Finite Group Databasebeta

← #49159201

Subgroups — #49159201

Order 37504.

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

normal subgroups only

This group has 342 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#34036636C211yesmaximal, characteristic
3#34036092C212yesmaximal, complemented
4#34035901C212yesmaximal, complemented
5#413×C293C211yesmaximal, characteristic, complemented
6C22×C4×D293C2211yescharacteristic
7#22627631—24no—
8C2×#17989786—22no—
9C2×#17989786C2212yes—
10#22627605C412yes—
11#22627600—21no—
12#22627595—21no—
13#22627592C412yes—
14C4×C22⋊D293—22no—
15#22627583C2211yescharacteristic
16#22627536—24no—
17C2×#41×C293C2211yescharacteristic
18C2×M16×C293C2212yes—
19#101×C293—21no—
20#100×C293—21no—
21C23×D293C2×C411yescharacteristic
22C2×C22⋊D293—22no—
23C22×Dic293C2×C411yescharacteristic
24C2×C4×D293—21no—
25C2×C4×D293—21no—
26C2×C4×D293—22no—
27C2×C4×D293—22no—
28#17989786—24no—
29#17989786—22no—
30#17989786—22no—
31C2×#14556873—24no—
32C2×#14556873D414yescomplemented
33C4⋊Dic293—21no—
34#17989781—22no—
35C22⋊Dic293—22no—
36C22⋊Dic293C2×C412yes—
37#17989779—22no—
38C4×Dic293—21no—
39#41×C293—22no—
40#41×C293—22no—
41C2×Q8×C293D411yescharacteristic
42C2×D4×C293C2×C412yes—
43C2×D4×C293D411yescharacteristic
44C22×C4×C293D412yes—
45C22×C4×C293C2311yescharacteristic
46M16×C293—22no—
47M16×C293—22no—
48C2×C8×C293—22no—
49C22×D293—21no—
50C22×D293—22no—
51C22×D293—42no—
52C22⋊D293—44no—
53C2×Dic293—22no—
54C2×Dic293—21no—
55C2×Dic293—22no—
56C4×D293—41no—
57C4×D293—42no—
58C4×D293—41no—
59#14556873—24no—
60C23×C293C22⋊C412yes—
61C23×C293C22×C411yescharacteristic
62Q8×C293—22no—
63D4×C293—24no—
64D4×C293—22no—
65C2×C4×C293—22no—
66C2×C4×C293—22no—
67C2×C4×C293C22⋊C412yes—
68C2×C4×C293C2×D412yes—
69C2×C4×C293C2×D411yescharacteristic
70C2×C4×C293C2×D411yescharacteristic
71C2×C4×C293#4112yes—
72C8×C293—42no—
73C22×C293—22no—
74C22×C293—41no—
75C22×C293—42no—
76C22×C293C2×C22⋊C411yescharacteristic
77C22×C293C4×D412yes—
78C2×D293—22no—
79C2×D293—24no—
80C2×D293—41no—
81C2×D293—42no—
82C2×D293—41no—
83C4×C293—22no—
84C4×C293#12011yescharacteristic
85C4×C293#12112yes—
86C4×C293#12311yescharacteristic
87Dic293—42no—
88Dic293—41no—
89C2×C293—21no—
90C2×C293—22no—
91C2×C293—42no—
92C2×C293#38611yescharacteristic
93D293—42no—
94C293#189111yescharacteristic, complemented, Sylow
95#1891—2931nomaximal, Sylow
96#568—2931no—
97#525—2931no—
98#413D29311yescharacteristic, complemented
99#407—2932no—
100#344—2932no—
101C2×#41C2×D29311yescharacteristic
102C23×C4—2931no—
103#131—5864no—
104C2×M16—2932no—
105C2×M16—5862no—
106C2×M16C2×D29312yes—
107#123—2932no—
108C22⋊Q8—5861no—
109#121—5861no—
110#120—2932no—
111C4×D4—5862no—
112#117—2931no—
113#101—21no—
114#100—21no—
115C22⋊C8—5864no—
116C24—2931no—
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D29311yescharacteristic
120C2×D4—5862no—
121C2×D4C4×D29312yes—
122C2×D4C22⋊D29311yescharacteristic
123C22×C4—2931no—
124C22×C4—5862no—
125C22×C4—5861no—
126C22×C4—5861no—
127C22×C4—5862no—
128C22×C4D117212yes—
129C22×C4C22×D29311yescharacteristic
130M16—22no—
131M16—22no—
132M16—5864no—
133M16—5862no—
134M16—5862no—
135C2×C8—22no—
136C2×C8—2934no—
137C2×C8—5864no—
138C4⋊C4—5862no—
139C4⋊C4—5861no—
140C22⋊C4—2932no—
141C22⋊C4—5862no—
142C22⋊C4—5862no—
143C42—5861no—
144C23—5862no—
145C23—5861no—
146C23—11722no—
147C23#1798977912yes—
148C23C2×C4×D29311yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—11724no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—5862no—
156C2×C4—5862no—
157C2×C4—5861no—
158C2×C4—11721no—
159C2×C4—11722no—
160C2×C4—11721no—
161C2×C4#1798977912yes—
162C2×C4D4×D29312yes—
163C2×C4C2×C22⋊D29311yescharacteristic
164C2×C4C2×D117211yescharacteristic
165C2×C4#1798980112yes—
166C8—42no—
167C8—5864no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—5862no—
172C22—5864no—
173C22—11721no—
174C22—11722no—
175C22—11721no—
176C22C2×#1798977911yescharacteristic
177C22#2262758412yes—
178C4—22no—
179C4—11722no—
180C4—11721no—
181C4#2262759411yescharacteristic
182C4#2262759612yes—
183C4#2262760811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—11722no—
188C2#3403600611yescharacteristic
subgroupG/H
centerC4#22627594
commutatorC23×C293C22×C4
FrattiniC22×C4C22×D293
Fitting#413×C293C2
socleC2×C293#386
prime radical#34036636C2
cosocleC22×C4×C293C23
hypercenter#413D293
hypocenterC293#1891
LCSG ▷ C23×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #34036896C2semidirect
3. #34036092C2semidirect
4. #34035901C2semidirect
5. #413×C293C2semidirect
32. C2×#14556873D4semidirect
94. C293#1891semidirect
98. #413D293semidirect

Maximal subgroups

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

subgroupindex
#340368962
#340366362
#340360922
#340359012
#413×C2932
#1891293