Finite Group Databasebeta

← #31797969

Subgroups — #31797969

Order 17536.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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#22578522C211yesmaximal, characteristic, complemented
2C2×#17984183C211yesmaximal, characteristic, complemented
3#22578504C212yesmaximal, complemented
4#22578503C212yesmaximal, complemented
5D137⋊#131C211yesmaximal, characteristic
6#17984185C2212yescomplemented
7#17984184—21no—
8#17984183—21no—
9#17984183—21no—
10#17984183C2212yescomplemented
11C2×#14555522—21no—
12#17984176C411yescharacteristic
13#17984176C2211yescharacteristic
14#17984175—21no—
15#17984175—21no—
16#17984175C411yescharacteristic
17#17984175C2211yescharacteristic
18D137⋊M16—21no—
19D137⋊M16C411yescharacteristic
20C2×#14555518—21no—
21C2×#14555518C2211yescharacteristic
22#41×D137C411yescharacteristic
23#14555522—22no—
24#14555522—22no—
25#14555520—21no—
26#14555520—21no—
27#14555520C2×C411yescharacteristic
28#14555520C2×C411yescharacteristic
29#14555519—21no—
30#14555519—21no—
31#14555519—21no—
32#14555518—21no—
33#14555518—21no—
34#14555518—21no—
35#14555518C2×C411yescharacteristic
36#14555518D411yescharacteristic
37#14555518D411yescharacteristic
38#14555518C2311yescharacteristic
39C2×#11764047—21no—
40C2×#11764047—21no—
41C2×#11764047D411yescharacteristic
42C2×#11764047D411yescharacteristic
43#14555511C2×C411yescharacteristic
44#14555510—21no—
45#14555510C811yescharacteristic
46#14555509—21no—
47#14555509C2×C411yescharacteristic
48Q8×D137C811yescharacteristic
49D4×D137—21no—
50D4×D137C811yescharacteristic
51C2×C4×D137—21no—
52C2×C4×D137C2×C411yescharacteristic
53#41×C137C811yescharacteristic
54#11764047—21no—
55#11764047—21no—
56#11764047C22⋊C411yescharacteristic
57#11764047C22⋊C411yescharacteristic
58#11764047C2×D411yescharacteristic
59#11764047C2×D411yescharacteristic
60C22×D137—21no—
61C22×D137M1611yescharacteristic
62Dic274—21no—
63Dic274C2×C811yescharacteristic
64D548—21no—
65D548C2×C811yescharacteristic
66C22⋊D137—21no—
67C22⋊D137—21no—
68C22⋊D137—21no—
69C2×Dic137—21no—
70C2×Dic137C22⋊C411yescharacteristic
71C4×D137—21no—
72C4×D137—21no—
73C4×D137—21no—
74C4×D137C22⋊C411yescharacteristic
75C4×D137C2×C811yescharacteristic
76C4×D137M1611yescharacteristic
77C4×D137C22×C411yescharacteristic
78Q8×C137C2×C811yescharacteristic
79D4×C137—21no—
80D4×C137C2×C811yescharacteristic
81C2×C4×C137—21no—
82C2×C4×C137C2×C811yescharacteristic
83C22×C137—21no—
84C22×C137C22⋊C811yescharacteristic
85C2×D137—22no—
86C2×D137—21no—
87C2×D137—42no—
88C2×D137C22⋊C811yescharacteristic
89C2×D137C2×M1611yescharacteristic
90C4×C137—21no—
91C4×C137C22⋊C811yescharacteristic
92C4×C137C22×C811yescharacteristic
93Dic137—21no—
94Dic137C22⋊C811yescharacteristic
95Dic137C2×C22⋊C411yescharacteristic
96C2×C137—21no—
97C2×C137—41no—
98C2×C137C2×C22⋊C811yescharacteristic
99D137—21no—
100D137—41no—
101D137#35012yescomplemented
102C137C2×#35011yescharacteristic, complemented, Sylow
103C2×#350—1371nomaximal, Sylow
104C2×#131—1371no—
105C2×M32—1371no—
106C22×C16—1371no—
107#350—1372no—
108#350—1372no—
109C2×#41—1371no—
110#131—2741no—
111#131—2741no—
112#131—1371no—
113#131—1371no—
114#131—1371no—
115#131—1371no—
116C2×M16—2741no—
117C2×M16—1371no—
118C22×C8—2741no—
119C22×C8—1371no—
120M32—2741no—
121M32—1372no—
122C2×C16—2741no—
123C2×C16—2741no—
124C2×C16—2741no—
125C2×C16—1372no—
126#41—2741no—
127#41—2741no—
128#41—1371no—
129#41—1371no—
130#41—1371no—
131#41#1176404911yescharacteristic
132C2×Q8—1371no—
133C2×D4—2741no—
134C2×D4—1371no—
135C22×C4—2741no—
136C22×C4—1371no—
137M16—2741no—
138M16—2741no—
139M16—2741no—
140M16—2741no—
141M16—2741no—
142M16—1371no—
143M16—1371no—
144C2×C8—2741no—
145C2×C8—2741no—
146C2×C8—2741no—
147C2×C8—2741no—
148C2×C8—2741no—
149C2×C8—1371no—
150C2×C8—1371no—
151C2×C8—1371no—
152C2×C8—1371no—
153C2×C8—1371no—
154C2×C8—1371no—
155C16—2742no—
156C16—2742no—
157C23—2741no—
158C23—1371no—
159Q8—2741no—
160Q8—1371no—
161Q8C2×#1176404911yescharacteristic
162D4—21no—
163D4—2741no—
164D4—2741no—
165D4—2741no—
166D4—2741no—
167D4—1371no—
168D4C2×#1176404911yescharacteristic
169C2×C4—21no—
170C2×C4—2741no—
171C2×C4—2741no—
172C2×C4—2741no—
173C2×C4—2741no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4C2×#1176404911yescharacteristic
180C8—2741no—
181C8—2741no—
182C8—1371no—
183C8—1371no—
184C8—1371no—
185C8—1371no—
186C22—21no—
187C22—2742no—
188C22—2741no—
189C22—5482no—
190C22—1371no—
191C22—1371no—
192C22C22⋊#1176404911yescharacteristic
193C4—21no—
194C4—2741no—
195C4—1371no—
196C4—1371no—
197C4C22⋊#1176404911yescharacteristic
198C4C22×#1176404911yescharacteristic
199C2—21no—
200C2—41no—
201C2—2741no—
202C2—5481no—
203C2—1372no—
204C2C2×C22⋊#1176404911yescharacteristic
subgroupG/H
centerC4C22⋊#11764049
commutatorC4×C137C22×C8
FrattiniC4C22×#11764049
Fitting#41×C137C8
socleC2×C137C2×C22⋊C8
prime radical#41×D137C4
cosocle#14555518C23
hypercenter#41#11764049
hypocenterC137C2×#350
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #22578522C2semidirect
2. C2×#17984183C2semidirect
3. #22578504C2semidirect
4. #22578503C2semidirect
6. #17984185C22semidirect
10. #17984183C22semidirect
101. D137#350semidirect
102. C137C2×#350semidirect

Maximal subgroups

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

subgroupindex
#225785222
C2×#179841832
#225785042
#225785032
D137⋊#1312
C2×#350137