Finite Group Databasebeta

← #36369074

Subgroups — #36369074

Order 22144.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012496C211yesmaximal, characteristic, complemented
2D173⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3#26012350C212yesmaximal, complemented
4#26012349C212yesmaximal, complemented
5#26012038C211yesmaximal, characteristic
6C2×#14583836—21no—
7C2×D173⋊C8—21no—
8#19149746—21no—
9#19149746C2212yes—
10#19149746C2212yes—
11C4⋊#13063443—21no—
12C2×#14583826—21no—
13#19149678C2211yescharacteristic
14#19149675C2211yescharacteristic
15#19149665—21no—
16#19149654C411yescharacteristic
17#19149652—21no—
18#19149650C411yescharacteristic
19C4×D692C411yescharacteristic
20C4×C22⋊D173—21no—
21C42×D173C2211yescharacteristic
22C4×D4×C173C411yescharacteristic
23#14583836—42no—
24D173⋊C8—42no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27#14583826—42no—
28#14583826—42no—
29C2×Dic346—21no—
30C2×D692—21no—
31C2×D692C2×C411yescharacteristic
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C4×D173—21no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C4⋊Dic173C2×C411yescharacteristic
40#14583805—21no—
41C22⋊Dic173—21no—
42#14583803—21no—
43#14583803—21no—
44C4×Dic173D411yescharacteristic
45C4×Dic173D411yescharacteristic
46C4×Dic173C2311yescharacteristic
47C2×D4×C173C2×C411yescharacteristic
48C22×C4×C173—21no—
49C4⋊C4×C173C2×C411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173C2×C411yescharacteristic
52#13063443—42no—
53#13063443—42no—
54C22×D173—21no—
55C22×D173C22⋊C411yescharacteristic
56Dic346—41no—
57Dic346—41no—
58D692—41no—
59D692—42no—
60C22⋊D173—42no—
61C22⋊D173—41no—
62C22⋊D173—41no—
63C22⋊D173—41no—
64C22⋊D173—41no—
65C2×Dic173—21no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C4×D173—21no—
70C4×D173—21no—
71C4×D173—41no—
72C4×D173—42no—
73C4×D173D812yes—
74C4×D173SD1612yes—
75C23×C173—21no—
76D4×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—42no—
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22⋊C411yescharacteristic
81C2×C4×C173C22×C411yescharacteristic
82C22×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173#10212yes—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—41no—
95C4×C173#10212yes—
96Dic173—21no—
97Dic173—41no—
98Dic173C2×D811yescharacteristic
99Dic173C2×SD1611yescharacteristic
100Dic173AGL(1,ℤ/8)11yescharacteristic
101Dic173#13711yescharacteristic
102C2×C173—81no—
103C2×C173#41311yescharacteristic
104C2×C173C2×#10211yescharacteristic
105C2×C173#41711yescharacteristic
106D173—22no—
107D173—81no—
108C173#154711yescharacteristic, complemented, Sylow
109#1547—1731nomaximal, Sylow
110#532—1731no—
111#423—1731no—
112C2×C4⋊C8—1731no—
113#332—1732no—
114#331—1732no—
115C2×#41—3461no—
116C2×M16—3461no—
117C22×C8—3461no—
118C4⋊Q8—1731no—
119#127—1731no—
120#124—3461no—
121#121—3461no—
122#121—1731no—
123#121—1731no—
124C4×D4—3461no—
125C4×D4—1731no—
126C4×D4#1062192411yescharacteristic
127C2×C42—1731no—
128C4⋊C8—3461no—
129C4⋊C8—3461no—
130C4⋊C8—1732no—
131C4⋊C8—1732no—
132#41—6922no—
133#41—6922no—
134C2×Q8—3461no—
135C2×D4—3461no—
136C2×D4—3461no—
137C2×D4—3461no—
138C2×D4—3461no—
139C2×D4—1731no—
140C2×D4C2×#1062192411yescharacteristic
141C22×C4—21no—
142C22×C4—3461no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—1731no—
146M16—6922no—
147C2×C8—3462no—
148C2×C8—3462no—
149C2×C8—6922no—
150C4⋊C4—3461no—
151C4⋊C4—1731no—
152C4⋊C4C2×#1062192411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C42—1731no—
158C42—1731no—
159C42—1731no—
160C42C2×#1062192411yescharacteristic
161C23—21no—
162C23—3461no—
163C23—1731no—
164Q8—6921no—
165Q8—6921no—
166D4—41no—
167D4—6921no—
168D4—6922no—
169D4—6921no—
170D4—6921no—
171D4—6921no—
172D4—6921no—
173D4—6922no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—1731no—
180C2×C4—1731no—
181C2×C4—1731no—
182C2×C4—1732no—
183C2×C4—1732no—
184C2×C4—6921no—
185C2×C4—6922no—
186C2×C4C22⋊#1062192411yescharacteristic
187C2×C4C22⋊#1062192411yescharacteristic
188C2×C4C22×#1062192411yescharacteristic
189C8—6922no—
190C8—6922no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—3461no—
195C22—3461no—
196C22—1732no—
197C22—6921no—
198C22—6921no—
199C22—6921no—
200C22C2×C22⋊#1062192411yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—3461no—
205C4—1731no—
206C4—1731no—
207C4—1731no—
208C4—1731no—
209C4—6921no—
210C4#1914974112yes—
211C2—81no—
212C2—3462no—
213C2—13841no—
214C2#2601247711yescharacteristic
215C2C2×#1914974111yescharacteristic
216C2#2601248211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radical#26012038C2
cosocleC4×Dic173C23
hypercenterC4×D4#10621924
hypocenterC173#1547
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012496C2semidirect
2. D173⋊(C4⋊C8)C2semidirect
3. #26012350C2semidirect
4. #26012349C2semidirect
108. C173#1547semidirect

Maximal subgroups

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

subgroupindex
#260124962
D173⋊(C4⋊C8)2
#260123502
#260123492
#260120382
#1547173