Finite Group Databasebeta

← #36364574

Subgroups — #36364574

Order 22144.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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#26012048C211yesmaximal, characteristic, complemented
2#26011886C212yesmaximal, complemented
3#26011850C212yesmaximal, complemented
4#26011698C212yesmaximal, complemented
5#26011688C212yesmaximal, complemented
6#26011653C214yesmaximal, complemented
7#26011548C211yesmaximal, characteristic, complemented
8#532×C173C211yesmaximal, characteristic
9C2×#14583816C2214yes—
10C2×#14583814C2214yes—
11#19149679C2211yescharacteristic
12#19149679C2211yescharacteristic
13#19149671—21no—
14#19149658—21no—
15#19149658C2212yes—
16C4×Dic346—21no—
17C4×Dic346C2212yes—
18#19149638—21no—
19C4⋊#13063436C2212yes—
20#19149603C2214yes—
21#19149600C2214yes—
22C22⋊#13063436C2212yes—
23#19149590C2212yes—
24C2×#41×C173C2212yes—
25C4⋊Q8×C173C2211yescharacteristic
26#127×C173C2211yescharacteristic
27#124×C173C2212yes—
28#121×C173—21no—
29#121×C173—21no—
30C4×D4×C173—21no—
31C4×D4×C173—21no—
32C2×C42×C173C2211yescharacteristic
33C2×Dic346—21no—
34C2×Dic346C2312yes—
35#14583816—44no—
36#14583814—44no—
37C2×#13063436C2314yes—
38C4⋊Dic173—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173—21no—
41C4⋊Dic173C2312yes—
42#14583805—22no—
43#14583805—22no—
44C22⋊Dic173—22no—
45C4×Dic173—22no—
46#41×C173—44no—
47C2×Q8×C173C2312yes—
48C2×D4×C173—22no—
49C2×D4×C173—21no—
50C2×D4×C173—21no—
51C2×D4×C173C2312yes—
52C22×C4×C173—22no—
53C22×C4×C173D411yescharacteristic
54C22×C4×C173D411yescharacteristic
55C22×C4×C173C2311yescharacteristic
56C4⋊C4×C173—21no—
57C4⋊C4×C173—21no—
58C22⋊C4×C173—22no—
59C22⋊C4×C173—22no—
60C42×C173D411yescharacteristic
61C42×C173D411yescharacteristic
62C42×C173C2311yescharacteristic
63C42×C173C2311yescharacteristic
64Dic346—22no—
65Dic346—22no—
66Dic346—42no—
67C2×Dic173—22no—
68C2×Dic173—22no—
69#13063436—24no—
70C23×C173—22no—
71C23×C173C2×D411yescharacteristic
72Q8×C173—24no—
73D4×C173—24no—
74D4×C173—42no—
75D4×C173—44no—
76D4×C173—42no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—22no—
80C2×C4×C173—41no—
81C2×C4×C173—44no—
82C2×C4×C173C2×D411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2×D411yescharacteristic
87C2×C4×C173C2411yescharacteristic
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—42no—
92C22×C173—42no—
93C22×C173—42no—
94C22×C173C22×D411yescharacteristic
95C4×C173—21no—
96C4×C173—21no—
97C4×C173—21no—
98C4×C173—21no—
99C4×C173—42no—
100C4×C173#13712yes—
101C4×C173H(5,𝔽2)12yes—
102Dic173—42no—
103Dic173—42no—
104C2×C173—41no—
105C2×C173—82no—
106C2×C173#53511yescharacteristic
107C2×C173C2×#13711yescharacteristic
108C2×C173#57511yescharacteristic
109C173#321611yescharacteristic, complemented, Sylow
110#3216—1731nomaximal, Sylow
111#533—1731no—
112#532D17311yescharacteristic
113#497—1732no—
114#489—1732no—
115#462—1732no—
116#460—1732no—
117#452—1734no—
118#432—1731no—
119C2×#41C2×D17312yes—
120C2×Q16—1734no—
121C2×SD16—1734no—
122C4⋊Q8—1731no—
123C4⋊Q8—1731no—
124C4⋊Q8C2×D17311yescharacteristic
125#127C2×D17311yescharacteristic
126#125—3461no—
127#124C2×D17312yes—
128C22⋊Q8—3461no—
129C22⋊Q8—1732no—
130#121—21no—
131#121—21no—
132C4×Q8—3461no—
133C4×Q8—1732no—
134C4×D4—21no—
135C4×D4—21no—
136#117—3461no—
137C2×C42C2×D17311yescharacteristic
138C4⋊C8—1732no—
139#103—1734no—
140#102—1734no—
141C22⋊C8—1732no—
142#97—1732no—
143#41—44no—
144C2×Q8—3461no—
145C2×Q8—1732no—
146C2×Q8C22×D17312yes—
147C2×D4—22no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C22×D17312yes—
151C22×C4—22no—
152C22×C4C22⋊D17311yescharacteristic
153C22×C4C22⋊D17311yescharacteristic
154C22×C4C22×D17311yescharacteristic
155Q16—6924no—
156SD16—6924no—
157C2×C8—1734no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C4⋊C4—3461no—
163C4⋊C4—3462no—
164C4⋊C4—3462no—
165C4⋊C4—1732no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—3462no—
169C42—3462no—
170C42C22⋊D17311yescharacteristic
171C42C22⋊D17311yescharacteristic
172C42C22×D17311yescharacteristic
173C42C22×D17311yescharacteristic
174C23—22no—
175C23C2×C22⋊D17311yescharacteristic
176Q8—24no—
177Q8—3462no—
178Q8—3462no—
179Q8—6922no—
180D4—24no—
181D4—42no—
182D4—44no—
183D4—42no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—44no—
189C2×C4—3462no—
190C2×C4—3462no—
191C2×C4C2×C22⋊D17311yescharacteristic
192C2×C4C2×C22⋊D17311yescharacteristic
193C2×C4C2×C22⋊D17311yescharacteristic
194C2×C4C2×C22⋊D17311yescharacteristic
195C2×C4C2×C22⋊D17311yescharacteristic
196C2×C4C23×D17311yescharacteristic
197C8—3464no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22C22×C22⋊D17311yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—6922no—
211C4—6922no—
212C4#1914970912yes—
213C4#1914972412yes—
214C2—41no—
215C2—82no—
216C2#2601205811yescharacteristic
217C2C2×#1914970911yescharacteristic
218C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#532×C173C2
socleC22×C173C22×D4
hypercenter#532D173
hypocenterC173#3216
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26012048C2semidirect
2. #26011886C2semidirect
3. #26011850C2semidirect
4. #26011698C2semidirect
5. #26011688C2semidirect
6. #26011653C2semidirect
7. #26011548C2semidirect
109. C173#3216semidirect

Maximal subgroups

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

subgroupindex
#260120482
#260118862
#260118502
#260116982
#260116882
#260116532
#260115482
#532×C1732
#3216173