Finite Group Databasebeta

← #36355557

Subgroups — #36355557

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#26011681C211yesmaximal, characteristic, complemented
3#26011659C211yesmaximal, characteristic
4#26011283C211yesmaximal, characteristic, complemented
5#26011281C211yesmaximal, characteristic, complemented
6#26011280C211yesmaximal, characteristic, complemented
7#458×C173C211yesmaximal, characteristic, complemented
8#19149704—21no—
9#19149675C2211yescharacteristic
10#19149665—21no—
11#19149665—21no—
12#19149664—21no—
13#19149660—21no—
14#19149660C2211yescharacteristic
15#19149644—21no—
16C4×C22⋊D173—21no—
17#19149639—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×D692D411yescharacteristic
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33#14583813—41no—
34#14583811—41no—
35#14583810—21no—
36#14583805—21no—
37#14583805—21no—
38#14583805—41no—
39C22⋊Dic173—21no—
40C22⋊Dic173—21no—
41C22⋊Dic173—41no—
42C22⋊Dic173D411yescharacteristic
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
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
62C22×D173—21no—
63Dic346—42no—
64D692—41no—
65C22⋊D173—42no—
66C22⋊D173—42no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C4×D173—81no—
73#13063436—41no—
74C23×C173—21no—
75C23×C173—21no—
76C23×C173C2×D411yescharacteristic
77C23×C173C2×D411yescharacteristic
78Q8×C173—21no—
79D4×C173—21no—
80D4×C173—41no—
81D4×C173—41no—
82D4×C173—41no—
83D4×C173—41no—
84C2×C4×C173—21no—
85C2×C4×C173—21no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173C2×D411yescharacteristic
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—41no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—81no—
98C22×C173—81no—
99C22×C173#12011yescharacteristic
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C4×C173—21no—
104C4×C173—41no—
105C4×C173—41no—
106C4×C173—41no—
107C4×C173—81no—
108Dic173—22no—
109Dic173—41no—
110Dic173—41no—
111Dic173—81no—
112C2×C173—21no—
113C2×C173—41no—
114C2×C173—41no—
115C2×C173—81no—
116C2×C173B(4,𝔽2)11yescharacteristic
117D173—81no—
118C173#219811yescharacteristic, complemented, Sylow
119#2198—1731nomaximal, Sylow
120#559—1731no—
121#458—1731no—
122#458D17311yescharacteristic, complemented
123#453—1731no—
124#353—1731no—
125#352—1731no—
126#352—1731no—
127H(5,𝔽2)C2×D17311yescharacteristic
128AGL(1,ℤ/8)—3461no—
129#127—1731no—
130#124—3461no—
131#124—3461no—
132#123—21no—
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#100—1731no—
142#99—21no—
143#99—3461no—
144#99—1731no—
145#99C2×D17311yescharacteristic
146#41—21no—
147#41—41no—
148C2×Q8—3461no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—3461no—
153C2×D4—3461no—
154C2×D4—1731no—
155C2×D4C22×D17311yescharacteristic
156C22×C4—3461no—
157C22×C4—1731no—
158C22×C4C22⋊D17311yescharacteristic
159SD16—6921no—
160D8—6921no—
161M16—3461no—
162C4⋊C4—21no—
163C4⋊C4—41no—
164C4⋊C4—3461no—
165C4⋊C4—3461no—
166C4⋊C4—6921no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—41no—
170C22⋊C4—3461no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—1731no—
176C22⋊C4—6921no—
177C22⋊C4C22⋊D17311yescharacteristic
178C42—3461no—
179C42—3461no—
180C42—1731no—
181C23—21no—
182C23—21no—
183C23—3461no—
184C23D4×D17311yescharacteristic
185C23C2×C22⋊D17311yescharacteristic
186Q8—21no—
187Q8—6922no—
188D4—21no—
189D4—41no—
190D4—41no—
191D4—41no—
192D4—41no—
193D4—6921no—
194D4—6922no—
195D4—6922no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—13841no—
207C2×C4—6921no—
208C2×C4—6921no—
209C2×C4D4×D17311yescharacteristic
210C8—6921no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—81no—
218C22—6921no—
219C22—6921no—
220C22—6921no—
221C22#1914964711yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—3462no—
228C4—13841no—
229C4—6921no—
230C4—6921no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—81no—
235C2—13841no—
236C2#2601167711yescharacteristic
subgroupG/H
centerC2#26011677
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#458×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011659C2
hypercenter#458D173
hypocenterC173#2198
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. #26012225C2semidirect
2. #26011681C2semidirect
4. #26011283C2semidirect
5. #26011281C2semidirect
6. #26011280C2semidirect
7. #458×C173C2semidirect
118. C173#2198semidirect
122. #458D173semidirect

Maximal subgroups

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

subgroupindex
#260122252
#260116812
#260116592
#260112832
#260112812
#260112802
#458×C1732
#2198173