Finite Group Databasebeta

← #49174201

Subgroups — #49174201

Order 37504.

286 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 58 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#34037101C211yesmaximal, characteristic, complemented
2#34037071C211yesmaximal, characteristic, complemented
3#34036982C212yesmaximal, complemented
4#34036980C212yesmaximal, complemented
5#34036639C211yesmaximal, characteristic
6C2×#17989811—21no—
7C2×C4⋊#11924078—21no—
8C2×C22⋊#11924078—21no—
9#22627681—21no—
10#22627681C2212yes—
11(C2×#11924078)⋊C4C2212yes—
12C2×#17989801C411yescharacteristic, complemented
13C22×C4×D293C2211yescharacteristic
14#22627606—21no—
15#22627604C2211yescharacteristic
16#22627603C411yescharacteristic, complemented
17#22627595C2211yescharacteristic
18#22627592—21no—
19D4×Dic293—21no—
20#22627584—21no—
21#22627582C411yescharacteristic, complemented
22#121×C293C411yescharacteristic, complemented
23C22×#11924078—22no—
24#17989811—42no—
25C2×#14556880—22no—
26C4⋊#11924078—42no—
27C22⋊#11924078—42no—
28C23×D293D411yescharacteristic
29#17989801—42no—
30C2×Dic586C2×C411yescharacteristic
31C2×C22⋊D293—21no—
32C2×C22⋊D293—21no—
33C22×Dic293—21no—
34C22×Dic293D411yescharacteristic
35C2×C4×D293—42no—
36C2×C4×D293C2×C411yescharacteristic
37C2×C4×D293D411yescharacteristic
38C2×C4×D293D411yescharacteristic
39C2×C4×D293C2311yescharacteristic
40C4⋊Dic293—21no—
41C4⋊Dic293C2×C411yescharacteristic
42#17989781—21no—
43C22⋊Dic293—21no—
44C22⋊Dic293—21no—
45#17989779—21no—
46#17989779—21no—
47C4×Dic293—21no—
48C2×D4×C293—21no—
49C2×D4×C293C2×C411yescharacteristic
50C22×C4×C293C2×C411yescharacteristic
51C4⋊C4×C293C2×C411yescharacteristic
52C22⋊C4×C293—21no—
53C2×#11924078—42no—
54C2×#11924078—42no—
55C2×#11924078—42no—
56#14556880—42no—
57C22×D293—21no—
58C22×D293—21no—
59C22×D293—41no—
60C22×D293—41no—
61C22×D293C22⋊C411yescharacteristic
62C22×D293C2×D411yescharacteristic
63Dic586—41no—
64C22⋊D293—42no—
65C22⋊D293—41no—
66C22⋊D293—41no—
67C2×Dic293—22no—
68C2×Dic293—21no—
69C2×Dic293—21no—
70C2×Dic293—42no—
71C2×Dic293C22⋊C411yescharacteristic
72C2×Dic293C2×D411yescharacteristic
73C4×D293—22no—
74C4×D293—22no—
75C4×D293—41no—
76C4×D293—41no—
77C23×C293—21no—
78C23×C293C22⋊C411yescharacteristic
79D4×C293—41no—
80D4×C293—42no—
81C2×C4×C293—21no—
82C2×C4×C293—41no—
83C2×C4×C293C22⋊C411yescharacteristic
84C2×C4×C293C22×C411yescharacteristic
85C22×C293—21no—
86C22×C293—21no—
87C22×C293—21no—
88C22×C293—41no—
89C22×C293—41no—
90C22×C293—41no—
91C22×C293C2×C22⋊C411yescharacteristic
92C2×D293—21no—
93C2×D293—21no—
94C2×D293—21no—
95C2×D293—21no—
96C2×D293—21no—
97C2×D293—42no—
98C2×D293—42no—
99C2×D293AGL(1,ℤ/8)11yescharacteristic
100C2×D293#13711yescharacteristic
101#11924078—82no—
102C4×C293—21no—
103C4×C293—21no—
104C4×C293—41no—
105Dic293—21no—
106Dic293—41no—
107Dic293—41no—
108Dic293#9912yes—
109C2×C293—41no—
110C2×C293—81no—
111C2×C293C2×#9911yescharacteristic
112C2×C293#41711yescharacteristic
113C2×C293#42111yescharacteristic
114D293—21no—
115D293—21no—
116D293—41no—
117C293#151311yescharacteristic, complemented, Sylow
118#1513—2931nomaximal, Sylow
119#525—2931no—
120#407—2931no—
121#380—2931no—
122#328—2932no—
123#327—2932no—
124C2×#41—2931no—
125C23×C4—2931no—
126C2×M16—5861no—
127#123—5861no—
128C22⋊Q8—2931no—
129C22⋊Q8—2931no—
130#121—2931no—
131#121#1192407811yescharacteristic, complemented
132#120—5861no—
133C4×D4—5861no—
134C4×D4—5861no—
135#117—2931no—
136C2×C4⋊C4—5861no—
137C2×C22⋊C4—5861no—
138C22⋊C8—2932no—
139C22⋊C8—5861no—
140#95—2932no—
141C24—2931no—
142#41—11722no—
143C2×Q8—2931no—
144C2×D4—21no—
145C2×D4—5861no—
146C2×D4—5861no—
147C2×D4C2×#1192407811yescharacteristic
148C22×C4—2931no—
149C22×C4—2931no—
150C22×C4—2931no—
151C22×C4—2931no—
152C22×C4—2931no—
153C22×C4—5861no—
154C22×C4—5862no—
155C22×C4—11722no—
156C22×C4C2×#1192407811yescharacteristic
157M16—11722no—
158C2×C8—5862no—
159C4⋊C4—2931no—
160C4⋊C4—5861no—
161C4⋊C4—5861no—
162C4⋊C4—11722no—
163C4⋊C4C2×#1192407811yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—5861no—
166C22⋊C4—5861no—
167C22⋊C4—5861no—
168C22⋊C4—5861no—
169C22⋊C4—11722no—
170C42—5861no—
171C23—21no—
172C23—2931no—
173C23—2931no—
174C23—5861no—
175C23—5861no—
176C23—11721no—
177C23—11721no—
178C23C22⋊#1192407811yescharacteristic
179Q8—11721no—
180D4—41no—
181D4—42no—
182D4—11721no—
183D4—11721no—
184D4—11722no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—2931no—
188C2×C4—2931no—
189C2×C4—5862no—
190C2×C4—5862no—
191C2×C4—5862no—
192C2×C4—5861no—
193C2×C4—5861no—
194C2×C4—11721no—
195C2×C4—11721no—
196C2×C4—11722no—
197C2×C4—11722no—
198C2×C4—11722no—
199C2×C4—11722no—
200C2×C4C22⋊#1192407811yescharacteristic
201C2×C4C22×#1192407811yescharacteristic
202C8—11722no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—2931no—
210C22—2931no—
211C22—5861no—
212C22—5861no—
213C22—5861no—
214C22—5861no—
215C22—5861no—
216C22—11722no—
217C22—11722no—
218C22C2×C22⋊#1192407811yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—2932no—
223C4—23442no—
224C4—5861no—
225C4—11721no—
226C4—11721no—
227C2—41no—
228C2—81no—
229C2—5861no—
230C2—5861no—
231C2—11721no—
232C2D293⋊#9911yescharacteristic
233C2#3403712011yescharacteristic
234C2#3403712711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#11924078
commutatorC2×C4×C293C22×C4
FrattiniC2×C4C22×#11924078
Fitting#121×C293C4
socleC22×C293C2×C22⋊C4
prime radical#34036639C2
cosocleC2×C4×D293C23
hypercenter#121#11924078
hypocenterC293#1513
LCSG ▷ C2×C4×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #34037101C2semidirect
2. #34037071C2semidirect
3. #34036982C2semidirect
4. #34036980C2semidirect
12. C2×#17989801C4semidirect
16. #22627603C4semidirect
21. #22627582C4semidirect
22. #121×C293C4semidirect
117. C293#1513semidirect
131. #121#11924078semidirect

Maximal subgroups

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

subgroupindex
#340371012
#340370712
#340369822
#340369802
#340366392
#1513293