Finite Group Databasebeta

← #49159166

Subgroups — #49159166

Order 37504.

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

normal subgroups only

This group has 332 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
1#34036896C211yesmaximal, characteristic, complemented
2#34036675C211yesmaximal, characteristic
3#34036147C212yesmaximal, complemented
4#34035888C212yesmaximal, complemented
5#410×C293C211yesmaximal, characteristic, complemented
6C2×#17989802C2211yescharacteristic
7#22627631—24no—
8C2×#17989786—22no—
9C2×#17989786C2212yes—
10#22627622C411yescharacteristic, complemented
11#22627620C411yescharacteristic, complemented
12#22627613C411yescharacteristic, complemented
13#22627611C411yescharacteristic, complemented
14#22627596—22no—
15C4×C22⋊D293—22no—
16C2×C4×Dic293C2211yescharacteristic
17#22627552—24no—
18C2×#41×C293C2211yescharacteristic
19#117×C293C2212yes—
20#99×C293—22no—
21#17989802—22no—
22#17989802—22no—
23C2×Dic586C2×C411yescharacteristic
24C2×D1172C2×C411yescharacteristic
25C2×C22⋊D293—22no—
26C2×C22⋊D293—21no—
27C22×Dic293—21no—
28C2×C4×D293—21no—
29#17989786—24no—
30#17989786—22no—
31#17989786D414yescomplemented
32C2×#14556873—24no—
33C2×#14556873—22no—
34#17989781—22no—
35C22⋊Dic293—22no—
36#17989779—22no—
37C4×Dic293—21no—
38C4×Dic293C2×C411yescharacteristic
39C4×Dic293C2×C411yescharacteristic
40#41×C293—22no—
41#41×C293—22no—
42C2×Q8×C293C2×C411yescharacteristic
43C2×D4×C293C2×C411yescharacteristic
44C2×D4×C293D412yes—
45C22×C4×C293D412yes—
46C22×C4×C293C2311yescharacteristic
47C4⋊C4×C293—22no—
48C22⋊C4×C293—22no—
49C42×C293—22no—
50C22×D293—21no—
51Dic586—22no—
52D1172—22no—
53C22⋊D293—44no—
54C22⋊D293—42no—
55C2×Dic293—21no—
56C2×Dic293—21no—
57C2×Dic293—21no—
58C2×Dic293—42no—
59C4×D293—41no—
60C4×D293—41no—
61#14556873—24no—
62C23×C293C22⋊C412yes—
63C23×C293C2×D411yescharacteristic
64Q8×C293—22no—
65D4×C293—24no—
66D4×C293—22no—
67C2×C4×C293—22no—
68C2×C4×C293—22no—
69C2×C4×C293—42no—
70C2×C4×C293C22⋊C412yes—
71C2×C4×C293C22×C411yescharacteristic
72C2×C4×C293C2×D412yes—
73C2×C4×C293C2×D411yescharacteristic
74C2×C4×C293#4112yes—
75C22×C293—22no—
76C22×C293—41no—
77C22×C293—42no—
78C22×C293C2×C22⋊C411yescharacteristic
79C22×C293#12112yes—
80C2×D293—42no—
81C2×D293—41no—
82C4×C293—22no—
83C4×C293—82no—
84C4×C293C4×D412yes—
85C4×C293#12011yescharacteristic
86C4×C293#12311yescharacteristic
87Dic293—22no—
88Dic293—41no—
89Dic293—41no—
90C2×C293—21no—
91C2×C293—22no—
92C2×C293—42no—
93C2×C293#38611yescharacteristic
94D293—81no—
95C293#188411yescharacteristic, complemented, Sylow
96#1884—2931nomaximal, Sylow
97#568—2931no—
98#532—2931no—
99#421—2932no—
100#410D29311yescharacteristic, complemented
101#339—2932no—
102C2×#41—2931no—
103C2×#41C2×D29311yescharacteristic
104#131—5864no—
105C2×M16—2932no—
106C2×M16—5862no—
107C4⋊Q8—2931no—
108#127—2931no—
109#124—2931no—
110#124—2931no—
111#121—5862no—
112C4×D4—5862no—
113#117C2×D29312yes—
114C2×C42—2931no—
115#104—5864no—
116#99—22no—
117#41—22no—
118#41—22no—
119#41—5862no—
120#41—5862no—
121C2×Q8—2931no—
122C2×Q8C4×D29311yescharacteristic
123C2×D4—2931no—
124C2×D4—5862no—
125C2×D4—5861no—
126C2×D4C4×D29311yescharacteristic
127C2×D4C22⋊D29312yes—
128C22×C4—5861no—
129C22×C4—5861no—
130C22×C4D117212yes—
131C22×C4C22×D29311yescharacteristic
132M16—2934no—
133M16—5864no—
134M16—5862no—
135C2×C8—5864no—
136C2×C8—5862no—
137C4⋊C4—22no—
138C4⋊C4—5862no—
139C22⋊C4—22no—
140C22⋊C4—5862no—
141C22⋊C4—5862no—
142C42—22no—
143C42—2931no—
144C42—2931no—
145C42—5861no—
146C23—5861no—
147C23#1798977912yes—
148C23C2×C22⋊D29311yescharacteristic
149Q8—22no—
150Q8—5862no—
151D4—24no—
152D4—22no—
153D4—5862no—
154D4—11724no—
155D4—11722no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—5861no—
160C2×C4—5861no—
161C2×C4—5861no—
162C2×C4—11722no—
163C2×C4—11721no—
164C2×C4—11721no—
165C2×C4#1798977912yes—
166C2×C4C2×C4×D29311yescharacteristic
167C2×C4D4×D29312yes—
168C2×C4C2×D117211yescharacteristic
169C2×C4#1798980112yes—
170C8—5864no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—11722no—
175C22—11721no—
176C22C2×#1798977911yescharacteristic
177C22#2262759612yes—
178C4—22no—
179C4—82no—
180C4—5862no—
181C4—11721no—
182C4—11721no—
183C4#2262758412yes—
184C4#2262759411yescharacteristic
185C4#2262760811yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—23441no—
190C2#3403600611yescharacteristic
subgroupG/H
centerC4#22627594
commutatorC2×C4×C293C22×C4
FrattiniC22×C4C22×D293
Fitting#410×C293C2
socleC2×C293#386
prime radical#34036675C2
cosocleC22×C4×C293C23
hypercenter#410D293
hypocenterC293#1884
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #34036896C2semidirect
3. #34036147C2semidirect
4. #34035888C2semidirect
5. #410×C293C2semidirect
10. #22627622C4semidirect
11. #22627620C4semidirect
12. #22627613C4semidirect
13. #22627611C4semidirect
31. #17989786D4semidirect
95. C293#1884semidirect
100. #410D293semidirect

Maximal subgroups

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

subgroupindex
#340368962
#340366752
#340361472
#340358882
#410×C2932
#1884293