Finite Group Databasebeta

← #31782460

Subgroups — #31782460

Order 17536.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#22577967C211yesmaximal, characteristic
2#22577377C211yesmaximal, characteristic, complemented
3#22577195C214yesmaximal, complemented
4#410×C137C211yesmaximal, characteristic, complemented
5C22×D548C411yescharacteristic, complemented
6#17984084C2212yes—
7#17984077C412yescomplemented
8#17984062—21no—
9#17984062—21no—
10#17984056—21no—
11#17984056—21no—
12#17984047—21no—
13#17984047C411yescharacteristic, complemented
14#17984003—21no—
15#17984003C2212yescomplemented
16C2×#41×C137C2211yescharacteristic
17#117×C137—21no—
18#99×C137—21no—
19#99×C137C2212yescomplemented
20C23×D137—21no—
21C2×D548—21no—
22C2×D548—41no—
23C2×D548—41no—
24C2×D548C2×C412yes—
25C2×C22⋊D137—22no—
26C2×C22⋊D137—22no—
27C4⋊Dic137—21no—
28C4⋊Dic137—41no—
29C22⋊Dic137—21no—
30C22⋊Dic137—22no—
31#14555487—22no—
32#14555487—22no—
33C4×Dic137—41no—
34C4×Dic137C2×C412yes—
35#41×C137—41no—
36#41×C137—41no—
37C2×Q8×C137C2×C411yescharacteristic
38C2×D4×C137D411yescharacteristic
39C2×D4×C137D411yescharacteristic
40C2×D4×C137C2311yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137D411yescharacteristic
43C22×C4×C137D411yescharacteristic
44C4⋊C4×C137—41no—
45C22⋊C4×C137—22no—
46C42×C137—41no—
47C22×D137—22no—
48C22×D137—42no—
49C22×D137—82no—
50D548—42no—
51D548—41no—
52D548—41no—
53C22⋊D137—82no—
54C22⋊D137—82no—
55C2×Dic137—22no—
56C2×Dic137—42no—
57C23×C137C22⋊C411yescharacteristic
58C23×C137C2×D411yescharacteristic
59C23×C137C2×D411yescharacteristic
60Q8×C137—41no—
61D4×C137—41no—
62D4×C137—41no—
63D4×C137—41no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22×C411yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—21no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—42no—
83C2×D137—42no—
84C2×D137—42no—
85C2×D137—84no—
86C4×C137—21no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—82no—
90C4×C137#9911yescharacteristic
91C4×C137#9911yescharacteristic
92Dic137—42no—
93Dic137—82no—
94C2×C137—21no—
95C2×C137—41no—
96C2×C137—41no—
97C2×C137—41no—
98C2×C137C2×#9911yescharacteristic
99D137—82no—
100C137#212811yescharacteristic, complemented, Sylow
101#2128—1371nomaximal, Sylow
102#535—1371no—
103#410—1371no—
104#410D13711yescharacteristic, complemented
105#353—1374no—
106C2×#41C2×D13711yescharacteristic
107C22×D4—1371no—
108#127—1372no—
109#124—1372no—
110#121—2741no—
111#121—2741no—
112#120—2741no—
113#120—2741no—
114#117—21no—
115#117—2741no—
116#117—1371no—
117#99—21no—
118#99—2741no—
119#99—1372no—
120#99C2×D13712yescomplemented
121C24—2741no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D13711yescharacteristic
125C2×D4—2742no—
126C2×D4—2742no—
127C2×D4—2741no—
128C2×D4—5481no—
129C2×D4—5481no—
130C2×D4—1372no—
131C2×D4C22⋊D13711yescharacteristic
132C2×D4D54811yescharacteristic
133C2×D4C22×D13711yescharacteristic
134C22×C4C4×D13711yescharacteristic
135C22×C4C22⋊D13711yescharacteristic
136C22×C4D54811yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—2741no—
139C4⋊C4—5481no—
140C22⋊C4—22no—
141C22⋊C4—2741no—
142C22⋊C4—2742no—
143C22⋊C4—2742no—
144C22⋊C4—2742no—
145C42—41no—
146C42—5481no—
147C42—1372no—
148C23—2742no—
149C23—5482no—
150C23—10962no—
151C23#1455548711yescharacteristic
152C23C2×C22⋊D13711yescharacteristic
153C23C2×D54811yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—5482no—
159D4—5481no—
160D4—5481no—
161D4—10962no—
162D4—10962no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—2742no—
171C2×C4—5482no—
172C2×C4#1455548711yescharacteristic
173C2×C4#1455548711yescharacteristic
174C2×C4#1455548711yescharacteristic
175C2×C4C2×C4×D13711yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—5482no—
183C22—5482no—
184C22—5482no—
185C22—10964no—
186C22C2×#1455548711yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—5482no—
192C4—10962no—
193C4#1798400211yescharacteristic
194C4#1798400211yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—10962no—
200C2C2×#1798400211yescharacteristic
subgroupG/H
centerC2C2×#17984002
commutatorC2×C4×C137C22×C4
FrattiniC2×D4C22×D137
Fitting#410×C137C2
socleC2×C137C2×#99
prime radical#22577967C2
cosocleC2×D4×C137C23
hypercenter#410D137
hypocenterC137#2128
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22577377C2semidirect
3. #22577195C2semidirect
4. #410×C137C2semidirect
5. C22×D548C4semidirect
7. #17984077C4semidirect
13. #17984047C4semidirect
15. #17984003C22semidirect
19. #99×C137C22semidirect
100. C137#2128semidirect
104. #410D137semidirect
120. #99C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779672
#225773772
#225771952
#410×C1372
#2128137