Finite Group Databasebeta

← #31779516

Subgroups — #31779516

Order 17536.

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

normal subgroups only

This group has 202 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#22577958C211yesmaximal, characteristic
2#22577459C211yesmaximal, characteristic
3#22577425C211yesmaximal, characteristic
4#22577152C211yesmaximal, characteristic, complemented
5#22577150C211yesmaximal, characteristic, complemented
6#22577148C211yesmaximal, characteristic, complemented
7#333×C137C211yesmaximal, characteristic, complemented
8C2×#14555493—21no—
9#17984088C2211yescharacteristic
10#17984079—21no—
11#17984068—21no—
12#17984064C412yes—
13Q8×Dic137C412yes—
14#17984053—21no—
15#17984046—21no—
16C42×D137C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984017—21no—
20#17984000—21no—
21#17983998—21no—
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×C4×D137C2×C411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137D411yescharacteristic
29#14555493—42no—
30C2×#11764040—21no—
31C2×#11764040—21no—
32C4⋊Dic137—22no—
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C412yes—
35#14555489—22no—
36#14555487—22no—
37C4×Dic137—22no—
38C4×Dic137C2×C411yescharacteristic
39C4×Dic137D411yescharacteristic
40C4×Dic137D411yescharacteristic
41C2×Q8×C137C2×C412yes—
42C2×C8×C137—21no—
43C2×C8×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137—21no—
46C42×C137C2311yescharacteristic
47C22×D137C22⋊C411yescharacteristic
48Dic274—41no—
49Dic274—41no—
50C2×Dic137—22no—
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C22⋊C411yescharacteristic
54C4×D137—21no—
55C4×D137—21no—
56C4×D137—21no—
57C4×D137—21no—
58C4×D137—41no—
59#11764040—41no—
60#11764040—41no—
61Q8×C137—42no—
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—21no—
72C4×C137—21no—
73C4×C137—42no—
74C4×C137AGL(1,ℤ/8)11yescharacteristic
75C4×C137#13711yescharacteristic
76C4×C137#13711yescharacteristic
77C4×C137#13711yescharacteristic
78Dic137—21no—
79Dic137—21no—
80Dic137—21no—
81Dic137—42no—
82C2×C137#41311yescharacteristic
83C2×C137#41711yescharacteristic
84C2×C137#41911yescharacteristic
85D137—41no—
86C137#155911yescharacteristic, complemented, Sylow
87#1559—1371nomaximal, Sylow
88#533—1371no—
89#432—1371no—
90#423—1371no—
91#333—1371no—
92#333D13711yescharacteristic, complemented
93#332—1371no—
94#332—1371no—
95C2×M16—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—
111#97—2741no—
112C2×Q8—2741no—
113C2×Q8C4×D13712yes—
114C22×C4—1371no—
115C22×C4—1371no—
116C22×C4—1371no—
117M16—5482no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—2741no—
121C2×C8—2741no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—2742no—
125C4⋊C4—2742no—
126C4⋊C4—2741no—
127C4⋊C4—1372no—
128C22⋊C4—2742no—
129C42—2742no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C42C22×D13711yescharacteristic
134C23—1371no—
135Q8—42no—
136Q8—5481no—
137Q8—5481no—
138C2×C4—22no—
139C2×C4—2741no—
140C2×C4—2741no—
141C2×C4—2741no—
142C2×C4—2741no—
143C2×C4—2742no—
144C2×C4—5481no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4C2×C4×D13711yescharacteristic
149C2×C4D4×D13711yescharacteristic
150C2×C4D4×D13711yescharacteristic
151C8—41no—
152C8—41no—
153C8—5481no—
154C8—5481no—
155C22—2741no—
156C22—2741no—
157C22—2741no—
158C22C22⋊C4×D13711yescharacteristic
159C4—21no—
160C4—42no—
161C4—2741no—
162C4—2741no—
163C4—2741no—
164C4—5482no—
165C4#1798411511yescharacteristic
166C4#1798411711yescharacteristic
167C4#1798412011yescharacteristic
168C4#1798412011yescharacteristic
169C2—5481no—
170C2#2257738411yescharacteristic
171C2#2257740311yescharacteristic
172C2#2257741011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#333×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC42×C137C23
hypercenter#333D137
hypocenterC137#1559
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
4. #22577152C2semidirect
5. #22577150C2semidirect
6. #22577148C2semidirect
7. #333×C137C2semidirect
86. C137#1559semidirect
92. #333D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#225774592
#225774252
#225771522
#225771502
#225771482
#333×C1372
#1559137