Finite Group Databasebeta

← #36352888

Subgroups — #36352888

Order 22144.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012235C211yesmaximal, characteristic
2#26011860C211yesmaximal, characteristic, complemented
3#26011851C211yesmaximal, characteristic, complemented
4#26011850C211yesmaximal, characteristic, complemented
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic, complemented
7#329×C173C211yesmaximal, characteristic, complemented
8#19149679C2211yescharacteristic
9#19149671C2211yescharacteristic
10#19149667C2211yescharacteristic
11#19149663—21no—
12#19149661—21no—
13#19149656—21no—
14#19149655—21no—
15#19149652—21no—
16#19149651—21no—
17#19149604—21no—
18#19149604—21no—
19#19149604—21no—
20#19149603—21no—
21#19149601—21no—
22#19149600—21no—
23#19149590C2211yescharacteristic
24#124×C173C2211yescharacteristic
25#97×C173C2211yescharacteristic
26#97×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C2×C4×D173—21no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173D411yescharacteristic
39C4⋊Dic173D411yescharacteristic
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43C22⋊Dic173—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47C2×Q8×C173D411yescharacteristic
48C2×D4×C173D411yescharacteristic
49C2×C8×C173—21no—
50C2×C8×C173—21no—
51C22⋊C4×C173—21no—
52C22⋊C4×C173—21no—
53C42×C173C2311yescharacteristic
54C22×D173—21no—
55Dic346—41no—
56Dic346—42no—
57D692—41no—
58C22⋊D173—81no—
59C22⋊D173—81no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—21no—
63C2×Dic173—41no—
64C2×Dic173—81no—
65C4×D173—81no—
66#13063436—41no—
67C23×C173—21no—
68Q8×C173—41no—
69D4×C173—41no—
70C2×C4×C173—21no—
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173C2×D411yescharacteristic
74C8×C173—41no—
75C8×C173—41no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173—41no—
79C22×C173#12011yescharacteristic
80C2×D173—41no—
81C2×D173—41no—
82C2×D173—41no—
83C4×C173—21no—
84C4×C173—21no—
85C4×C173—21no—
86C4×C173—41no—
87Dic173—41no—
88Dic173—41no—
89Dic173—41no—
90Dic173—81no—
91C2×C173—81no—
92C2×C173#45411yescharacteristic
93C2×C173#45511yescharacteristic
94C2×C173#45511yescharacteristic
95D173—81no—
96C173#166211yescharacteristic, complemented, Sylow
97#1662—1731nomaximal, Sylow
98#562—1731no—
99#491—1731no—
100#489—1731no—
101#489—1731no—
102#330—1731no—
103#329—1731no—
104#329D17311yescharacteristic, complemented
105C4⋊Q8—1731no—
106#125—1731no—
107#124—1731no—
108#124C2×D17311yescharacteristic
109#123—3461no—
110#123—3461no—
111C22⋊Q8—3461no—
112C22⋊Q8—3461no—
113#121—3461no—
114#121—3461no—
115#103—3461no—
116#103—3461no—
117#103—3461no—
118#103—3461no—
119#102—3461no—
120#102—3461no—
121#97—1731no—
122#97C2×D17311yescharacteristic
123#97C2×D17311yescharacteristic
124C2×Q8—3461no—
125C2×Q8—1731no—
126C2×Q8D69211yescharacteristic
127C2×D4—3461no—
128C2×D4—3461no—
129C2×D4—1731no—
130C2×D4D69211yescharacteristic
131C22×C4—3461no—
132C22×C4—3461no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3461no—
136C4⋊C4—3461no—
137C4⋊C4—3461no—
138C4⋊C4—3461no—
139C4⋊C4—3461no—
140C4⋊C4—3461no—
141C4⋊C4—3461no—
142C4⋊C4—1731no—
143C4⋊C4—1731no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—3461no—
147C22⋊C4—3461no—
148C22⋊C4—3461no—
149C22⋊C4—3461no—
150C42C22×D17311yescharacteristic
151C23—21no—
152C23—3461no—
153Q8—41no—
154Q8—6922no—
155Q8—6921no—
156D4—41no—
157D4—13841no—
158D4—13841no—
159D4—6921no—
160C2×C4—21no—
161C2×C4—3461no—
162C2×C4—3461no—
163C2×C4—3461no—
164C2×C4—13841no—
165C2×C4—13841no—
166C2×C4—6921no—
167C2×C4D4×D17311yescharacteristic
168C2×C4D4×D17311yescharacteristic
169C2×C4C2×D69211yescharacteristic
170C8—41no—
171C8—41no—
172C8—6921no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—6921no—
177C22—6921no—
178C22—6921no—
179C22#1914964911yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—13841no—
185C4—6921no—
186C4—6921no—
187C4—6921no—
188C2—81no—
189C2—13841no—
190C2#2601166511yescharacteristic
191C2#2601167011yescharacteristic
192C2#2601167111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173#120
prime radical#26012235C2
hypercenter#329D173
hypocenterC173#1662
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26011860C2semidirect
3. #26011851C2semidirect
4. #26011850C2semidirect
5. #26011232C2semidirect
6. #26011230C2semidirect
7. #329×C173C2semidirect
96. C173#1662semidirect
104. #329D173semidirect

Maximal subgroups

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

subgroupindex
#260122352
#260118602
#260118512
#260118502
#260112322
#260112302
#329×C1732
#1662173