Finite Group Databasebeta

← #36352535

Subgroups — #36352535

Order 22144.

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#26012120C211yesmaximal, characteristic
2#26011647C211yesmaximal, characteristic
3#26011410C211yesmaximal, characteristic
4#26011229C211yesmaximal, characteristic
5#26011228C211yesmaximal, characteristic
6#26011216C211yesmaximal, characteristic
7#451×C173C211yesmaximal, characteristic
8C2×#14583816—21no—
9#19149678—21no—
10#19149677—21no—
11#19149669—21no—
12#19149669—21no—
13#19149658C2211yescharacteristic
14#19149656—21no—
15#19149635—21no—
16C2×#14583805—21no—
17C2×#14583805C2211yescharacteristic
18#19149603—21no—
19C22⋊#13063436C2211yescharacteristic
20#19149586—21no—
21#19149586C2211yescharacteristic
22C22×Q8×C173C2211yescharacteristic
23C2×Q16×C173—21no—
24C22⋊Q8×C173C2211yescharacteristic
25#103×C173—21no—
26C22⋊C8×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C22×Dic173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32#14583816—42no—
33C2×#13063436—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D411yescharacteristic
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—42no—
44C22⋊Dic173—21no—
45C4×Dic173—21no—
46C4×Dic173—21no—
47C2×Q8×C173—21no—
48C2×Q8×C173—21no—
49C2×Q8×C173—42no—
50C2×Q8×C173D411yescharacteristic
51C22×C4×C173—21no—
52C22×C4×C173C2311yescharacteristic
53Q16×C173—42no—
54C2×C8×C173—21no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173D411yescharacteristic
57C22⋊C4×C173—21no—
58Dic346—41no—
59Dic346—81no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—22no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—41no—
67C2×Dic173—41no—
68C2×Dic173—41no—
69C2×Dic173Q1612yescomplemented
70#13063436—41no—
71C23×C173C2×D411yescharacteristic
72Q8×C173—22no—
73Q8×C173—22no—
74Q8×C173—41no—
75Q8×C173—41no—
76Q8×C173—41no—
77C2×C4×C173—22no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—41no—
81C2×C4×C173—41no—
82C2×C4×C173—41no—
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C8×C173—41no—
86C22×C173—21no—
87C22×C173—21no—
88C22×C173#12011yescharacteristic
89C22×C173C2×Q1611yescharacteristic
90C22×C173#13711yescharacteristic
91C4×C173—21no—
92C4×C173—21no—
93C4×C173—41no—
94C4×C173—42no—
95C4×C173—41no—
96Dic173—22no—
97Dic173—42no—
98Dic173—41no—
99Dic173—41no—
100Dic173—41no—
101C2×C173—21no—
102C2×C173—21no—
103C2×C173#45111yescharacteristic
104C2×C173#45611yescharacteristic
105C2×C173B(4,𝔽2)11yescharacteristic
106C173#160511yescharacteristic, complemented, Sylow
107#1605—1731nomaximal, Sylow
108#544—1731no—
109#451—1731no—
110#451D17311yescharacteristic
111#395—1731no—
112#328—1731no—
113#328—1731no—
114#324—1731no—
115C22×Q8C2×D17311yescharacteristic
116C2×Q16—21no—
117C2×Q16—3461no—
118C4⋊Q8—3461no—
119C4⋊Q8—3461no—
120#125—3461no—
121#125—3461no—
122C22⋊Q8—3461no—
123C22⋊Q8—1731no—
124C22⋊Q8C2×D17311yescharacteristic
125#117—3461no—
126C2×C4⋊C4—3461no—
127C2×C4⋊C4—1731no—
128#103—21no—
129#103—3461no—
130C22⋊C8—1731no—
131C22⋊C8C2×D17311yescharacteristic
132#95—3461no—
133#95—1731no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—3461no—
138C2×Q8—1731no—
139C2×Q8C22⋊D17311yescharacteristic
140C22×C4—21no—
141C22×C4—3461no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4C22×D17311yescharacteristic
145Q16—42no—
146Q16—6922no—
147C2×C8—21no—
148C2×C8—3461no—
149C4⋊C4—21no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—1731no—
159C4⋊C4—6922no—
160C4⋊C4C22⋊D17311yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3461no—
163C42—3461no—
164C42—3461no—
165C23D4×D17311yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—41no—
171Q8—13841no—
172Q8—6921no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3462no—
182C2×C4—3461no—
183C2×C4—3461no—
184C2×C4—3461no—
185C2×C4—1732no—
186C2×C4—6921no—
187C2×C4—6921no—
188C2×C4—6921no—
189C2×C4D4×D17311yescharacteristic
190C2×C4C2×C22⋊D17311yescharacteristic
191C8—41no—
192C8—6921no—
193C22—21no—
194C22—21no—
195C22#1914964711yescharacteristic
196C22Q16×D17311yescharacteristic
197C22#1914971111yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—3462no—
204C4—6922no—
205C4—6921no—
206C4—6921no—
207C4—6921no—
208C2—21no—
209C2—21no—
210C2#2601164511yescharacteristic
211C2#2601167211yescharacteristic
212C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#451×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#451D173
hypocenterC173#1605
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
69. C2×Dic173Q16semidirect
106. C173#1605semidirect

Maximal subgroups

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

subgroupindex
#260121202
#260116472
#260114102
#260112292
#260112282
#260112162
#451×C1732
#1605173