Finite Group Databasebeta

← #36355577

Subgroups — #36355577

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#26011682C211yesmaximal, characteristic
3#26011680C211yesmaximal, characteristic
4#26011671C211yesmaximal, characteristic
5#26011285C211yesmaximal, characteristic, complemented
6#26011282C211yesmaximal, characteristic, complemented
7#352×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149712—21no—
10#19149678—21no—
11#19149672—21no—
12#19149672—21no—
13#19149669—21no—
14#19149666C2211yescharacteristic
15#19149663—21no—
16#19149662—21no—
17#19149660C2211yescharacteristic
18#19149658—21no—
19#19149656—21no—
20#19149605—21no—
21#19149594C2211yescharacteristic
22#19149593—21no—
23#19149593—21no—
24#123×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27#14583826—21no—
28#14583825—41no—
29C2×Dic346—21no—
30C2×Dic346D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33D4×D173—41no—
34C22×Dic173D411yescharacteristic
35Dic692—41no—
36#14583815—41no—
37C4⋊Dic173—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173—41no—
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45#14583805—41no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173D411yescharacteristic
51#14583803—41no—
52#14583803—41no—
53C4×Dic173—21no—
54C4×Dic173D411yescharacteristic
55C2×D4×C173C2311yescharacteristic
56C22×C4×C173D411yescharacteristic
57M16×C173—21no—
58C4⋊C4×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173D411yescharacteristic
62C22×D173—21no—
63C22×D173—21no—
64Dic346—21no—
65Dic346—41no—
66Dic346—81no—
67D692—21no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C22⋊D173—41no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C4×D173—41no—
80C23×C173C2×D411yescharacteristic
81C23×C173C2×D411yescharacteristic
82D4×C173—41no—
83C2×C4×C173—21no—
84C2×C4×C173—21no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173C2×D411yescharacteristic
88C8×C173—41no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173—41no—
92C22×C173—41no—
93C22×C173#12011yescharacteristic
94C2×D173—41no—
95C2×D173—81no—
96C2×D173—81no—
97C4×C173—21no—
98C4×C173—41no—
99C4×C173—41no—
100C4×C173—81no—
101Dic173—41no—
102Dic173—41no—
103Dic173—41no—
104Dic173—41no—
105Dic173—41no—
106Dic173—81no—
107C2×C173—21no—
108C2×C173—41no—
109C2×C173—41no—
110C2×C173B(4,𝔽2)11yescharacteristic
111D173—81no—
112C173#220111yescharacteristic, complemented, Sylow
113#2201—1731nomaximal, Sylow
114#563—1731no—
115#458—1731no—
116#458—1731no—
117#455—1731no—
118#354—1731no—
119#352—1731no—
120#352D17311yescharacteristic, complemented
121H(5,𝔽2)—1731no—
122#137—3461no—
123C4⋊Q8—3461no—
124#126—3461no—
125#126—3461no—
126#125—3461no—
127#124—1731no—
128#123—3461no—
129#123—3461no—
130#123—1731no—
131#123C2×D17311yescharacteristic
132C22⋊Q8—3461no—
133C22⋊Q8—3461no—
134#104—3461no—
135#100C2×D17311yescharacteristic
136#99—3461no—
137#99—3461no—
138#99—1731no—
139#99C2×D17311yescharacteristic
140#41—3461no—
141#41—6921no—
142C2×Q8—3461no—
143C2×Q8—1731no—
144C2×D4—3461no—
145C2×D4—3461no—
146C2×D4—6921no—
147C2×D4C22×D17311yescharacteristic
148C22×C4—1731no—
149C22×C4D69211yescharacteristic
150Q16—6921no—
151SD16—6921no—
152M16—21no—
153C4⋊C4—21no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3461no—
157C4⋊C4—3461no—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—6921no—
162C4⋊C4—6921no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—3461no—
166C22⋊C4—3461no—
167C22⋊C4—3461no—
168C22⋊C4—3461no—
169C22⋊C4—1731no—
170C22⋊C4—6921no—
171C22⋊C4—6921no—
172C22⋊C4D69211yescharacteristic
173C42—3461no—
174C42—1731no—
175C23—3461no—
176C23—3461no—
177C23D4×D17311yescharacteristic
178C23C2×D69211yescharacteristic
179Q8—3461no—
180Q8—13841no—
181Q8—6921no—
182D4—41no—
183D4—3461no—
184D4—6921no—
185D4—6921no—
186D4—6921no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—3461no—
192C2×C4—3461no—
193C2×C4—3461no—
194C2×C4—3461no—
195C2×C4—3461no—
196C2×C4—3461no—
197C2×C4—6921no—
198C2×C4—6921no—
199C2×C4—6921no—
200C2×C4D4×D17311yescharacteristic
201C8—41no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—13841no—
207C22—13841no—
208C22—6921no—
209C22#1914964911yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—81no—
214C4—13841no—
215C4—6921no—
216C4—6921no—
217C4—6921no—
218C4—6921no—
219C4—6921no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—13841no—
224C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#352×C173C2
socleC2×C173B(4,𝔽2)
prime radical#19149724C22
hypercenter#352D173
hypocenterC173#2201
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #26012244C2semidirect
5. #26011285C2semidirect
6. #26011282C2semidirect
7. #352×C173C2semidirect
112. C173#2201semidirect
120. #352D173semidirect

Maximal subgroups

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

subgroupindex
#260122442
#260116822
#260116802
#260116712
#260112852
#260112822
#352×C1732
#2201173