Finite Group Databasebeta

← #37507972

Subgroups — #37507972

Order 22912.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#26039068C211yesmaximal, characteristic
2#26038595C211yesmaximal, characteristic
3#26038358C211yesmaximal, characteristic
4#26038177C211yesmaximal, characteristic
5#26038176C211yesmaximal, characteristic
6#26038164C211yesmaximal, characteristic
7#451×C179C211yesmaximal, characteristic
8C2×#15668249—21no—
9#19153112—21no—
10#19153111—21no—
11#19153103—21no—
12#19153103—21no—
13#19153092C2211yescharacteristic
14#19153090—21no—
15#19153069—21no—
16C2×Dic179⋊C4—21no—
17C2×Dic179⋊C4C2211yescharacteristic
18#19153037—21no—
19C22⋊(C179⋊C8)C2211yescharacteristic
20C179⋊#95—21no—
21C179⋊#95C2211yescharacteristic
22C22×Q8×C179C2211yescharacteristic
23C2×Q16×C179—21no—
24C22⋊Q8×C179C2211yescharacteristic
25#103×C179—21no—
26C22⋊C8×C179C2211yescharacteristic
27C2×Dic358—21no—
28C2×Dic358D411yescharacteristic
29C22×Dic179—21no—
30C22×Dic179D411yescharacteristic
31C22×Dic179D411yescharacteristic
32#15668249—42no—
33C2×C179⋊C8—21no—
34C4⋊Dic179—21no—
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—21no—
37Dic179⋊C4—21no—
38Dic179⋊C4—21no—
39Dic179⋊C4—21no—
40Dic179⋊C4—21no—
41Dic179⋊C4—21no—
42Dic179⋊C4—21no—
43Dic179⋊C4—42no—
44C22⋊Dic179—21no—
45C4×Dic179—21no—
46C4×Dic179—21no—
47C2×Q8×C179—21no—
48C2×Q8×C179—21no—
49C2×Q8×C179—42no—
50C2×Q8×C179D411yescharacteristic
51C22×C4×C179—21no—
52C22×C4×C179C2311yescharacteristic
53Q16×C179—42no—
54C2×C8×C179—21no—
55C4⋊C4×C179—21no—
56C4⋊C4×C179D411yescharacteristic
57C22⋊C4×C179—21no—
58Dic358—41no—
59Dic358—81no—
60C2×Dic179—21no—
61C2×Dic179—21no—
62C2×Dic179—22no—
63C2×Dic179—21no—
64C2×Dic179—21no—
65C2×Dic179—21no—
66C2×Dic179—41no—
67C2×Dic179—41no—
68C2×Dic179—41no—
69C2×Dic179Q1612yescomplemented
70C179⋊C8—41no—
71C23×C179C2×D411yescharacteristic
72Q8×C179—22no—
73Q8×C179—22no—
74Q8×C179—41no—
75Q8×C179—41no—
76Q8×C179—41no—
77C2×C4×C179—22no—
78C2×C4×C179—21no—
79C2×C4×C179—21no—
80C2×C4×C179—41no—
81C2×C4×C179—41no—
82C2×C4×C179—41no—
83C2×C4×C179C2×D411yescharacteristic
84C2×C4×C179C2×D411yescharacteristic
85C8×C179—41no—
86C22×C179—21no—
87C22×C179—21no—
88C22×C179#12011yescharacteristic
89C22×C179C2×Q1611yescharacteristic
90C22×C179#13711yescharacteristic
91C4×C179—21no—
92C4×C179—21no—
93C4×C179—41no—
94C4×C179—42no—
95C4×C179—41no—
96Dic179—22no—
97Dic179—42no—
98Dic179—41no—
99Dic179—41no—
100Dic179—41no—
101C2×C179—21no—
102C2×C179—21no—
103C2×C179#45111yescharacteristic
104C2×C179#45611yescharacteristic
105C2×C179B(4,𝔽2)11yescharacteristic
106C179#160511yescharacteristic, complemented, Sylow
107#1605—1791nomaximal, Sylow
108#544—1791no—
109#451—1791no—
110#451D17911yescharacteristic
111#395—1791no—
112#328—1791no—
113#328—1791no—
114#324—1791no—
115C22×Q8C2×D17911yescharacteristic
116C2×Q16—21no—
117C2×Q16—3581no—
118C4⋊Q8—3581no—
119C4⋊Q8—3581no—
120#125—3581no—
121#125—3581no—
122C22⋊Q8—3581no—
123C22⋊Q8—1791no—
124C22⋊Q8C2×D17911yescharacteristic
125#117—3581no—
126C2×C4⋊C4—3581no—
127C2×C4⋊C4—1791no—
128#103—21no—
129#103—3581no—
130C22⋊C8—1791no—
131C22⋊C8C2×D17911yescharacteristic
132#95—3581no—
133#95—1791no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—3581no—
138C2×Q8—1791no—
139C2×Q8C22⋊D17911yescharacteristic
140C22×C4—21no—
141C22×C4—3581no—
142C22×C4—1791no—
143C22×C4—1791no—
144C22×C4C22×D17911yescharacteristic
145Q16—42no—
146Q16—7162no—
147C2×C8—21no—
148C2×C8—3581no—
149C4⋊C4—21no—
150C4⋊C4—3581no—
151C4⋊C4—3581no—
152C4⋊C4—3581no—
153C4⋊C4—3581no—
154C4⋊C4—3581no—
155C4⋊C4—3581no—
156C4⋊C4—3581no—
157C4⋊C4—3581no—
158C4⋊C4—1791no—
159C4⋊C4—7162no—
160C4⋊C4C22⋊D17911yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3581no—
163C42—3581no—
164C42—3581no—
165C23D4×D17911yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—41no—
171Q8—14321no—
172Q8—7161no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3581no—
180C2×C4—3581no—
181C2×C4—3582no—
182C2×C4—3581no—
183C2×C4—3581no—
184C2×C4—3581no—
185C2×C4—1792no—
186C2×C4—7161no—
187C2×C4—7161no—
188C2×C4—7161no—
189C2×C4D4×D17911yescharacteristic
190C2×C4C2×C22⋊D17911yescharacteristic
191C8—41no—
192C8—7161no—
193C22—21no—
194C22—21no—
195C22#1915308111yescharacteristic
196C22Q16×D17911yescharacteristic
197C22#1915314511yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—3582no—
204C4—7162no—
205C4—7161no—
206C4—7161no—
207C4—7161no—
208C2—21no—
209C2—21no—
210C2#2603859311yescharacteristic
211C2#2603862011yescharacteristic
212C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#451×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#451D179
hypocenterC179#1605
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
69. C2×Dic179Q16semidirect
106. C179#1605semidirect

Maximal subgroups

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

subgroupindex
#260390682
#260385952
#260383582
#260381772
#260381762
#260381642
#451×C1792
#1605179