Finite Group Databasebeta

← #36355573

Subgroups — #36355573

Order 22144.

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

normal subgroups only

This group has 260 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#26012228C211yesmaximal, characteristic, complemented
2#26011682C211yesmaximal, characteristic, complemented
3#26011679C211yesmaximal, characteristic, complemented
4#26011658C211yesmaximal, characteristic, complemented
5#26011280C211yesmaximal, characteristic, complemented
6#26011277C211yesmaximal, characteristic, complemented
7#353×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149705—21no—
10#19149673—21no—
11#19149663—21no—
12#19149660—21no—
13#19149660C2211yescharacteristic
14#19149651—21no—
15#19149651—21no—
16#19149651—21no—
17#19149649—21no—
18#19149647—21no—
19#19149647C2211yescharacteristic
20#19149607—21no—
21#19149596C2211yescharacteristic
22#19149593—21no—
23#19149593—21no—
24#127×C173C2211yescharacteristic
25#99×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583826—21no—
29#14583825—41no—
30C2×D692—41no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36D4×D173—41no—
37C22×Dic173—21no—
38C22×Dic173D411yescharacteristic, complemented
39#14583814—41no—
40#14583811—41no—
41#14583810—21no—
42C4⋊Dic173—41no—
43#14583805—21no—
44#14583805—21no—
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48C22⋊Dic173D411yescharacteristic
49C22⋊Dic173D411yescharacteristic, complemented
50#14583803—21no—
51#14583803—21no—
52#14583803—21no—
53#14583803—41no—
54#14583803—41no—
55C2×D4×C173—21no—
56C2×D4×C173—21no—
57C2×D4×C173D411yescharacteristic, complemented
58C2×D4×C173C2311yescharacteristic
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C42×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—41no—
66C22×D173—81no—
67Dic346—21no—
68D692—21no—
69D692—82no—
70C22⋊D173—41no—
71C22⋊D173—41no—
72C22⋊D173—41no—
73C22⋊D173—81no—
74C22⋊D173—81no—
75C22⋊D173—81no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—21no—
81C2×Dic173—41no—
82C2×Dic173—81no—
83C4×D173—41no—
84#13063436—41no—
85C23×C173—21no—
86C23×C173C2×D411yescharacteristic
87C23×C173C2×D411yescharacteristic
88D4×C173—41no—
89D4×C173—41no—
90D4×C173—81no—
91D4×C173—81no—
92C2×C4×C173—21no—
93C2×C4×C173—41no—
94C2×C4×C173—41no—
95C2×C4×C173C2×D411yescharacteristic
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173—41no—
102C22×C173—41no—
103C22×C173#12011yescharacteristic
104C2×D173—41no—
105C2×D173—41no—
106C2×D173—42no—
107C2×D173—41no—
108C2×D173—41no—
109C2×D173—81no—
110C2×D173—81no—
111C2×D173—81no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—81no—
115C4×C173—81no—
116Dic173—41no—
117Dic173—41no—
118Dic173—41no—
119Dic173—41no—
120C2×C173—21no—
121C2×C173—41no—
122C2×C173—41no—
123C2×C173—81no—
124C2×C173B(4,𝔽2)11yescharacteristic
125D173—81no—
126D173—81no—
127C173#220011yescharacteristic, complemented, Sylow
128#2200—1731nomaximal, Sylow
129#560—1731no—
130#458—1731no—
131B(4,𝔽2)—1731no—
132#453—1731no—
133#353D17311yescharacteristic, complemented
134#352—1731no—
135AΣL(1,𝔽16)—1731no—
136H(5,𝔽2)—1731no—
137AGL(1,ℤ/8)—3461no—
138#127C2×D17311yescharacteristic
139#126—3461no—
140#123—3461no—
141#123—3461no—
142#123—1731no—
143#121—3461no—
144#121—3461no—
145#121—3461no—
146#120—3461no—
147#120—3461no—
148#120—1731no—
149#104—3461no—
150#100—1731no—
151#99—3461no—
152#99—3461no—
153#99C2×D17311yescharacteristic
154#99C2×D17311yescharacteristic
155C24—1731no—
156#41—3461no—
157#41—6921no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—3461no—
161C2×D4—3461no—
162C2×D4—3461no—
163C2×D4—3461no—
164C2×D4—3461no—
165C2×D4—6921no—
166C2×D4—6921no—
167C2×D4D69211yescharacteristic, complemented
168C2×D4C22×D17311yescharacteristic
169C22×C4—3461no—
170C22×C4—1731no—
171SD16—6921no—
172D8—6921no—
173M16—3461no—
174C4⋊C4—3461no—
175C4⋊C4—3461no—
176C4⋊C4—3461no—
177C4⋊C4—6921no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—3461no—
181C22⋊C4—3461no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—3461no—
185C22⋊C4—1731no—
186C22⋊C4—1731no—
187C22⋊C4—6921no—
188C22⋊C4—6921no—
189C42D69211yescharacteristic, complemented
190C23—21no—
191C23—3461no—
192C23—3461no—
193C23—3461no—
194C23—13841no—
195C23—6921no—
196C23D4×D17311yescharacteristic
197C23D4×D17311yescharacteristic
198Q8—3461no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—3461no—
204D4—13842no—
205D4—13841no—
206D4—13841no—
207D4—13841no—
208D4—6921no—
209D4—6921no—
210D4—6921no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3461no—
215C2×C4—3461no—
216C2×C4—3461no—
217C2×C4—3461no—
218C2×C4—3461no—
219C2×C4—13841no—
220C2×C4—6921no—
221C2×C4—6921no—
222C2×C4C2×D69211yescharacteristic
223C8—6921no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—13841no—
232C22—13841no—
233C22—13841no—
234C22—6921no—
235C22—6921no—
236C22—6922no—
237C22—6921no—
238C22—6921no—
239C22#1914964911yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4—6921no—
245C4—6921no—
246C4—6921no—
247C4—6921no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—13841no—
253C2—13841no—
254C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#353×C173C2
socleC2×C173B(4,𝔽2)
hypercenter#353D173
hypocenterC173#2200
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #26012228C2semidirect
2. #26011682C2semidirect
3. #26011679C2semidirect
4. #26011658C2semidirect
5. #26011280C2semidirect
6. #26011277C2semidirect
7. #353×C173C2semidirect
38. C22×Dic173D4semidirect
49. C22⋊Dic173D4semidirect
57. C2×D4×C173D4semidirect
61. C42×C173D4semidirect
127. C173#2200semidirect
133. #353D173semidirect
167. C2×D4D692semidirect
189. C42D692semidirect

Maximal subgroups

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

subgroupindex
#260122282
#260116822
#260116792
#260116582
#260112802
#260112772
#353×C1732
#2200173