Finite Group Databasebeta

← #49157705

Subgroups — #49157705

Order 37504.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#34036691C211yesmaximal, characteristic, complemented
2#34036265C212yesmaximal, complemented
3#34036038C211yesmaximal, characteristic, complemented
4#34035864C212yesmaximal, complemented
5#323×C293C211yesmaximal, characteristic, complemented
6C22×D1172C2211yescharacteristic, complemented
7C22×C22⋊D293—21no—
8C22×C22⋊D293—21no—
9C2×D2344—22no—
10#22627608—22no—
11#22627598—21no—
12#22627598C2212yes—
13#22627594—22no—
14C2×#17989779—21no—
15#22627546—22no—
16#22627531C2211yescharacteristic, complemented
17C2×C22⋊C4×C293C2211yescharacteristic
18C22⋊C8×C293C2212yescomplemented
19C23×D293—21no—
20C23×D293—21no—
21C2×D1172—21no—
22C2×D1172—22no—
23C2×D1172—42no—
24C2×D1172D412yescomplemented
25C2×C22⋊D293—22no—
26C2×C22⋊D293—22no—
27C2×C22⋊D293—42no—
28C2×C22⋊D293—42no—
29C2×C22⋊D293—42no—
30C2×C22⋊D293—42no—
31C2×C22⋊D293—42no—
32C22×Dic293—21no—
33C22×Dic293—21no—
34D2344—44no—
35C4⋊Dic293—21no—
36C4⋊Dic293D412yescomplemented
37#17989779—22no—
38#17989779—22no—
39#17989779—42no—
40C24×C293D411yescharacteristic
41C22×C4×C293D411yescharacteristic
42C22×C4×C293C2311yescharacteristic
43C2×C8×C293—22no—
44C22⋊C4×C293—22no—
45C22×D293—22no—
46C22×D293—22no—
47C22×D293—41no—
48C22×D293—41no—
49C22×D293—42no—
50C22×D293—41no—
51C22×D293—41no—
52C22×D293—41no—
53C22×D293—41no—
54C22×D293—81no—
55C22×D293—81no—
56D1172—24no—
57D1172—42no—
58D1172—42no—
59D1172—81no—
60C22⋊D293—44no—
61C22⋊D293—48no—
62C22⋊D293—42no—
63C22⋊D293—42no—
64C2×Dic293—22no—
65C2×Dic293—22no—
66C2×Dic293—41no—
67C2×Dic293—42no—
68C2×Dic293—41no—
69C23×C293—21no—
70C23×C293—21no—
71C23×C293—41no—
72C23×C293—41no—
73C23×C293D812yescomplemented
74C23×C293C2×D411yescharacteristic
75C2×C4×C293—21no—
76C2×C4×C293—41no—
77C2×C4×C293—41no—
78C2×C4×C293C2×D412yes—
79C8×C293—42no—
80C22×C293—21no—
81C22×C293—21no—
82C22×C293—22no—
83C22×C293—42no—
84C22×C293—41no—
85C22×C293—41no—
86C22×C293—81no—
87C22×C293#12011yescharacteristic
88C22×C293C2×D811yescharacteristic
89C22×C293AGL(1,ℤ/8)11yescharacteristic
90C2×D293—42no—
91C2×D293—42no—
92C2×D293—42no—
93C2×D293—42no—
94C2×D293—42no—
95C2×D293—42no—
96C2×D293—81no—
97C2×D293—81no—
98C2×D293—84no—
99C2×D293—81no—
100C2×D293—81no—
101C4×C293—22no—
102C4×C293—41no—
103Dic293—42no—
104Dic293—42no—
105C2×C293—21no—
106C2×C293—21no—
107C2×C293—42no—
108C2×C293#44711yescharacteristic
109C2×C293#45311yescharacteristic
110C2×C293B(4,𝔽2)11yescharacteristic
111D293—82no—
112D293—82no—
113C293#159511yescharacteristic, complemented, Sylow
114#1595—2931nomaximal, Sylow
115#534—2931no—
116#447—2932no—
117#392—2931no—
118#327—2932no—
119#323D29311yescharacteristic, complemented
120C22×D4—2931no—
121C22×D4—5861no—
122C22×D4—5861no—
123C2×D8—5862no—
124#123—5862no—
125#121—2932no—
126#121—5861no—
127#120—5862no—
128C2×C22⋊C4—5861no—
129C2×C22⋊C4C2×D29311yescharacteristic
130#102—5862no—
131C22⋊C8C2×D29312yescomplemented
132#95—2931no—
133C24—5861no—
134C24—5861no—
135C24D117211yescharacteristic
136C2×D4—2932no—
137C2×D4—5861no—
138C2×D4—5862no—
139C2×D4—5862no—
140C2×D4—5862no—
141C2×D4—11722no—
142C2×D4—11722no—
143C2×D4—11722no—
144C2×D4—11722no—
145C2×D4—11722no—
146C2×D4—11722no—
147C22×C4—5861no—
148C22×C4—5861no—
149C22×C4D117211yescharacteristic
150C22×C4C22×D29311yescharacteristic
151D8—11724no—
152C2×C8—22no—
153C4⋊C4—2932no—
154C4⋊C4—5861no—
155C22⋊C4—22no—
156C22⋊C4—5862no—
157C22⋊C4—5862no—
158C22⋊C4—11722no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—23441no—
164C23—23441no—
165C23—5862no—
166C23—5862no—
167C23—11721no—
168C23—11721no—
169C23—11721no—
170C23—11722no—
171C23—11721no—
172C23—11721no—
173C23—11721no—
174C23D234412yescomplemented
175C23C2×D117211yescharacteristic
176D4—23441no—
177D4—5864no—
178D4—11722no—
179D4—11722no—
180D4—11724no—
181D4—11728no—
182D4—11722no—
183D4—11722no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5862no—
188C2×C4—5862no—
189C2×C4—11721no—
190C2×C4—11722no—
191C2×C4—11721no—
192C2×C4D4×D29312yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—23441no—
202C22—23441no—
203C22—23444no—
204C22—23441no—
205C22—23441no—
206C22—11722no—
207C22—11722no—
208C22—11722no—
209C22—11722no—
210C22—11722no—
211C22—11722no—
212C22#2262759411yescharacteristic
213C22C2×D234411yescharacteristic
214C22#2262765211yescharacteristic
215C4—22no—
216C4—41no—
217C4—11722no—
218C4—11722no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—23442no—
223C2—23442no—
224C2#3403626511yescharacteristic
225C2#3403629911yescharacteristic
226C2#3403631711yescharacteristic
subgroupG/H
centerC22#22627594
commutator, cosocleC22×C4×C293C23
FrattiniC22×C4C22×D293
Fitting#323×C293C2
socleC22×C293#120
hypercenter#323D293
hypocenterC293#1595
LCSG ▷ C22×C4×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #34036691C2semidirect
2. #34036265C2semidirect
3. #34036038C2semidirect
4. #34035864C2semidirect
5. #323×C293C2semidirect
6. C22×D1172C22semidirect
16. #22627531C22semidirect
18. C22⋊C8×C293C22semidirect
24. C2×D1172D4semidirect
36. C4⋊Dic293D4semidirect
73. C23×C293D8semidirect
113. C293#1595semidirect
119. #323D293semidirect
131. C22⋊C8C2×D293semidirect
174. C23D2344semidirect

Maximal subgroups

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

subgroupindex
#340366912
#340362652
#340360382
#340358642
#323×C2932
#1595293