Finite Group Databasebeta

← #36352883

Subgroups — #36352883

Order 22144.

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

normal subgroups only

This group has 208 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#26011900C211yesmaximal, characteristic, complemented
3#26011885C211yesmaximal, characteristic, complemented
4#26011850C211yesmaximal, characteristic, complemented
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic, complemented
7#329×C173C211yesmaximal, characteristic, complemented
8C2×Dic692—21no—
9C2×#14583815—21no—
10#19149679C2211yescharacteristic
11#19149671C2211yescharacteristic
12#19149667C2211yescharacteristic
13#19149663—21no—
14#19149661—21no—
15#19149656—21no—
16#19149655—21no—
17#19149652—21no—
18#19149651—21no—
19#19149613—21no—
20#19149611—21no—
21#19149603—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—
34Dic692—42no—
35#14583815—42no—
36C2×#13063436—21no—
37C4⋊Dic173—21no—
38C4⋊Dic173—21no—
39C4⋊Dic173—21no—
40C4⋊Dic173D411yescharacteristic
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43#14583805—21no—
44#14583805—21no—
45C22⋊Dic173—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49C2×Q8×C173D411yescharacteristic
50C2×D4×C173D411yescharacteristic
51C2×C8×C173—21no—
52C2×C8×C173—21no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C42×C173C2311yescharacteristic
56C22×D173—21no—
57Dic346—41no—
58Dic346—41no—
59Dic346—41no—
60D692—41no—
61C22⋊D173—81no—
62C22⋊D173—81no—
63C2×Dic173—21no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173—41no—
67C2×Dic173—81no—
68C4×D173—81no—
69#13063436—41no—
70C23×C173—21no—
71Q8×C173—41no—
72D4×C173—41no—
73C2×C4×C173—21no—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173C2×D411yescharacteristic
77C8×C173—22no—
78C8×C173—22no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173—41no—
82C22×C173#12011yescharacteristic
83C2×D173—41no—
84C2×D173—41no—
85C2×D173—41no—
86C4×C173—21no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—41no—
90Dic173—41no—
91Dic173—41no—
92Dic173—41no—
93Dic173—81no—
94C2×C173—81no—
95C2×C173#45311yescharacteristic
96C2×C173#45511yescharacteristic
97C2×C173#45611yescharacteristic
98D173—81no—
99C173#166111yescharacteristic, complemented, Sylow
100#1661—1731nomaximal, Sylow
101#562—1731no—
102#501—1731no—
103#497—1731no—
104#489—1731no—
105#330—1731no—
106#329—1731no—
107#329D17311yescharacteristic, complemented
108C2×Q16—3461no—
109C2×SD16—3461no—
110C4⋊Q8—1731no—
111#125—1731no—
112#124—1731no—
113#124C2×D17311yescharacteristic
114#123—3461no—
115#123—3461no—
116C22⋊Q8—3461no—
117C22⋊Q8—3461no—
118#121—3461no—
119#121—3461no—
120#107—3461no—
121#106—3461no—
122#103—3461no—
123#102—3461no—
124#97—1731no—
125#97C2×D17311yescharacteristic
126#97C2×D17311yescharacteristic
127C2×Q8—3461no—
128C2×Q8—1731no—
129C2×Q8D69211yescharacteristic
130C2×D4—3461no—
131C2×D4—3461no—
132C2×D4—1731no—
133C2×D4D69211yescharacteristic
134C22×C4—3461no—
135C22×C4—3461no—
136Q16—6922no—
137SD16—6922no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3461no—
141C4⋊C4—3461no—
142C4⋊C4—3461no—
143C4⋊C4—3461no—
144C4⋊C4—3461no—
145C4⋊C4—3461no—
146C4⋊C4—3461no—
147C4⋊C4—1731no—
148C4⋊C4—1731no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—3461no—
152C22⋊C4—3461no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C42C22×D17311yescharacteristic
156C23—21no—
157C23—3461no—
158Q8—41no—
159Q8—6921no—
160Q8—6921no—
161Q8—6921no—
162D4—41no—
163D4—13841no—
164D4—13841no—
165D4—6921no—
166C2×C4—21no—
167C2×C4—3461no—
168C2×C4—3461no—
169C2×C4—3461no—
170C2×C4—13841no—
171C2×C4—13841no—
172C2×C4—6921no—
173C2×C4D4×D17311yescharacteristic
174C2×C4D4×D17311yescharacteristic
175C2×C4C2×D69211yescharacteristic
176C8—22no—
177C8—22no—
178C8—6921no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—6921no—
183C22—6921no—
184C22—6921no—
185C22#1914964911yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—13841no—
191C4—6921no—
192C4—6921no—
193C4—6921no—
194C2—81no—
195C2—13841no—
196C2#2601166011yescharacteristic
197C2#2601167111yescharacteristic
198C2#2601167511yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173#120
prime radical#26012235C2
hypercenter#329D173
hypocenterC173#1661
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. #26011900C2semidirect
3. #26011885C2semidirect
4. #26011850C2semidirect
5. #26011232C2semidirect
6. #26011230C2semidirect
7. #329×C173C2semidirect
99. C173#1661semidirect
107. #329D173semidirect

Maximal subgroups

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

subgroupindex
#260122352
#260119002
#260118852
#260118502
#260112322
#260112302
#329×C1732
#1661173