Finite Group Databasebeta

← #31780035

Subgroups — #31780035

Order 17536.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#22578068C211yesmaximal, characteristic
2#22577680C211yesmaximal, characteristic
3#22577610C211yesmaximal, characteristic, complemented
4#22577149C211yesmaximal, characteristic, complemented
5#22577135C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic
7#500×C137C211yesmaximal, characteristic, complemented
8C2×Q8×D137—21no—
9C2×#14555500—21no—
10#17984087—21no—
11#17984066—22no—
12#17984064—21no—
13C4×Dic274C2211yescharacteristic
14C4×D548C2211yescharacteristic
15#17984050—21no—
16#17984021—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984012—21no—
20#17984008—21no—
21C4⋊Q8×C137C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23#107×C137—22no—
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—22no—
26C2×Dic274D411yescharacteristic
27Q8×D137—42no—
28Q8×D137—42no—
29C2×D548D411yescharacteristic
30C2×C4×D137—22no—
31C2×C4×D137—21no—
32#14555500—42no—
33C2×#11764040—21no—
34C2×#11764040—21no—
35C4⋊Dic137D411yescharacteristic
36C4⋊Dic137D411yescharacteristic
37#14555489—22no—
38#14555489—21no—
39#14555487—22no—
40#14555487—21no—
41C4×Dic137—21no—
42C2×Q8×C137—21no—
43C2×Q8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137D412yes—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51Dic274—21no—
52Dic274—44no—
53Dic274D812yes—
54D548—21no—
55D548Q1612yescomplemented
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C4×D137—44no—
60C4×D137—41no—
61C4×D137—41no—
62#11764040—41no—
63#11764040—41no—
64Q8×C137—41no—
65Q8×C137—41no—
66Q8×C137—41no—
67Q8×C137—41no—
68C2×C4×C137—22no—
69C2×C4×C137—41no—
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—22no—
74C22×C137#12011yescharacteristic
75C2×D137—22no—
76C2×D137—41no—
77C2×D137—41no—
78C4×C137—21no—
79C4×C137—42no—
80C4×C137—81no—
81C4×C137C2×D811yescharacteristic
82C4×C137C2×Q1611yescharacteristic
83C4×C137AGL(1,ℤ/8)11yescharacteristic
84C4×C137#13711yescharacteristic
85Dic137—22no—
86Dic137—41no—
87Dic137—41no—
88C2×C137#44711yescharacteristic
89C2×C137#45111yescharacteristic
90C2×C137#45611yescharacteristic
91D137—42no—
92C137#164911yescharacteristic, complemented, Sylow
93#1649—1371nomaximal, Sylow
94D4×Q8—1371no—
95#500D13711yescharacteristic, complemented
96#474—1371no—
97#462—1371no—
98#332—1371no—
99Q8⋊C8—1371no—
100D4⋊C8—1371no—
101C22×Q8—2741no—
102C2×Q16—2741no—
103C4⋊Q8—2741no—
104C4⋊Q8C2×D13711yescharacteristic
105C4⋊Q8C2×D13711yescharacteristic
106C22⋊Q8—2741no—
107C22⋊Q8—2742no—
108C4×Q8—1371no—
109C4×D4—2741no—
110C4×D4—1371no—
111#107—22no—
112#107—2741no—
113C4⋊C8—1371no—
114C4⋊C8—1371no—
115#103—2741no—
116#102—2741no—
117C4×C8C2×D13711yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—2742no—
121C2×Q8—5482no—
122C2×Q8—5482no—
123C2×Q8—1371no—
124C2×D4—1371no—
125C22×C4—2742no—
126C22×C4—2741no—
127Q16—5482no—
128C2×C8—21no—
129C2×C8—2741no—
130C2×C8—2741no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2742no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4—1371no—
138C4⋊C4C22⋊D13712yes—
139C22⋊C4—2742no—
140C22⋊C4—2741no—
141C42—2741no—
142C42C22×D13711yescharacteristic
143C23—2741no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—2741no—
149Q8—5484no—
150Q8—1372no—
151D4—2741no—
152D4—1372no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—5484no—
159C2×C4—5481no—
160C2×C4—5481no—
161C2×C4D4×D13711yescharacteristic
162C2×C4D4×D13711yescharacteristic
163C2×C4C2×C22⋊D13711yescharacteristic
164C8—22no—
165C8—5481no—
166C8—5481no—
167C22—2742no—
168C22—5481no—
169C22—5481no—
170C22#1798405611yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—2742no—
175C4—5481no—
176C4—5481no—
177C4D8×D13711yescharacteristic
178C4Q16×D13711yescharacteristic
179C4#1798411211yescharacteristic
180C4#1798412011yescharacteristic
181C2—5482no—
182C2#2257753511yescharacteristic
183C2#2257755711yescharacteristic
184C2#2257758511yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#500×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#500D137
hypocenterC137#1649
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #22577610C2semidirect
4. #22577149C2semidirect
5. #22577135C2semidirect
7. #500×C137C2semidirect
55. D548Q16semidirect
92. C137#1649semidirect
95. #500D137semidirect

Maximal subgroups

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

subgroupindex
#225780682
#225776802
#225776102
#225771492
#225771352
#225771322
#500×C1372
#1649137