×C179—21no—
47C4⋊C4×C179—21no—
48C4⋊C4×C179D412yes—
49C42×C179C2311yescharacteristic
50C22×D179—21no—
51Dic358—21no—
52Dic358—44no—
53Dic358—41no—
54Dic358—41no—
55Dic358D812yes—
56D716—21no—
57D716Q1612yescomplemented
58C2×Dic179—21no—
59C2×Dic179—21no—
60C2×Dic179—21no—
61C2×Dic179—41no—
62C4×D179—44no—
63C4×D179—41no—
64C4×D179—41no—
65C179⋊C8—22no—
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—
78C4×C179—21no—
79C4×C179—42no—
80C4×C179C2×D811yescharacteristic
81C4×C179C2×Q1611yescharacteristic
82C4×C179AGL(1,ℤ/8)11yescharacteristic
83C4×C179#13711yescharacteristic
84Dic179—22no—
85Dic179—41no—
86Dic179—41no—
87Dic179—81no—
88C2×C179#44711yescharacteristic
89C2×C179#45111yescharacteristic
90C2×C179#45611yescharacteristic
91D179—42no—
92C179#164911yescharacteristic, complemented, Sylow
93#1649—1791nomaximal, Sylow
94D4×Q8—1791no—
95#500—1791no—
96#474—1791no—
97#462—1791no—
98#332D17911yescharacteristic, complemented
99Q8⋊C8—1791no—
100D4⋊C8—1791no—
101C22×Q8—3581no—
102C2×Q16—3581no—
103C4⋊Q8—3581no—
104C4⋊Q8—1791no—
105C4⋊Q8C2×D17911yescharacteristic
106C22⋊Q8—3581no—
107C22⋊Q8—3582no—
108C4×Q8—1791no—
109C4×D4—3581no—
110C4×D4—1791no—
111#107—3581no—
112#107—3582no—
113C4⋊C8C2×D17911yescharacteristic
114C4⋊C8C2×D17911yescharacteristic
115#103—3581no—
116#102—3581no—
117C4×C8—1791no—
118C2×Q8—21no—
119C2×Q8—3582no—
120C2×Q8—3581no—
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—21no—
130C2×C8—3581no—
131C4⋊C4—21no—
132C4⋊C4—3582no—
133C4⋊C4—3581no—
134C4⋊C4—3581no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1791no—
138C4⋊C4D71612yes—
139C22⋊C4—3582no—
140C22⋊C4—3581no—
141C42—3581no—
142C42C22×D17911yescharacteristic
143C23—3581no—
144Q8—41no—
145Q8—41no—
146Q8—3581no—
147Q8—1792no—
148Q8—7164no—
149Q8—7161no—
150Q8—7161no—
151D4—3581no—
152D4—1792no—
153C2×C4—22no—
154C2×C4—3581no—
155C2×C4—3581no—
156C2×C4—3581no—
157C2×C4—7164no—
158C2×C4—7161no—
159C2×C4—7161no—
160C2×C4—7161no—
161C2×C4D4×D17911yescharacteristic
162C2×C4D4×D17911yescharacteristic
163C2×C4C2×D71611yescharacteristic
164C8—41no—
165C8—41no—
166C8—3582no—
167C22—3582no—
168C22—7161no—
169C22—7161no—
170C22#1915308311yescharacteristic
171C4—21no—
172C4—42no—
173C4—3582no—
174C4—14321no—
175C4—7161no—
176C4—7161no—
177C4D8×D17911yescharacteristic
178C4Q16×D17911yescharacteristic
179C4#1915314011yescharacteristic
180C4#1915314211yescharacteristic
181C2—7162no—
182C2#2603857411yescharacteristic
183C2#2603859611yescharacteristic
184C2#2603862411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#332×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#332D179
hypocenterC179#1649
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. #26038842C2semidirect
4. #26038650C2semidirect
5. #26038169C2semidirect
7. #332×C179C2semidirect
57. D716Q16semidirect
92. C179#1649semidirect
98. #332D179semidirect

Maximal subgroups

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

subgroupindex
#260391042
#260388422
#260387202
#260386502
#260381692
#260381662
#332×C1792
#1649179