Finite Group Databasebeta

← #49174223

Subgroups — #49174223

Order 37504.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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×#22627681C211yesmaximal, characteristic
2#34037071C211yesmaximal, characteristic
3#34036983C212yesmaximal, complemented
4#34036982C212yesmaximal, complemented
5C22⋊Q8×D293C211yesmaximal, characteristic
6C22×#14556880—21no—
7C2×C4⋊#11924078—21no—
8C2×C22⋊#11924078—21no—
9#22627681—21no—
10#22627681C2212yes—
11(C2×#11924078)⋊C4C2212yes—
12C2×Q8×D293C411yescharacteristic, complemented
13C22×C4×D293C2211yescharacteristic
14#22627604C2211yescharacteristic
15#22627603C411yescharacteristic, complemented
16#22627602—21no—
17#22627601—21no—
18#22627600C2211yescharacteristic
19C4⋊C4×D293—21no—
20C4⋊C4×D293C411yescharacteristic, complemented
21C22⋊C4×D293—21no—
22C22⋊Q8×C293C411yescharacteristic, complemented
23C22×#11924078—22no—
24C2×#14556880—22no—
25C2×#14556880—42no—
26C4⋊#11924078—42no—
27C22⋊#11924078—42no—
28C23×D293D411yescharacteristic
29C2×Dic586—21no—
30C2×Dic586C2×C411yescharacteristic
31Q8×D293—42no—
32C22×Dic293D411yescharacteristic
33C2×C4×D293—21no—
34C2×C4×D293—21no—
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—
43#17989781—21no—
44#17989781—21no—
45C22⋊Dic293—21no—
46#17989779—21no—
47#17989779—21no—
48C2×Q8×C293C2×C411yescharacteristic
49C22×C4×C293C2×C411yescharacteristic
50C4⋊C4×C293—21no—
51C4⋊C4×C293C2×C411yescharacteristic
52C22⋊C4×C293—21no—
53C2×#11924078—42no—
54C2×#11924078—42no—
55C2×#11924078—42no—
56#14556880—42no—
57C22×D293—41no—
58C22×D293—41no—
59C22×D293C22⋊C411yescharacteristic
60C22×D293SD1612yes—
61C22×D293Q1612yes—
62C22×D293C2×D411yescharacteristic
63Dic586—42no—
64Dic586—41no—
65C2×Dic293—22no—
66C2×Dic293—21no—
67C2×Dic293—21no—
68C2×Dic293C22⋊C411yescharacteristic
69C2×Dic293C2×D411yescharacteristic
70C4×D293—22no—
71C4×D293—22no—
72C4×D293—41no—
73C4×D293—41no—
74C4×D293—42no—
75C4×D293—42no—
76C23×C293C22⋊C411yescharacteristic
77Q8×C293—41no—
78C2×C4×C293—21no—
79C2×C4×C293—21no—
80C2×C4×C293—41no—
81C2×C4×C293C22⋊C411yescharacteristic
82C2×C4×C293C22×C411yescharacteristic
83C22×C293—21no—
84C22×C293—21no—
85C22×C293#10312yes—
86C22×C293C2×C22⋊C411yescharacteristic
87C2×D293—21no—
88C2×D293—21no—
89C2×D293—21no—
90C2×D293—21no—
91C2×D293—42no—
92C2×D293—42no—
93C2×D293#10312yes—
94C2×D293C2×SD1611yescharacteristic
95C2×D293C2×Q1611yescharacteristic
96#11924078—82no—
97C4×C293—21no—
98C4×C293—21no—
99C4×C293—41no—
100C4×C293—41no—
101Dic293—21no—
102Dic293—41no—
103Dic293—41no—
104Dic293#9912yes—
105C2×C293—22no—
106C2×C293C2×#9911yescharacteristic
107C2×C293C2×#10311yescharacteristic
108C2×C293#42111yescharacteristic
109D293—21no—
110D293—22no—
111D293—21no—
112C293#152011yescharacteristic, complemented, Sylow
113#1520—2931nomaximal, Sylow
114C2×C22⋊Q8—2931no—
115C2×C22⋊C8—2931no—
116#380—2931no—
117#328—2932no—
118#328—2932no—
119C22×Q8—2931no—
120C23×C4—2931no—
121C22×C8—5861no—
122C22⋊Q8—2931no—
123C22⋊Q8—2931no—
124C22⋊Q8—2931no—
125C22⋊Q8—5861no—
126C22⋊Q8—5861no—
127C22⋊Q8#1192407811yescharacteristic, complemented
128C2×C4⋊C4—2931no—
129C2×C4⋊C4—5861no—
130C2×C4⋊C4—5861no—
131C2×C22⋊C4—5861no—
132C2×C22⋊C4—5861no—
133C22⋊C8—2932no—
134C22⋊C8—5861no—
135#95—2932no—
136C24—2931no—
137C2×Q8—2931no—
138C2×Q8—5861no—
139C2×Q8—11722no—
140C2×Q8C2×#1192407811yescharacteristic
141C22×C4—2931no—
142C22×C4—2931no—
143C22×C4—2931no—
144C22×C4—2931no—
145C22×C4—2931no—
146C22×C4—5861no—
147C22×C4—5861no—
148C22×C4—5862no—
149C22×C4—11722no—
150C22×C4C2×#1192407811yescharacteristic
151C2×C8—5862no—
152C2×C8—11722no—
153C4⋊C4—21no—
154C4⋊C4—2931no—
155C4⋊C4—5861no—
156C4⋊C4—5861no—
157C4⋊C4—5861no—
158C4⋊C4—5861no—
159C4⋊C4—11722no—
160C4⋊C4C2×#1192407811yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—5861no—
163C22⋊C4—5861no—
164C22⋊C4—5861no—
165C22⋊C4—11722no—
166C23—2931no—
167C23—2931no—
168C23—2932no—
169C23—2932no—
170C23—11721no—
171C23—11721no—
172C23C22⋊#1192407811yescharacteristic
173Q8—41no—
174Q8—11721no—
175Q8—11722no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—2931no—
180C2×C4—2931no—
181C2×C4—5862no—
182C2×C4—5862no—
183C2×C4—5862no—
184C2×C4—5861no—
185C2×C4—5861no—
186C2×C4—11721no—
187C2×C4—11721no—
188C2×C4—11722no—
189C2×C4—11722no—
190C2×C4—11722no—
191C2×C4—11722no—
192C2×C4—11722no—
193C2×C4C22⋊#1192407811yescharacteristic
194C2×C4C22×#1192407811yescharacteristic
195C8—11722no—
196C22—21no—
197C22—21no—
198C22—2931no—
199C22—2932no—
200C22—2931no—
201C22—5861no—
202C22—5861no—
203C22—5861no—
204C22—5861no—
205C22—11722no—
206C22—11722no—
207C22#2262768712yes—
208C22C2×C22⋊#1192407811yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—2932no—
214C4—23442no—
215C4—5861no—
216C4—11721no—
217C4—11721no—
218C2—22no—
219C2—5861no—
220C2—5862no—
221C2—5861no—
222C2D293⋊#9911yescharacteristic
223C2C2×#2262768711yescharacteristic
224C2#3403712611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#11924078
commutatorC2×C4×C293C22×C4
FrattiniC2×C4C22×#11924078
FittingC22⋊Q8×C293C4
socleC22×C293C2×C22⋊C4
prime radicalC23×D293D4
cosocleC2×C4×D293C23
hypercenterC22⋊Q8#11924078
hypocenterC293#1520
LCSG ▷ C2×C4×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #34036983C2semidirect
4. #34036982C2semidirect
12. C2×Q8×D293C4semidirect
15. #22627603C4semidirect
20. C4⋊C4×D293C4semidirect
22. C22⋊Q8×C293C4semidirect
112. C293#1520semidirect
127. C22⋊Q8#11924078semidirect

Maximal subgroups

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

subgroupindex
C2×#226276812
#340370712
#340369832
#340369822
C22⋊Q8×D2932
#1520293