Finite Group Databasebeta

← #31779445

Subgroups — #31779445

Order 17536.

212 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 73 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#22577958C211yesmaximal, characteristic
2#22577455C211yesmaximal, characteristic
3C4⋊C8×D137C211yesmaximal, characteristic
4#22577151C211yesmaximal, characteristic, complemented
5#22577151C211yesmaximal, characteristic, complemented
6#22577149C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×C8×D137—21no—
9#17984088C2211yescharacteristic
10#17984079—21no—
11#17984068C412yes—
12#17984064—21no—
13Q8×Dic137—21no—
14#17984053C412yes—
15#17984046—21no—
16C42×D137C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984017—21no—
20#17984000—21no—
21C8×Dic137—21no—
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274C2×C412yes—
26C2×C4×D137C2×C411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C2×#11764040—21no—
30C2×#11764040—21no—
31C8×D137—42no—
32C4⋊Dic137—22no—
33C4⋊Dic137—21no—
34C4⋊Dic137—21no—
35#14555489—22no—
36#14555487—22no—
37C4×Dic137—22no—
38C4×Dic137C2×C411yescharacteristic
39C4×Dic137D411yescharacteristic
40C4×Dic137D411yescharacteristic
41C2×Q8×C137—21no—
42C2×C8×C137—21no—
43C2×C8×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137C2×C412yes—
46C42×C137C2311yescharacteristic
47C22×D137C22⋊C411yescharacteristic
48Dic274—42no—
49C2×Dic137—22no—
50C2×Dic137C22⋊C411yescharacteristic
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137C22⋊C411yescharacteristic
53C4×D137—21no—
54C4×D137—21no—
55C4×D137—41no—
56C4×D137SD1612yes—
57C4×D137Q1612yes—
58#11764040—41no—
59#11764040—41no—
60Q8×C137—41no—
61Q8×C137—41no—
62C2×C4×C137—22no—
63C2×C4×C137C22×C411yescharacteristic
64C2×C4×C137C2×D411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C8×C137—41no—
67C8×C137—41no—
68C22×C137C2×C22⋊C411yescharacteristic
69C2×D137—21no—
70C2×D137—21no—
71C2×D137#10312yes—
72C4×C137—21no—
73C4×C137—42no—
74C4×C137C2×SD1611yescharacteristic
75C4×C137C2×Q1611yescharacteristic
76C4×C137#13511yescharacteristic
77C4×C137#13511yescharacteristic
78Dic137—21no—
79Dic137—21no—
80Dic137—42no—
81Dic137#10312yes—
82C2×C137#41311yescharacteristic
83C2×C137C2×#10311yescharacteristic
84C2×C137#41611yescharacteristic
85D137—22no—
86C137#154211yescharacteristic, complemented, Sylow
87#1542—1371nomaximal, Sylow
88#533—1371no—
89#431—1371no—
90C2×C4⋊C8—1371no—
91#333—1371no—
92#333—1371no—
93#332—1371no—
94#332D13711yescharacteristic, complemented
95C22×C8—2741no—
96C4⋊Q8—1371no—
97C4⋊Q8C2×D13711yescharacteristic
98#125—2741no—
99C22⋊Q8—2741no—
100C22⋊Q8—1372no—
101C4×Q8—2741no—
102C4×Q8—1372no—
103#117—2741no—
104C2×C42—1371no—
105C4⋊C8—2741no—
106C4⋊C8—1371no—
107C4⋊C8—1371no—
108C4⋊C8C2×D13711yescharacteristic
109C4⋊C8C2×D13711yescharacteristic
110C22⋊C8—2741no—
111C4×C8—2741no—
112C2×Q8—21no—
113C2×Q8—1372no—
114C22×C4—1371no—
115C22×C4—1371no—
116C22×C4—1371no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—2741no—
120C2×C8—2741no—
121C2×C8—5482no—
122C4⋊C4—21no—
123C4⋊C4—2742no—
124C4⋊C4—2742no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4C4×D13712yes—
128C22⋊C4—2742no—
129C42—2742no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C42C22×D13711yescharacteristic
134C23—1371no—
135Q8—41no—
136Q8—41no—
137Q8—5482no—
138C2×C4—22no—
139C2×C4—2741no—
140C2×C4—2741no—
141C2×C4—2742no—
142C2×C4—5481no—
143C2×C4—1371no—
144C2×C4—1371no—
145C2×C4—1371no—
146C2×C4—1372no—
147C2×C4—1372no—
148C2×C4C2×C4×D13711yescharacteristic
149C2×C4D4×D13711yescharacteristic
150C2×C4D4×D13711yescharacteristic
151C8—41no—
152C8—41no—
153C8—5481no—
154C8—5481no—
155C22—2741no—
156C22—2741no—
157C22—1372no—
158C22C22⋊C4×D13711yescharacteristic
159C4—21no—
160C4—42no—
161C4—2741no—
162C4—2741no—
163C4—5482no—
164C4—1372no—
165C4SD16×D13711yescharacteristic
166C4Q16×D13711yescharacteristic
167C4#1798410711yescharacteristic
168C4#1798410911yescharacteristic
169C2—2742no—
170C2#2257738611yescharacteristic
171C2#103×D13711yescharacteristic
172C2#2257739611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC42×C137C23
hypercenter#332D137
hypocenterC137#1542
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #22577151C2semidirect
5. #22577151C2semidirect
6. #22577149C2semidirect
7. #332×C137C2semidirect
86. C137#1542semidirect
94. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#225774552
C4⋊C8×D1372
#225771512
#225771492
#332×C1372
#1542137