Finite Group Databasebeta

← #31779949

Subgroups — #31779949

Order 17536.

188 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578131C211yesmaximal, characteristic
2#22577754C211yesmaximal, characteristic
3#22577699C211yesmaximal, characteristic
4#22577698C211yesmaximal, characteristic
5#22577148C211yesmaximal, characteristic
6#22577135C211yesmaximal, characteristic
7Q8⋊C8×C137C211yesmaximal, characteristic
8#17984088C2211yescharacteristic
9#17984088C2211yescharacteristic
10#17984086—21no—
11#17984080—21no—
12#17984078—22no—
13C4×Dic274—21no—
14C4×Dic274C2211yescharacteristic
15#17984053—21no—
16#17984022—21no—
17#17984021—21no—
18C4⋊#11764040C2211yescharacteristic
19#17984013—21no—
20#17984013—22no—
21#17984012—21no—
22C4×Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×Dic274—21no—
27C2×Dic274D411yescharacteristic
28C2×#11764040—21no—
29C4⋊Dic137—21no—
30C4⋊Dic137—21no—
31C4⋊Dic137—21no—
32C4⋊Dic137D412yes—
33C4⋊Dic137D411yescharacteristic
34#14555489—22no—
35#14555489—22no—
36#14555489—22no—
37#14555489—21no—
38C4×Dic137—22no—
39C4×Dic137—21no—
40C2×Q8×C137D411yescharacteristic
41C2×C8×C137—21no—
42C2×C8×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137D411yescharacteristic
45C42×C137—21no—
46C42×C137C2311yescharacteristic
47Dic274—21no—
48Dic274—41no—
49Dic274—41no—
50Dic274—42no—
51Dic274Q1612yes—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—22no—
55C2×Dic137—41no—
56#11764040—41no—
57Q8×C137—21no—
58Q8×C137—21no—
59C2×C4×C137—21no—
60C2×C4×C137—21no—
61C2×C4×C137C2×D411yescharacteristic
62C2×C4×C137C2×D411yescharacteristic
63C2×C4×C137C2×D411yescharacteristic
64C8×C137—41no—
65C8×C137—41no—
66C22×C137#12011yescharacteristic
67C4×C137—21no—
68C4×C137—41no—
69C4×C137—41no—
70C4×C137C2×Q1611yescharacteristic
71C4×C137#13511yescharacteristic
72C4×C137#13711yescharacteristic
73C4×C137#13711yescharacteristic
74Dic137—22no—
75Dic137—42no—
76Dic137—41no—
77Dic137—81no—
78C2×C137#45111yescharacteristic
79C2×C137#45211yescharacteristic
80C2×C137#45411yescharacteristic
81C137#163611yescharacteristic, complemented, Sylow
82#1636—1371nomaximal, Sylow
83#557—1371no—
84#487—1371no—
85#477—1371no—
86#477—1371no—
87#332—1371no—
88Q8⋊C8—1371no—
89Q8⋊C8D13711yescharacteristic
90C4⋊Q8—2741no—
91C4⋊Q8—1371no—
92C4⋊Q8—1371no—
93#125—2741no—
94#125—2742no—
95C4×Q8—2741no—
96C4×Q8—2741no—
97C4×Q8—1371no—
98C4×Q8C2×D13711yescharacteristic
99#107—2741no—
100#107—2741no—
101C4⋊C8—1371no—
102C4⋊C8C2×D13711yescharacteristic
103#103—2741no—
104#103—2741no—
105#103—2742no—
106C4×C8C2×D13711yescharacteristic
107C2×Q8—2741no—
108C2×Q8—2741no—
109C2×Q8—1371no—
110C2×Q8D54811yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—2741no—
114C4⋊C4—21no—
115C4⋊C4—2742no—
116C4⋊C4—2742no—
117C4⋊C4—2742no—
118C4⋊C4—2741no—
119C4⋊C4—2741no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—1372no—
123C4⋊C4—1371no—
124C4⋊C4D54811yescharacteristic
125C42—21no—
126C42—2742no—
127C42—2741no—
128C42C22×D13711yescharacteristic
129Q8—21no—
130Q8—21no—
131Q8—2741no—
132Q8—5482no—
133Q8—5481no—
134Q8—5481no—
135Q8—1372no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—2741no—
139C2×C4—2741no—
140C2×C4—2742no—
141C2×C4—5481no—
142C2×C4D4×D13711yescharacteristic
143C2×C4D4×D13711yescharacteristic
144C2×C4C2×D54811yescharacteristic
145C8—41no—
146C8—41no—
147C8—5481no—
148C22#1798405811yescharacteristic
149C4—21no—
150C4—41no—
151C4—41no—
152C4—2742no—
153C4—5482no—
154C4—5481no—
155C4—10961no—
156C4Q16×D13711yescharacteristic
157C4#1798411011yescharacteristic
158C4#1798411711yescharacteristic
159C4#1798412111yescharacteristic
160C2#2257756011yescharacteristic
161C2#2257756711yescharacteristic
162C2#2257757711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socle, prime radicalC22×C137#120
hypercenterQ8⋊C8D137
hypocenterC137#1636
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
81. C137#1636semidirect

Maximal subgroups

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

subgroupindex
#225781312
#225777542
#225776992
#225776982
#225771482
#225771352
Q8⋊C8×C1372
#1636137