Finite Group Databasebeta

← #31780823

Subgroups — #31780823

Order 17536.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×#14555493C211yesmaximal, characteristic
2C2×#17984020C211yesmaximal, characteristic, complemented
3#22577260C212yesmaximal
4#22577165C212yesmaximal, complemented
5C2×#107×C137C211yesmaximal, characteristic, complemented
6C22×C4×D137C2211yescharacteristic
7C2×#14555493—22no—
8C2×#14555493—22no—
9C2×#14555493C414yescomplemented
10C22×#11764040C2211yescharacteristic
11C2×C4⋊Dic137C2212yes—
12C2×#14555487—22no—
13#17984020—22no—
14#17983995—22no—
15C22×C8×C137C2211yescharacteristic
16C2×C4⋊C4×C137C2212yes—
17#107×C137—22no—
18C23×D137D411yescharacteristic
19C22×Dic137—22no—
20C22×Dic137D411yescharacteristic
21C2×C4×D137—21no—
22C2×C4×D137—21no—
23C2×C4×D137—21no—
24C2×C4×D137—21no—
25C2×C4×D137C2×C412yes—
26C2×C4×D137Q812yes—
27#14555493—28no—
28C2×#11764040—21no—
29C2×#11764040—21no—
30C2×#11764040C2×C412yes—
31C4⋊Dic137—22no—
32C4⋊Dic137—22no—
33#14555487—44no—
34C22×C4×C137—22no—
35C22×C4×C137C2311yescharacteristic
36C2×C8×C137C2×C412yes—
37C2×C8×C137D412yes—
38C2×C8×C137D412yes—
39C4⋊C4×C137—22no—
40C4⋊C4×C137—22no—
41C22×D137—21no—
42C22×D137—21no—
43C22×D137—22no—
44C22×D137—22no—
45C22×D137C4⋊C412yes—
46C2×Dic137—21no—
47C2×Dic137—21no—
48C2×Dic137—42no—
49C2×Dic137—42no—
50C2×Dic137—42no—
51C2×Dic137C4⋊C412yes—
52C4×D137—22no—
53C4×D137—22no—
54C4×D137—22no—
55C4×D137—22no—
56#11764040—22no—
57C23×C137C2×D411yescharacteristic
58C2×C4×C137—42no—
59C2×C4×C137—42no—
60C2×C4×C137—42no—
61C2×C4×C137C22×C411yescharacteristic
62C2×C4×C137C2×D411yescharacteristic
63C2×C4×C137C2×D411yescharacteristic
64C2×C4×C137C2×Q811yescharacteristic
65C2×C4×C137#4111yescharacteristic
66C2×C4×C137#4111yescharacteristic
67C8×C137C22⋊C414yescomplemented
68C22×C137C2×C4⋊C411yescharacteristic
69C22×C137#12111yescharacteristic
70C22×C137#12111yescharacteristic
71C22×C137AGL(1,ℤ/8)12yes—
72C22×C137#13712yes—
73C2×D137—22no—
74C2×D137—24no—
75C2×D137—42no—
76C2×D137—41no—
77C2×D137—41no—
78C4×C137—82no—
79C4×C137C2×C22⋊C411yescharacteristic
80C4×C137#11711yescharacteristic
81C4×C137C22⋊Q811yescharacteristic
82C4×C137C22⋊Q811yescharacteristic
83Dic137—22no—
84Dic137—82no—
85C2×C137#38011yescharacteristic
86C2×C137#42812yes—
87C2×C137#46812yes—
88C2×C137#46911yescharacteristic
89C2×C137#47011yescharacteristic
90D137—42no—
91C137#180911yescharacteristic, complemented, Sylow
92#1809—1371nomaximal, Sylow
93C22×M16—1371no—
94C2×#107D13711yescharacteristic, complemented
95C2×#106—1371no—
96#380—1372no—
97#340—1372no—
98C23×C4—1371no—
99C2×M16—2742no—
100C2×M16—2742no—
101C2×M16—1374no—
102C22×C8—1371no—
103C22×C8C2×D13711yescharacteristic
104C2×C4⋊C4—1372no—
105C2×C4⋊C4C2×D13712yes—
106C2×C22⋊C4—2742no—
107#107—22no—
108#106—2742no—
109#95—2742no—
110C24—1371no—
111C22×C4—22no—
112C22×C4—2741no—
113C22×C4—2741no—
114C22×C4—2741no—
115C22×C4—2741no—
116C22×C4—2742no—
117C22×C4—1371no—
118C22×C4—1372no—
119C22×C4—1372no—
120C22×C4C22×D13711yescharacteristic
121M16—2748no—
122C2×C8—2741no—
123C2×C8—2741no—
124C2×C8—1372no—
125C2×C8C4×D13712yes—
126C2×C8C22⋊D13712yes—
127C2×C8D54812yes—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—2742no—
131C4⋊C4—2742no—
132C22⋊C4—5484no—
133C23—2741no—
134C23—2741no—
135C23—2742no—
136C23—2742no—
137C23—1372no—
138C23D4×D13711yescharacteristic
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—2742no—
145C2×C4—2742no—
146C2×C4—2742no—
147C2×C4—2742no—
148C2×C4—5482no—
149C2×C4—5482no—
150C2×C4—5482no—
151C2×C4—1372no—
152C2×C4C2×C4×D13711yescharacteristic
153C2×C4C2×C22⋊D13711yescharacteristic
154C2×C4C2×D54811yescharacteristic
155C2×C4Q8×D13711yescharacteristic
156C2×C4#1455550911yescharacteristic
157C2×C4#1455551111yescharacteristic
158C8—2742no—
159C8#1455548714yescomplemented
160C22—2744no—
161C22—2742no—
162C22—5482no—
163C22—5481no—
164C22—5481no—
165C22C4⋊C4×D13711yescharacteristic
166C22#1798405911yescharacteristic
167C22#1798406311yescharacteristic
168C22#1798411212yes—
169C22#1798412012yes—
170C4—82no—
171C4—2742no—
172C4—10962no—
173C4C2×#1455548711yescharacteristic
174C4#1798404611yescharacteristic
175C4#1798406411yescharacteristic
176C4#1798406811yescharacteristic
177C2—5482no—
178C2#2257726011yescharacteristic
179C2#2257744412yes—
180C2#2257765012yes—
181C2#2257765311yescharacteristic
182C2#2257765811yescharacteristic
subgroupG/H
centerC23D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
FittingC2×#107×C137C2
socleC23×C137C2×D4
prime radicalC23×D137D4
cosocleC22×C4×C137C23
hypercenterC2×#107D137
hypocenterC137#1809
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#17984020C2semidirect
4. #22577165C2semidirect
5. C2×#107×C137C2semidirect
9. C2×#14555493C4semidirect
67. C8×C137C22⋊C4semidirect
91. C137#1809semidirect
94. C2×#107D137semidirect
159. C8#14555487semidirect

Maximal subgroups

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

subgroupindex
C22×#145554932
C2×#179840202
#225772602
#225771652
C2×#107×C1372
#1809137