Finite Group Databasebeta

← #36352611

Subgroups — #36352611

Order 22144.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#26011824C211yesmaximal, characteristic, complemented
2#26011807C211yesmaximal, characteristic, complemented
3#26011427C211yesmaximal, characteristic, complemented
4#26011228C211yesmaximal, characteristic, complemented
5#26011225C211yesmaximal, characteristic, complemented
6#26011215C211yesmaximal, characteristic, complemented
7#538×C173C211yesmaximal, characteristic
8C2×C4⋊Dic173C2211yescharacteristic
9C2×#14583805—21no—
10C2×C22⋊Dic173—21no—
11#19149612—21no—
12#19149610—21no—
13#19149603—21no—
14#19149600—21no—
15C22⋊#13063436C2211yescharacteristic
16C22⋊#13063436C2211yescharacteristic
17#19149586C2211yescharacteristic
18#126×C173—21no—
19#124×C173—21no—
20#123×C173—21no—
21C22⋊Q8×C173C2211yescharacteristic
22#121×C173—21no—
23#121×C173C2211yescharacteristic
24#120×C173—21no—
25C4×D4×C173—21no—
26C2×C22⋊C4×C173C2211yescharacteristic
27C22×Dic173—21no—
28C22×Dic173—21no—
29C2×#13063436—21no—
30C2×#13063436—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33#14583805—42no—
34C22⋊Dic173—42no—
35C24×C173D411yescharacteristic
36C2×Q8×C173D411yescharacteristic
37C2×D4×C173—21no—
38C2×D4×C173—21no—
39C2×D4×C173—21no—
40C2×D4×C173D411yescharacteristic
41C22×C4×C173—21no—
42C22×C4×C173D411yescharacteristic
43C22×C4×C173C2311yescharacteristic
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173D411yescharacteristic
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C22⋊C4×C173—21no—
50C22⋊C4×C173—21no—
51C22⋊C4×C173—21no—
52C22⋊C4×C173—21no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C42×C173—21no—
56C2×Dic173—22no—
57C2×Dic173—41no—
58C2×Dic173—41no—
59C2×Dic173—41no—
60C2×Dic173—41no—
61C2×Dic173—41no—
62#13063436—41no—
63#13063436—41no—
64C23×C173—21no—
65C23×C173—21no—
66C23×C173—21no—
67C23×C173—21no—
68C23×C173—41no—
69C23×C173—41no—
70C23×C173C2×D411yescharacteristic
71Q8×C173—41no—
72D4×C173—41no—
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—81no—
76D4×C173—81no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—21no—
81C2×C4×C173—41no—
82C2×C4×C173—41no—
83C2×C4×C173—81no—
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C22×C173—21no—
87C22×C173—21no—
88C22×C173—42no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173—41no—
95C22×C173—81no—
96C22×C173#12011yescharacteristic
97C22×C173#13511yescharacteristic
98C22×C173#13711yescharacteristic
99C4×C173—21no—
100C4×C173—21no—
101C4×C173—41no—
102C4×C173—41no—
103C4×C173—41no—
104C4×C173—41no—
105Dic173—42no—
106Dic173—81no—
107C2×C173—21no—
108C2×C173—21no—
109C2×C173—81no—
110C2×C173—81no—
111C2×C173#45211yescharacteristic
112C2×C173#45311yescharacteristic
113C2×C173#45811yescharacteristic
114C173#161811yescharacteristic, complemented, Sylow
115#1618—1731nomaximal, Sylow
116#538D17311yescharacteristic
117#484—1731no—
118#481—1731no—
119#399—1731no—
120#328—1731no—
121#327—1731no—
122#323—1731no—
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C2×D17311yescharacteristic
127#121—21no—
128#121C2×D17311yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—3461no—
132C2×C4⋊C4—1731no—
133C2×C22⋊C4—3461no—
134C2×C22⋊C4C2×D17311yescharacteristic
135#107—3461no—
136#106—3461no—
137#103—3461no—
138#102—3461no—
139C22⋊C8—1731no—
140C22⋊C8—1731no—
141#95—1731no—
142C24C22⋊D17311yescharacteristic
143C2×Q8C22⋊D17311yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C22⋊D17311yescharacteristic
148C22×C4—21no—
149C22×C4—3461no—
150C22×C4—3461no—
151C22×C4C22⋊D17311yescharacteristic
152C22×C4C22×D17311yescharacteristic
153C2×C8—3461no—
154C2×C8—3461no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—6922no—
160C4⋊C4C22⋊D17311yescharacteristic
161C4⋊C4C22⋊D17311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—6922no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C22⋊D17311yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—3462no—
192C2×C4—6921no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4—6921no—
197C2×C4C2×C22⋊D17311yescharacteristic
198C2×C4C2×C22⋊D17311yescharacteristic
199C8—6921no—
200C8—6921no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#1914964811yescharacteristic
212C22#1914969911yescharacteristic
213C22#1914970911yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—13841no—
221C4—6922no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2601165311yescharacteristic
227C2#2601165911yescharacteristic
228C2#2601168111yescharacteristic
subgroupG/H
centerC22#19149648
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting, prime radical#538×C173C2
socleC22×C173#120
hypercenter#538D173
hypocenterC173#1618
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #26011824C2semidirect
2. #26011807C2semidirect
3. #26011427C2semidirect
4. #26011228C2semidirect
5. #26011225C2semidirect
6. #26011215C2semidirect
114. C173#1618semidirect

Maximal subgroups

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

subgroupindex
#260118242
#260118072
#260114272
#260112282
#260112252
#260112152
#538×C1732
#1618173