Finite Group Databasebeta

← #31782440

Subgroups — #31782440

Order 17536.

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

normal subgroups only

This group has 266 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#22577191C214yesmaximal, complemented
4#413×C137C211yesmaximal, characteristic, complemented
5C22×D548C411yescharacteristic, complemented
6#17984084—21no—
7#17984077—21no—
8#17984062—21no—
9#17984062—21no—
10#17984056C412yescomplemented
11#17984056C2212yes—
12#17984047—21no—
13#17984047C411yescharacteristic, complemented
14#17984003—21no—
15#17984003C2212yes—
16C2×#41×C137C2211yescharacteristic
17C2×M16×C137—21no—
18#101×C137—21no—
19#100×C137C2212yes—
20C23×D137C2×C412yes—
21C2×D548—21no—
22C2×D548—21no—
23C2×D548—41no—
24C2×D548—41no—
25C2×C22⋊D137—22no—
26C2×C22⋊D137—22no—
27C4⋊Dic137—21no—
28C4⋊Dic137—41no—
29C22⋊Dic137—22no—
30C22⋊Dic137C2×C412yes—
31#14555487—22no—
32#14555487—22no—
33C4×Dic137—21no—
34C4×Dic137—41no—
35#41×C137—41no—
36#41×C137—41no—
37C2×Q8×C137D411yescharacteristic
38C2×D4×C137C2×C411yescharacteristic
39C2×D4×C137D411yescharacteristic
40C2×D4×C137C2311yescharacteristic
41C22×C4×C137C2×C411yescharacteristic
42C22×C4×C137D411yescharacteristic
43C22×C4×C137D411yescharacteristic
44M16×C137—22no—
45M16×C137—41no—
46C2×C8×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×C137C22×C411yescharacteristic
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×C137C22⋊C411yescharacteristic
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C8×C137—42no—
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#9911yescharacteristic
90C4×C137#9911yescharacteristic
91Dic137—42no—
92Dic137—82no—
93C2×C137—21no—
94C2×C137—41no—
95C2×C137—41no—
96C2×C137—41no—
97C2×C137C2×#9911yescharacteristic
98D137—82no—
99C137#212211yescharacteristic, complemented, Sylow
100#2122—1371nomaximal, Sylow
101#535—1371no—
102#413D13711yescharacteristic, complemented
103#410—1371no—
104AΣL(1,𝔽16)—1374no—
105C2×#41C2×D13711yescharacteristic
106C22×D4—1371no—
107C2×M16—21no—
108#127—2741no—
109#124—2741no—
110#121—2741no—
111#121—2741no—
112#120—1372no—
113#120—1372no—
114#117—2741no—
115#117—1371no—
116#101—21no—
117#100C2×D13712yes—
118#99—2741no—
119#99—1372no—
120C24—1372no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D13711yescharacteristic
124C2×D4—2742no—
125C2×D4—2742no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—5481no—
129C2×D4—5481no—
130C2×D4C4×D13711yescharacteristic
131C2×D4D54811yescharacteristic
132C2×D4C22×D13711yescharacteristic
133C22×C4C4×D13711yescharacteristic
134C22×C4C22⋊D13711yescharacteristic
135C22×C4D54811yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—2741no—
140C4⋊C4—5481no—
141C22⋊C4—2742no—
142C22⋊C4—2742no—
143C22⋊C4—2742no—
144C22⋊C4—1372no—
145C42—2741no—
146C42—5481no—
147C23—2742no—
148C23—5482no—
149C23—10962no—
150C23#1455548711yescharacteristic
151C23C2×C4×D13711yescharacteristic
152C23C2×D54811yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—5482no—
158D4—5481no—
159D4—5481no—
160D4—10962no—
161D4—10962no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—2742no—
169C2×C4—5482no—
170C2×C4#1455548711yescharacteristic
171C2×C4#1455548711yescharacteristic
172C2×C4#1455548711yescharacteristic
173C2×C4C2×C22⋊D13711yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—5482no—
182C22—5482no—
183C22—5482no—
184C22—10964no—
185C22C2×#1455548711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—5482no—
190C4—10962no—
191C4#1798400211yescharacteristic
192C4#1798400211yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—10962no—
198C2C2×#1798400211yescharacteristic
subgroupG/H
centerC2C2×#17984002
commutatorC23×C137C22×C4
FrattiniC2×D4C22×D137
Fitting#413×C137C2
socleC2×C137C2×#99
prime radical#22577967C2
cosocleC2×D4×C137C23
hypercenter#413D137
hypocenterC137#2122
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22577377C2semidirect
3. #22577191C2semidirect
4. #413×C137C2semidirect
5. C22×D548C4semidirect
10. #17984056C4semidirect
13. #17984047C4semidirect
99. C137#2122semidirect
102. #413D137semidirect

Maximal subgroups

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

subgroupindex
#225779672
#225773772
#225771912
#413×C1372
#2122137