Finite Group Databasebeta

← #37510165

Subgroups — #37510165

Order 22912.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#26039065C211yesmaximal, characteristic, complemented
2#26038294C212yesmaximal
3#26038294C211yesmaximal, characteristic
4#26038205C212yesmaximal, complemented
5#342×C179C211yesmaximal, characteristic, complemented
6C24×D179C2211yescharacteristic
7#19153092—22no—
8#19153090—24no—
9C2×C22⋊Dic179C2211yescharacteristic
10C2×C22⋊Dic179C2212yes—
11C2×(C2×D179)⋊C4—24no—
12C2×(C2×D179)⋊C4—22no—
13C2×C22⋊C4×C179C2212yes—
14C2×C22⋊C4×C179C2211yescharacteristic
15C23×D179—22no—
16C23×D179—21no—
17C23×D179—21no—
18C23×D179—22no—
19C23×D179—44no—
20C23×D179Q812yescomplemented
21C2×Dic358—22no—
22C22×Dic179—21no—
23C22×Dic179D414yes—
24C4⋊Dic179—22no—
25Dic179⋊C4—24no—
26C22⋊Dic179—22no—
27C22⋊Dic179—22no—
28C22⋊Dic179—22no—
29(C2×D179)⋊C4—44no—
30(C2×D179)⋊C4—48no—
31C24×C179C2311yescharacteristic
32C22×C4×C179—22no—
33C22×C4×C179D412yes—
34C22⋊C4×C179—22no—
35C22⋊C4×C179—22no—
36C22⋊C4×C179—22no—
37C22×D179—28no—
38C22×D179—24no—
39C22×D179—24no—
40C22×D179—22no—
41C22×D179—44no—
42C22×D179—44no—
43C22×D179—48no—
44C22×D179—44no—
45C22×D179—42no—
46C22×D179—41no—
47C22×D179—42no—
48C22×D179—41no—
49Dic358—82no—
50C2×Dic179—24no—
51C2×Dic179—44no—
52C2×Dic179—42no—
53C2×Dic179—41no—
54C23×C179—42no—
55C23×C179C2×D412yes—
56C23×C179C2×D411yescharacteristic
57C23×C179C2×Q811yescharacteristic
58C23×C179#4111yescharacteristic
59C23×C179#4112yes—
60C2×C4×C179—22no—
61C2×C4×C179—42no—
62C2×C4×C179—44no—
63C2×C4×C179—42no—
64C22×C179—24no—
65C22×C179—22no—
66C22×C179—41no—
67C22×C179—41no—
68C22×C179—42no—
69C22×C179#12011yescharacteristic
70C22×C179C22⋊Q812yes—
71C22×C179C22⋊Q811yescharacteristic
72C22×C179#12412yes—
73C22×C179#12411yescharacteristic
74C2×D179—28no—
75C2×D179—24no—
76C2×D179—44no—
77C2×D179—48no—
78C2×D179—44no—
79C2×D179—42no—
80C2×D179—42no—
81C2×D179—44no—
82C4×C179—42no—
83C4×C179—82no—
84Dic179—44no—
85Dic179—81no—
86C2×C179—21no—
87C2×C179—22no—
88C2×C179—22no—
89C2×C179—21no—
90C2×C179#39311yescharacteristic
91C2×C179B(4,𝔽2)12yes—
92D179—44no—
93C179#202911yescharacteristic, complemented, Sylow
94#2029—1791nomaximal, Sylow
95#543—1791no—
96#379—1792no—
97#379—1791no—
98#342—1792no—
99#342D17911yescharacteristic, complemented
100C25—1791no—
101C22⋊Q8—3584no—
102C22⋊Q8—3582no—
103C2×C22⋊C4—3584no—
104C2×C22⋊C4—3582no—
105C2×C22⋊C4—1791no—
106C2×C22⋊C4—1792no—
107C2×C22⋊C4C2×D17912yes—
108C2×C22⋊C4C2×D17911yescharacteristic
109C24—3582no—
110C24—3581no—
111C24—3581no—
112C24—3582no—
113C24—1792no—
114C24—7164no—
115C24C22×D17911yescharacteristic
116C2×Q8—3582no—
117C22×C4—22no—
118C22×C4—3581no—
119C22×C4—1794no—
120C22×C4D71612yes—
121C4⋊C4—3584no—
122C4⋊C4—3582no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—3582no—
127C22⋊C4—3582no—
128C22⋊C4—3582no—
129C22⋊C4—7164no—
130C22⋊C4—7168no—
131C23—42no—
132C23—3588no—
133C23—3584no—
134C23—3584no—
135C23—3582no—
136C23—7164no—
137C23—7162no—
138C23—7164no—
139C23—7164no—
140C23—7168no—
141C23—7161no—
142C23—7162no—
143C23—7161no—
144C23D4×D17912yes—
145C23C2×D71611yescharacteristic
146C23Q8×D17911yescharacteristic
147C23#1566825811yescharacteristic
148C23#1566825912yes—
149Q8—14322no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—3584no—
155C2×C4—7162no—
156C2×C4—7161no—
157C2×C4—7164no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—3588no—
164C22—3584no—
165C22—7168no—
166C22—7164no—
167C22—7164no—
168C22—7162no—
169C22—7162no—
170C22—7164no—
171C22#1915308311yescharacteristic
172C22#1915309112yes—
173C22#1915309311yescharacteristic
174C22#1915309912yes—
175C22#1915310111yescharacteristic
176C4—42no—
177C4—82no—
178C4—14321no—
179C4—7164no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—7164no—
185C2#2603835211yescharacteristic
186C2#2603862712yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC24×C179C23
FrattiniC24C22×D179
Fitting#342×C179C2
socleC22×C179#120
prime radicalC24×D179C22
hypercenter#342D179
hypocenterC179#2029
LCSG ▷ C24×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26039065C2semidirect
4. #26038205C2semidirect
5. #342×C179C2semidirect
20. C23×D179Q8semidirect
93. C179#2029semidirect
99. #342D179semidirect

Maximal subgroups

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

subgroupindex
#260390652
#260382942
#260382052
#342×C1792
#2029179