Finite Group Databasebeta

← #36355574

Subgroups — #36355574

Order 22144.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012244C211yesmaximal, characteristic, complemented
2#26011681C211yesmaximal, characteristic
3#26011669C211yesmaximal, characteristic
4#26011285C211yesmaximal, characteristic, complemented
5#26011281C211yesmaximal, characteristic, complemented
6#26011280C211yesmaximal, characteristic, complemented
7#458×C173C211yesmaximal, characteristic
8#19149709—21no—
9#19149678—21no—
10#19149672—21no—
11#19149672—21no—
12#19149669—21no—
13#19149666C2211yescharacteristic
14#19149660—21no—
15#19149660C2211yescharacteristic
16#19149658—21no—
17#19149656—21no—
18#19149606—21no—
19#19149596C2211yescharacteristic
20#19149594—21no—
21#19149594C2211yescharacteristic
22H(5,𝔽2)×C173C2211yescharacteristic
23#123×C173—21no—
24#123×C173C2211yescharacteristic
25#99×C173—21no—
26#99×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C22×Dic173D411yescharacteristic
30#14583816—41no—
31#14583814—41no—
32#14583810—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35#14583805—21no—
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—41no—
41C22⋊Dic173—21no—
42C22⋊Dic173—21no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—41no—
46C22⋊Dic173D411yescharacteristic
47C4×Dic173—21no—
48C4×Dic173D411yescharacteristic
49#41×C173—21no—
50#41×C173—41no—
51C2×D4×C173—21no—
52C2×D4×C173—21no—
53C2×D4×C173—41no—
54C2×D4×C173C2311yescharacteristic
55C22×C4×C173D411yescharacteristic
56C4⋊C4×C173—21no—
57C4⋊C4×C173—41no—
58C22⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—41no—
61C22⋊C4×C173D411yescharacteristic
62Dic346—41no—
63Dic346—81no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—41no—
69C2×Dic173—41no—
70#13063436—41no—
71C23×C173—21no—
72C23×C173—21no—
73C23×C173C2×D411yescharacteristic
74C23×C173C2×D411yescharacteristic
75Q8×C173—21no—
76D4×C173—21no—
77D4×C173—41no—
78D4×C173—41no—
79D4×C173—41no—
80D4×C173—41no—
81C2×C4×C173—21no—
82C2×C4×C173—21no—
83C2×C4×C173—21no—
84C2×C4×C173—21no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173C2×D411yescharacteristic
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173—41no—
94C22×C173—81no—
95C22×C173—81no—
96C22×C173#12011yescharacteristic
97C4×C173—21no—
98C4×C173—41no—
99C4×C173—41no—
100C4×C173—41no—
101C4×C173—81no—
102Dic173—41no—
103Dic173—41no—
104Dic173—41no—
105Dic173—41no—
106Dic173—81no—
107C2×C173—21no—
108C2×C173—41no—
109C2×C173—41no—
110C2×C173—81no—
111C2×C173B(4,𝔽2)11yescharacteristic
112C173#220111yescharacteristic, complemented, Sylow
113#2201—1731nomaximal, Sylow
114#563—1731no—
115#458—1731no—
116#458D17311yescharacteristic
117#455—1731no—
118#354—1731no—
119#352—1731no—
120#352—1731no—
121H(5,𝔽2)C2×D17311yescharacteristic
122#137—3461no—
123C4⋊Q8—3461no—
124#126—3461no—
125#126—3461no—
126#125—3461no—
127#124—1731no—
128#123—21no—
129#123—3461no—
130#123—1731no—
131#123C2×D17311yescharacteristic
132C22⋊Q8—3461no—
133C22⋊Q8—3461no—
134#104—3461no—
135#100—1731no—
136#99—21no—
137#99—3461no—
138#99—1731no—
139#99C2×D17311yescharacteristic
140#41—21no—
141#41—41no—
142C2×Q8—3461no—
143C2×Q8—1731no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—41no—
147C2×D4C22×D17311yescharacteristic
148C22×C4—1731no—
149C22×C4C22⋊D17311yescharacteristic
150Q16—6921no—
151SD16—6921no—
152M16—3461no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C4⋊C4—6921no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—41no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—3461no—
170C22⋊C4—1731no—
171C22⋊C4—6921no—
172C22⋊C4C22⋊D17311yescharacteristic
173C42—3461no—
174C42—1731no—
175C23—21no—
176C23—21no—
177C23D4×D17311yescharacteristic
178C23C2×C22⋊D17311yescharacteristic
179Q8—21no—
180Q8—13841no—
181Q8—6921no—
182D4—21no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—41no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—3461no—
198C2×C4—6921no—
199C2×C4—6921no—
200C2×C4D4×D17311yescharacteristic
201C8—6921no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22—81no—
209C22#1914964711yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—13841no—
216C4—6921no—
217C4—6921no—
218C4—6921no—
219C4—6921no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—81no—
224C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#458×C173C2
socleC2×C173B(4,𝔽2)
prime radicalH(5,𝔽2)×C173C22
hypercenter#458D173
hypocenterC173#2201
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #26012244C2semidirect
4. #26011285C2semidirect
5. #26011281C2semidirect
6. #26011280C2semidirect
112. C173#2201semidirect

Maximal subgroups

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

subgroupindex
#260122442
#260116812
#260116692
#260112852
#260112812
#260112802
#458×C1732
#2201173