Finite Group Databasebeta

← #31780868

Subgroups — #31780868

Order 17536.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C4⋊C4×D137C211yesmaximal, characteristic
2#22577366C212yesmaximal
3#22577172C211yesmaximal, characteristic, complemented
4#22577171C212yesmaximal, complemented
5#343×C137C211yesmaximal, characteristic, complemented
6C22×C4×D137C412yes—
7C22×C4×D137C2211yescharacteristic
8C2×#14555494C2212yes—
9C2×C4⋊Dic137C2211yescharacteristic
10C2×#14555489C412yes—
11C4⋊C4×D137—22no—
12C4⋊C4×D137—22no—
13#17984000—22no—
14#17984000—22no—
15C2×M16×C137C2212yes—
16C2×C4⋊C4×C137C2211yescharacteristic
17C23×D137C2×C411yescharacteristic
18C22×Dic137C2×C411yescharacteristic
19C22×Dic137C2×C412yes—
20C2×C4×D137—22no—
21C2×C4×D137—22no—
22C2×C4×D137—21no—
23C2×C4×D137—22no—
24C2×C4×D137—21no—
25C2×C4×D137—44no—
26C2×C4×D137D412yes—
27C2×C4×D137D412yes—
28#14555494—22no—
29#14555494—22no—
30C2×#11764040—22no—
31C4⋊Dic137—22no—
32#14555489—24no—
33C22×C4×C137C2×C412yes—
34C22×C4×C137C2311yescharacteristic
35M16×C137—22no—
36M16×C137—22no—
37C2×C8×C137—22no—
38C4⋊C4×C137—22no—
39C22×D137—21no—
40C22×D137—21no—
41C22×D137—22no—
42C22×D137—22no—
43C22×D137C22⋊C411yescharacteristic
44C22×D137C22⋊C411yescharacteristic
45C2×Dic137—21no—
46C2×Dic137—21no—
47C2×Dic137—22no—
48C2×Dic137—42no—
49C2×Dic137C22⋊C412yes—
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—22no—
53C4×D137—22no—
54C4×D137—44no—
55C4×D137—44no—
56#11764040—42no—
57C23×C137C22×C411yescharacteristic
58C2×C4×C137—22no—
59C2×C4×C137—42no—
60C2×C4×C137C2×D412yes—
61C2×C4×C137#4111yescharacteristic
62C2×C4×C137#4112yes—
63C2×C4×C137#4111yescharacteristic
64C8×C137—42no—
65C22×C137—21no—
66C22×C137—21no—
67C22×C137C2×C22⋊C411yescharacteristic
68C22×C137#11711yescharacteristic
69C22×C137#11711yescharacteristic
70C2×D137—24no—
71C2×D137—24no—
72C2×D137—21no—
73C2×D137—21no—
74C2×D137—21no—
75C2×D137—21no—
76C2×D137#10012yes—
77C2×D137#10112yes—
78C4×C137—42no—
79C4×C137#12312yes—
80C4×C137#12312yes—
81Dic137—22no—
82Dic137—42no—
83C2×C137—21no—
84C2×C137—21no—
85C2×C137#38111yescharacteristic
86C2×C137C2×#10011yescharacteristic
87C2×C137C2×#10111yescharacteristic
88D137—22no—
89D137—22no—
90C137#182111yescharacteristic, complemented, Sylow
91#1821—1371nomaximal, Sylow
92C22×C4⋊C4—1371no—
93#407—1372no—
94#343—1372no—
95#343—1371no—
96#343D13711yescharacteristic, complemented
97C23×C4—1371no—
98C23×C4—1372no—
99C2×M16—1372no—
100C2×M16C2×D13712yes—
101C2×C4⋊C4—2742no—
102C2×C4⋊C4—2742no—
103C2×C4⋊C4—1372no—
104C2×C4⋊C4—1371no—
105C2×C4⋊C4C2×D13711yescharacteristic
106C22⋊C8—2742no—
107C22⋊C8—2742no—
108C24—1371no—
109C22×C4—2742no—
110C22×C4—2742no—
111C22×C4—2741no—
112C22×C4—2742no—
113C22×C4—2741no—
114C22×C4—5484no—
115C22×C4—1371no—
116C22×C4—1372no—
117C22×C4—1372no—
118C22×C4—1372no—
119C22×C4C4×D13712yes—
120C22×C4C22×D13711yescharacteristic
121M16—22no—
122M16—22no—
123M16—2742no—
124M16—2742no—
125C2×C8—22no—
126C2×C8—2742no—
127C4⋊C4—22no—
128C4⋊C4—2744no—
129C4⋊C4—2742no—
130C23—2742no—
131C23—2742no—
132C23—2741no—
133C23—2741no—
134C23—1371no—
135C23—1371no—
136C23C2×C4×D13711yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—2741no—
140C2×C4—2741no—
141C2×C4—2742no—
142C2×C4—2742no—
143C2×C4—2742no—
144C2×C4—2742no—
145C2×C4—2742no—
146C2×C4—5484no—
147C2×C4—5484no—
148C2×C4—5482no—
149C2×C4—1372no—
150C2×C4D4×D13712yes—
151C2×C4#1455550911yescharacteristic
152C2×C4#1455551012yes—
153C2×C4#1455551111yescharacteristic
154C8—42no—
155C8—5482no—
156C22—21no—
157C22—21no—
158C22—2744no—
159C22—2744no—
160C22—2741no—
161C22—2741no—
162C22—2741no—
163C22—2741no—
164C22—1372no—
165C22—1372no—
166C22C22⋊C4×D13711yescharacteristic
167C22#1798404511yescharacteristic
168C22#1798404611yescharacteristic
169C4—42no—
170C4—2742no—
171C4—5482no—
172C4#1798407012yes—
173C4#1798407312yes—
174C2—21no—
175C2—21no—
176C2—2742no—
177C2—2742no—
178C2#2257726511yescharacteristic
179C2#100×D13711yescharacteristic
180C2#101×D13711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC23×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#343×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC23×D137C2×C4
cosocleC22×C4×C137C23
hypercenter#343D137
hypocenterC137#1821
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #22577172C2semidirect
4. #22577171C2semidirect
5. #343×C137C2semidirect
90. C137#1821semidirect
96. #343D137semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1372
#225773662
#225771722
#225771712
#343×C1372
#1821137