Finite Group Databasebeta

← #49158985

Subgroups — #49158985

Order 37504.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#22627582C211yesmaximal, characteristic
2C2×#22627557C211yesmaximal, characteristic, complemented
3C2×#22627556C211yesmaximal, characteristic, complemented
4C2×#22627536C211yesmaximal, characteristic
5#34035986C211yesmaximal, characteristic
6#34035891C211yesmaximal, characteristic, complemented
7#340×C293C211yesmaximal, characteristic, complemented
8C22×C4×D293C2211yescharacteristic
9C22×#14556873C2211yescharacteristic
10#22627582—22no—
11#22627582—22no—
12C2×C4⋊Dic293C2211yescharacteristic
13C2×C4⋊Dic293C2211yescharacteristic
14C2×#17989779—21no—
15C2×#17989779—21no—
16C2×C4×Dic293—21no—
17#22627557—22no—
18#22627556—22no—
19#22627536C414yescomplemented
20#22627531—21no—
21C22×C8×C293C2211yescharacteristic
22C2×C4⋊C4×C293C2211yescharacteristic
23C2×C4⋊C4×C293C2211yescharacteristic
24C23×D293D411yescharacteristic
25C22×Dic293—21no—
26C22×Dic293—21no—
27C22×Dic293D411yescharacteristic
28C2×C4×D293—21no—
29C2×C4×D293—21no—
30C2×C4×D293—21no—
31C2×C4×D293—21no—
32C2×C4×D293C2×C412yes—
33C2×C4×D293Q812yes—
34C2×#14556873—21no—
35C2×#14556873—21no—
36C2×#14556873C2×C412yes—
37C4⋊Dic293—21no—
38C4⋊Dic293—21no—
39C4⋊Dic293—21no—
40C4⋊Dic293D412yes—
41#17989779—24no—
42#17989779—42no—
43C4×Dic293—24no—
44C22×C4×C293—21no—
45C22×C4×C293—21no—
46C22×C4×C293C2311yescharacteristic
47C2×C8×C293—21no—
48C2×C8×C293—21no—
49C2×C8×C293C2×C412yes—
50C4⋊C4×C293—21no—
51C4⋊C4×C293—21no—
52C4⋊C4×C293—21no—
53C4⋊C4×C293D412yes—
54C22×D293—21no—
55C22×D293—21no—
56C22×D293—21no—
57C22×D293—21no—
58C22×D293—21no—
59C22×D293—21no—
60C22×D293C4⋊C412yes—
61C2×Dic293—21no—
62C2×Dic293—21no—
63C2×Dic293—22no—
64C2×Dic293—41no—
65C2×Dic293—41no—
66C2×Dic293—41no—
67C2×Dic293—41no—
68C2×Dic293—41no—
69C2×Dic293C4⋊C412yes—
70C4×D293—22no—
71C4×D293—22no—
72C4×D293—22no—
73C4×D293—22no—
74#14556873—22no—
75C23×C293C2×D411yescharacteristic
76C2×C4×C293—22no—
77C2×C4×C293—41no—
78C2×C4×C293—41no—
79C2×C4×C293—41no—
80C2×C4×C293—41no—
81C2×C4×C293—41no—
82C2×C4×C293C22×C411yescharacteristic
83C2×C4×C293C2×D411yescharacteristic
84C2×C4×C293C2×D411yescharacteristic
85C2×C4×C293C2×Q811yescharacteristic
86C2×C4×C293#4111yescharacteristic
87C2×C4×C293#4111yescharacteristic
88C8×C293—22no—
89C22×C293C2×C4⋊C411yescharacteristic
90C22×C293#12011yescharacteristic
91C22×C293#12311yescharacteristic
92C22×C293#13511yescharacteristic
93C22×C293#13511yescharacteristic
94C22×C293AGL(1,ℤ/8)11yescharacteristic
95C22×C293#13711yescharacteristic
96C2×D293—22no—
97C2×D293—22no—
98C2×D293—22no—
99C2×D293—41no—
100C2×D293—41no—
101C2×D293—41no—
102C2×D293—41no—
103C4×C293—42no—
104C4×C293—81no—
105C4×C293C4×D411yescharacteristic
106C4×C293C4×D411yescharacteristic
107C4×C293C22⋊Q811yescharacteristic
108C4×C293C22⋊Q811yescharacteristic
109Dic293—22no—
110Dic293—42no—
111Dic293—81no—
112C2×C293#38511yescharacteristic
113C2×C293#42711yescharacteristic
114C2×C293#42811yescharacteristic
115C2×C293#44911yescharacteristic
116C2×C293#45211yescharacteristic
117C2×C293#48211yescharacteristic
118C2×C293#48511yescharacteristic
119D293—42no—
120C293#185411yescharacteristic, complemented, Sylow
121#1854—2931nomaximal, Sylow
122C2×#117—2931no—
123C2×#107—2931no—
124C2×#106—2931no—
125C2×C22⋊C8—2931no—
126#380—2931no—
127#340—2931no—
128#340D29311yescharacteristic, complemented
129C23×C4—2931no—
130C22×C8—2931no—
131C22×C8C2×D29311yescharacteristic
132#117—5862no—
133#117—5862no—
134C2×C4⋊C4—2931no—
135C2×C4⋊C4—2931no—
136C2×C4⋊C4C2×D29311yescharacteristic
137C2×C4⋊C4C2×D29311yescharacteristic
138C2×C22⋊C4—5861no—
139C2×C22⋊C4—5861no—
140C2×C42—5861no—
141#107—5862no—
142#106—5862no—
143C22⋊C8—2934no—
144#95—5861no—
145C24—2931no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2931no—
149C22×C4—2932no—
150C22×C4—2932no—
151C22×C4—5861no—
152C22×C4—5861no—
153C22×C4—5861no—
154C22×C4—5861no—
155C22×C4—5861no—
156C22×C4—5861no—
157C22×C4C22×D29311yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—2932no—
161C2×C8—5861no—
162C2×C8—5861no—
163C2×C8C4×D29312yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—2932no—
168C4⋊C4—5861no—
169C4⋊C4—5861no—
170C4⋊C4—5861no—
171C4⋊C4D117212yes—
172C22⋊C4—5864no—
173C22⋊C4—11722no—
174C42—5864no—
175C23—2932no—
176C23—5861no—
177C23—5861no—
178C23—5861no—
179C23—5861no—
180C23—5861no—
181C23—5861no—
182C23D4×D29311yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2932no—
190C2×C4—5861no—
191C2×C4—5861no—
192C2×C4—5862no—
193C2×C4—5862no—
194C2×C4—5862no—
195C2×C4—5862no—
196C2×C4—5862no—
197C2×C4—11721no—
198C2×C4—11721no—
199C2×C4—11721no—
200C2×C4—11721no—
201C2×C4—11721no—
202C2×C4C2×C4×D29311yescharacteristic
203C2×C4D4×D29311yescharacteristic
204C2×C4C2×D117211yescharacteristic
205C2×C4Q8×D29311yescharacteristic
206C2×C4#1798980111yescharacteristic
207C2×C4#1798980211yescharacteristic
208C8—22no—
209C8—5862no—
210C22—5862no—
211C22—5862no—
212C22—5862no—
213C22—11721no—
214C22—11721no—
215C22—11721no—
216C22—11721no—
217C22C4⋊C4×D29311yescharacteristic
218C22#2262759411yescharacteristic
219C22#2262760611yescharacteristic
220C22#2262764511yescharacteristic
221C22#2262764511yescharacteristic
222C22#2262764811yescharacteristic
223C22#2262765611yescharacteristic
224C4—42no—
225C4—81no—
226C4—23441no—
227C4—5862no—
228C4—11722no—
229C4#2262758411yescharacteristic
230C4C4×D117211yescharacteristic
231C4#2262760211yescharacteristic
232C4#2262760411yescharacteristic
233C2—11722no—
234C2#3403600311yescharacteristic
235C2#3403616411yescharacteristic
236C2#3403617011yescharacteristic
237C2#3403627511yescharacteristic
238C2#3403629411yescharacteristic
239C2#3403645211yescharacteristic
240C2#3403646811yescharacteristic
subgroupG/H
centerC23D4×D293
commutatorC2×C4×C293C22×C4
FrattiniC22×C4C22×D293
Fitting#340×C293C2
socleC23×C293C2×D4
prime radicalC23×D293D4
cosocleC22×C4×C293C23
hypercenter#340D293
hypocenterC293#1854
LCSG ▷ C2×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×#22627556C2semidirect
6. #34035891C2semidirect
7. #340×C293C2semidirect
19. #22627536C4semidirect
120. C293#1854semidirect
128. #340D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226275822
C2×#226275572
C2×#226275562
C2×#226275362
#340359862
#340358912
#340×C2932
#1854293