Finite Group Databasebeta

← #36355560

Subgroups — #36355560

Order 22144.

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

normal subgroups only

This group has 246 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#26012225C211yesmaximal, characteristic, complemented
2#26011682C211yesmaximal, characteristic, complemented
3#26011680C211yesmaximal, characteristic, complemented
4#26011661C211yesmaximal, characteristic
5#26011283C211yesmaximal, characteristic, complemented
6#26011282C211yesmaximal, characteristic, complemented
7#352×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149707—21no—
10#19149675C2211yescharacteristic
11#19149665—21no—
12#19149665—21no—
13#19149664—21no—
14#19149663—21no—
15#19149662—21no—
16#19149660C2211yescharacteristic
17#19149644—21no—
18C4×C22⋊D173—21no—
19#19149639—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×D692D411yescharacteristic
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35D4×D173—41no—
36C22×Dic173D411yescharacteristic
37C2×C4×D173—21no—
38#14583815—41no—
39D1384—41no—
40C4⋊Dic173—41no—
41#14583805—21no—
42#14583805—21no—
43#14583805—41no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173D411yescharacteristic
47#14583803—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—41no—
51#14583803—41no—
52C4×Dic173—21no—
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—
64C22×D173—21no—
65Dic346—21no—
66Dic346—42no—
67D692—21no—
68D692—41no—
69C22⋊D173—42no—
70C22⋊D173—42no—
71C22⋊D173—41no—
72C22⋊D173—41no—
73C22⋊D173—41no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—41no—
80C2×Dic173—41no—
81C4×D173—41no—
82C4×D173—81no—
83C23×C173C2×D411yescharacteristic
84C23×C173C2×D411yescharacteristic
85D4×C173—41no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173C2×D411yescharacteristic
91C8×C173—41no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173#12011yescharacteristic
97C2×D173—41no—
98C2×D173—41no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—81no—
102C2×D173—81no—
103C4×C173—21no—
104C4×C173—41no—
105C4×C173—41no—
106C4×C173—81no—
107Dic173—22no—
108Dic173—41no—
109Dic173—41no—
110Dic173—41no—
111Dic173—81no—
112C2×C173—21no—
113C2×C173—41no—
114C2×C173—41no—
115C2×C173B(4,𝔽2)11yescharacteristic
116D173—81no—
117D173—81no—
118C173#219811yescharacteristic, complemented, Sylow
119#2198—1731nomaximal, Sylow
120#559—1731no—
121#458—1731no—
122#458—1731no—
123#453—1731no—
124#353—1731no—
125#352—1731no—
126#352D17311yescharacteristic, complemented
127H(5,𝔽2)—1731no—
128AGL(1,ℤ/8)—3461no—
129#127—1731no—
130#124—3461no—
131#124—3461no—
132#123—3461no—
133#123—3461no—
134#123—3461no—
135#123—1731no—
136#123C2×D17311yescharacteristic
137C4×Q8—3461no—
138C4×D4—3461no—
139C4×D4—3461no—
140#104—3461no—
141#100C2×D17311yescharacteristic
142#99—3461no—
143#99—3461no—
144#99—1731no—
145#99C2×D17311yescharacteristic
146#41—3461no—
147#41—6921no—
148C2×Q8—3461no—
149C2×D4—3461no—
150C2×D4—3461no—
151C2×D4—3461no—
152C2×D4—3461no—
153C2×D4—1731no—
154C2×D4—6921no—
155C2×D4C22×D17311yescharacteristic
156C22×C4—3461no—
157C22×C4—1731no—
158C22×C4D69211yescharacteristic
159SD16—6921no—
160D8—6921no—
161M16—21no—
162C4⋊C4—21no—
163C4⋊C4—3461no—
164C4⋊C4—3461no—
165C4⋊C4—6921no—
166C4⋊C4—6921no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—3461no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—1731no—
175C22⋊C4—6921no—
176C22⋊C4—6921no—
177C22⋊C4D69211yescharacteristic
178C42—3461no—
179C42—3461no—
180C42—1731no—
181C23—3461no—
182C23—3461no—
183C23—3461no—
184C23D4×D17311yescharacteristic
185C23C2×D69211yescharacteristic
186Q8—3461no—
187Q8—6922no—
188D4—41no—
189D4—3461no—
190D4—6921no—
191D4—6922no—
192D4—6922no—
193D4—6921no—
194D4—6921no—
195D4—6921no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—13841no—
206C2×C4—6921no—
207C2×C4—6921no—
208C2×C4—6921no—
209C2×C4D4×D17311yescharacteristic
210C8—41no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—13841no—
216C22—13841no—
217C22—6921no—
218C22—6921no—
219C22—6921no—
220C22—6921no—
221C22#1914964911yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—81no—
226C4—3462no—
227C4—13841no—
228C4—6921no—
229C4—6921no—
230C4—6921no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—13841no—
235C2—13841no—
236C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#352×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011661C2
hypercenter#352D173
hypocenterC173#2198
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. #26012225C2semidirect
2. #26011682C2semidirect
3. #26011680C2semidirect
5. #26011283C2semidirect
6. #26011282C2semidirect
7. #352×C173C2semidirect
118. C173#2198semidirect
126. #352D173semidirect

Maximal subgroups

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

subgroupindex
#260122252
#260116822
#260116802
#260116612
#260112832
#260112822
#352×C1732
#2198173