Finite Group Databasebeta

← #36369826

Subgroups — #36369826

Order 22144.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#26012478C211yesmaximal, characteristic
2#26012471C211yesmaximal, characteristic
3#26012394C212yesmaximal, complemented
4#26012393C212yesmaximal, complemented
5#26012063C211yesmaximal, characteristic
6C2×#14583835—21no—
7#19149758—21no—
8#19149740—21no—
9#19149739C2212yes—
10#19149737—21no—
11#19149737C2212yes—
12C2×#14583825C2211yescharacteristic
13C22×Dic346C411yescharacteristic, complemented
14#19149678—21no—
15#19149665—21no—
16#19149663C2211yescharacteristic
17#19149662C2211yescharacteristic
18#19149660C411yescharacteristic, complemented
19#19149657—21no—
20#19149656—21no—
21#19149635C411yescharacteristic, complemented
22#123×C173C411yescharacteristic, complemented
23#14583835—22no—
24#14583835—41no—
25C2×#13063443—41no—
26C4⋊#10621924—41no—
27C22⋊#10621924—22no—
28C4×#10621924—41no—
29#14583825—41no—
30#14583825—41no—
31C2×Dic346—21no—
32C2×Dic346—21no—
33C2×Dic346—41no—
34C2×Dic346—41no—
35C2×Dic346D411yescharacteristic
36C2×C22⋊D173D411yescharacteristic
37C2×C22⋊D173C2311yescharacteristic
38C22×Dic173C2×C411yescharacteristic
39C22×Dic173C2×C411yescharacteristic
40C22×Dic173D411yescharacteristic
41C2×C4×D173D411yescharacteristic
42C4⋊Dic173—21no—
43#14583805—21no—
44#14583805—21no—
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173C2×C411yescharacteristic
48#14583803—21no—
49#14583803—21no—
50C4×Dic173—21no—
51C2×D4×C173C2×C411yescharacteristic
52C22×C4×C173C2×C411yescharacteristic
53C4⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173C2×C411yescharacteristic
56C2×#10621924—42no—
57#13063443—42no—
58C22×D173C2×D411yescharacteristic
59Dic346—41no—
60Dic346—42no—
61Dic346—41no—
62Dic346—41no—
63C22⋊D173—41no—
64C22⋊D173—41no—
65C2×Dic173—21no—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—41no—
72C2×Dic173C22⋊C411yescharacteristic
73C2×Dic173C22⋊C411yescharacteristic
74C2×Dic173C2×D411yescharacteristic
75C4×D173—21no—
76C4×D173—21no—
77C23×C173C22⋊C411yescharacteristic
78C23×C173C22×C411yescharacteristic
79D4×C173—41no—
80C2×C4×C173—21no—
81C2×C4×C173—21no—
82C2×C4×C173—41no—
83C2×C4×C173C22⋊C411yescharacteristic
84C22×C173—21no—
85C22×C173—21no—
86C22×C173—41no—
87C22×C173—41no—
88C22×C173C2×C22⋊C411yescharacteristic
89C2×D173—21no—
90C2×D173—41no—
91#10621924—82no—
92C4×C173—21no—
93C4×C173—41no—
94C4×C173—41no—
95Dic173—21no—
96Dic173—21no—
97Dic173—41no—
98Dic173—41no—
99Dic173#9911yescharacteristic
100Dic173#9911yescharacteristic
101C2×C173—21no—
102C2×C173—41no—
103C2×C173—41no—
104C2×C173C2×#9911yescharacteristic
105D173—41no—
106C173#212311yescharacteristic, complemented, Sylow
107#2123—1731nomaximal, Sylow
108#536—1731no—
109#413—1731no—
110#410—1731no—
111#352—1732no—
112#352—1732no—
113C2×#41—1731no—
114C22×Q8—1731no—
115C2×M16—3461no—
116C4⋊Q8—3461no—
117#124—3461no—
118#123—1731no—
119#123—1731no—
120#123—1731no—
121#123#1062192411yescharacteristic, complemented
122C22⋊Q8—3461no—
123C22⋊Q8—3461no—
124#117—3461no—
125#117—1731no—
126#101—3461no—
127#100—1732no—
128#99—3461no—
129#99—1732no—
130#41—6921no—
131#41—6921no—
132C2×Q8—3461no—
133C2×Q8—3461no—
134C2×Q8—1731no—
135C2×Q8—6921no—
136C2×Q8—6921no—
137C2×D4—1731no—
138C2×D4—1731no—
139C2×D4C2×#1062192411yescharacteristic
140C22×C4—1731no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4C2×#1062192411yescharacteristic
145M16—3462no—
146M16—6921no—
147C2×C8—6921no—
148C4⋊C4—21no—
149C4⋊C4—3461no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—3461no—
153C4⋊C4—6921no—
154C22⋊C4—21no—
155C22⋊C4—3461no—
156C22⋊C4—3461no—
157C22⋊C4—3461no—
158C22⋊C4—3462no—
159C22⋊C4—1731no—
160C22⋊C4C2×#1062192411yescharacteristic
161C42—3461no—
162C42—6921no—
163C23—1731no—
164C23C22⋊#1062192411yescharacteristic
165C23C22×#1062192411yescharacteristic
166Q8—6921no—
167Q8—6922no—
168Q8—6921no—
169Q8—6921no—
170D4—41no—
171D4—6921no—
172D4—6921no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—3461no—
179C2×C4—3461no—
180C2×C4—3461no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—3461no—
184C2×C4—1731no—
185C2×C4—1731no—
186C2×C4—1731no—
187C2×C4—6922no—
188C2×C4—6921no—
189C2×C4C22⋊#1062192411yescharacteristic
190C8—6922no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—3461no—
196C22—6921no—
197C22C2×C22⋊#1062192411yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—3461no—
202C4—3461no—
203C4—13842no—
204C4—1731no—
205C4—1731no—
206C4—6921no—
207C4—6921no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—6921no—
212C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#123×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583825C22
cosocleC2×C22⋊D173C23
hypercenter#123#10621924
hypocenterC173#2123
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #26012394C2semidirect
4. #26012393C2semidirect
13. C22×Dic346C4semidirect
18. #19149660C4semidirect
21. #19149635C4semidirect
22. #123×C173C4semidirect
106. C173#2123semidirect
121. #123#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124782
#260124712
#260123942
#260123932
#260120632
#2123173