Finite Group Databasebeta

← #49159991

Subgroups — #49159991

Order 37504.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#22627604C211yesmaximal, characteristic
2C2×#22627557C211yesmaximal, characteristic, complemented
3C2×#22627549C211yesmaximal, characteristic, complemented
4C2×#22627547C211yesmaximal, characteristic, complemented
5C2×#22627536C211yesmaximal, characteristic
6#34035986C211yesmaximal, characteristic
7#340×C293C211yesmaximal, characteristic, complemented
8C22×Dic586C2211yescharacteristic
9C22×C4×D293C2211yescharacteristic
10C22×#14556873C2211yescharacteristic
11#22627604—22no—
12#22627604—22no—
13C2×C4⋊Dic293—21no—
14C2×C4⋊Dic293C2211yescharacteristic
15C2×#17989779—21no—
16C2×#17989779—21no—
17#22627557—22no—
18#22627549—22no—
19#22627547—22no—
20#22627536—22no—
21#22627531—21no—
22C22×C8×C293C2211yescharacteristic
23C2×C4⋊C4×C293C2211yescharacteristic
24C2×C4⋊C4×C293C2211yescharacteristic
25C23×D293D411yescharacteristic
26C2×Dic586—21no—
27C2×Dic586—21no—
28C2×Dic586—21no—
29C2×Dic586—21no—
30C2×Dic586—21no—
31C2×Dic586D412yes—
32C22×Dic293—21no—
33C22×Dic293—21no—
34C22×Dic293D411yescharacteristic
35C2×C4×D293—21no—
36C2×C4×D293—21no—
37C2×C4×D293—21no—
38C2×C4×D293—21no—
39C2×C4×D293—21no—
40C2×C4×D293—21no—
41C2×#14556873—21no—
42C2×#14556873—21no—
43C2×#14556873D412yes—
44C4⋊Dic293—24no—
45C4⋊Dic293—21no—
46C4⋊Dic293—21no—
47#17989779—24no—
48#17989779—42no—
49C22×C4×C293—21no—
50C22×C4×C293—21no—
51C22×C4×C293C2311yescharacteristic
52C2×C8×C293—21no—
53C2×C8×C293—21no—
54C2×C8×C293—21no—
55C4⋊C4×C293—21no—
56C4⋊C4×C293—21no—
57C4⋊C4×C293—21no—
58C4⋊C4×C293D412yes—
59C22×D293—21no—
60C22×D293—21no—
61C22×D293—21no—
62C22×D293—21no—
63C22×D293—21no—
64C22×D293—21no—
65C22×D293Q1612yes—
66Dic586—22no—
67Dic586—22no—
68Dic586—41no—
69Dic586—41no—
70C2×Dic293—22no—
71C2×Dic293—21no—
72C2×Dic293—21no—
73C2×Dic293—21no—
74C2×Dic293—41no—
75C2×Dic293—41no—
76C2×Dic293—41no—
77C2×Dic293—41no—
78C2×Dic293—41no—
79C4×D293—41no—
80C4×D293—41no—
81C4×D293—41no—
82C4×D293—41no—
83#14556873—22no—
84C23×C293C2×D411yescharacteristic
85C2×C4×C293—22no—
86C2×C4×C293—41no—
87C2×C4×C293—41no—
88C2×C4×C293—41no—
89C2×C4×C293—41no—
90C2×C4×C293—41no—
91C2×C4×C293C2×D411yescharacteristic
92C2×C4×C293C2×D411yescharacteristic
93C2×C4×C293C2×D411yescharacteristic
94C2×C4×C293#4111yescharacteristic
95C2×C4×C293#4111yescharacteristic
96C2×C4×C293#4111yescharacteristic
97C8×C293—41no—
98C22×C293#12011yescharacteristic
99C22×C293#12111yescharacteristic
100C22×C293#12311yescharacteristic
101C22×C293C2×Q1611yescharacteristic
102C22×C293#13511yescharacteristic
103C22×C293AGL(1,ℤ/8)11yescharacteristic
104C22×C293#13711yescharacteristic
105C2×D293—22no—
106C2×D293—22no—
107C2×D293—22no—
108C2×D293—41no—
109C2×D293—41no—
110C2×D293—41no—
111C2×D293—41no—
112C4×C293—42no—
113C4×C293—81no—
114C4×C293#12111yescharacteristic
115C4×C293#12111yescharacteristic
116C4×C293#12311yescharacteristic
117C4×C293#12411yescharacteristic
118Dic293—41no—
119Dic293—42no—
120Dic293—81no—
121C2×C293#39411yescharacteristic
122C2×C293#44911yescharacteristic
123C2×C293#45111yescharacteristic
124C2×C293#46711yescharacteristic
125C2×C293#46811yescharacteristic
126C2×C293#48411yescharacteristic
127C2×C293#48511yescharacteristic
128D293—42no—
129C293#203611yescharacteristic, complemented, Sylow
130#2036—2931nomaximal, Sylow
131C2×C22⋊Q8—2931no—
132C2×#107—2931no—
133C2×#103—2931no—
134C2×#103—2931no—
135C2×C22⋊C8—2931no—
136#380—2931no—
137#340D29311yescharacteristic, complemented
138C22×Q8—2931no—
139C23×C4—2931no—
140C22×C8—2931no—
141C22×C8C2×D29311yescharacteristic
142C22⋊Q8—5862no—
143C22⋊Q8—5862no—
144C2×C4⋊C4—2931no—
145C2×C4⋊C4—5861no—
146C2×C4⋊C4C2×D29311yescharacteristic
147C2×C4⋊C4C2×D29311yescharacteristic
148C2×C22⋊C4—5861no—
149C2×C22⋊C4—5861no—
150#107—5862no—
151#103—5862no—
152#103—5862no—
153C22⋊C8—5862no—
154#95—5861no—
155C24—2931no—
156C2×Q8—2932no—
157C2×Q8—5861no—
158C2×Q8—5861no—
159C2×Q8—5861no—
160C2×Q8—5861no—
161C2×Q8—5861no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—2931no—
165C22×C4—5861no—
166C22×C4—5861no—
167C22×C4—5861no—
168C22×C4—5861no—
169C22×C4—5861no—
170C22×C4—5861no—
171C22×C4—5861no—
172C22×C4—5861no—
173C22×C4C22×D29311yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2932no—
178C2×C8—5861no—
179C2×C8—5861no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—5864no—
184C4⋊C4—5861no—
185C4⋊C4—5861no—
186C4⋊C4D117212yes—
187C22⋊C4—5864no—
188C22⋊C4—11722no—
189C23—2932no—
190C23—5861no—
191C23—5861no—
192C23—5861no—
193C23—5861no—
194C23—5861no—
195C23—5861no—
196C23D4×D29311yescharacteristic
197Q8—5862no—
198Q8—5862no—
199Q8—11721no—
200Q8—11721no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—5862no—
208C2×C4—5861no—
209C2×C4—5861no—
210C2×C4—5861no—
211C2×C4—11721no—
212C2×C4—11721no—
213C2×C4—11721no—
214C2×C4—11721no—
215C2×C4—11721no—
216C2×C4—11721no—
217C2×C4—11721no—
218C2×C4—11721no—
219C2×C4—11721no—
220C2×C4D4×D29311yescharacteristic
221C2×C4D4×D29311yescharacteristic
222C2×C4C2×D117211yescharacteristic
223C2×C4#1798980211yescharacteristic
224C2×C4#1798980211yescharacteristic
225C2×C4#1798980311yescharacteristic
226C8—41no—
227C8—5862no—
228C22—5862no—
229C22—5862no—
230C22—5862no—
231C22—11721no—
232C22—11721no—
233C22—11721no—
234C22—11721no—
235C22#2262759411yescharacteristic
236C22#2262759711yescharacteristic
237C22#2262760911yescharacteristic
238C22Q16×D29311yescharacteristic
239C22#2262764511yescharacteristic
240C22#2262764811yescharacteristic
241C22#2262765311yescharacteristic
242C4—42no—
243C4—81no—
244C4—23441no—
245C4—11721no—
246C4—11722no—
247C4#2262759711yescharacteristic
248C4#2262759911yescharacteristic
249C4#2262760911yescharacteristic
250C4#2262761211yescharacteristic
251C2—11722no—
252C2#3403604711yescharacteristic
253C2#3403627511yescharacteristic
254C2#3403628611yescharacteristic
255C2#3403636911yescharacteristic
256C2#3403637411yescharacteristic
257C2#3403646611yescharacteristic
258C2#3403647311yescharacteristic
subgroupG/H
centerC23D4×D293
commutator, cosocleC22×C4×C293C23
FrattiniC22×C4C22×D293
Fitting#340×C293C2
socleC23×C293C2×D4
prime radicalC23×D293D4
hypercenter#340D293
hypocenterC293#2036
LCSG ▷ C22×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#22627557C2semidirect
3. C2×#22627549C2semidirect
4. C2×#22627547C2semidirect
7. #340×C293C2semidirect
129. C293#2036semidirect
137. #340D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276042
C2×#226275572
C2×#226275492
C2×#226275472
C2×#226275362
#340359862
#340×C2932
#2036293