Finite Group Databasebeta

← #31779955

Subgroups — #31779955

Order 17536.

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

normal subgroups only

This group has 228 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#22577755C211yesmaximal, characteristic, complemented
3#22577699C211yesmaximal, characteristic, complemented
4#22577694C211yesmaximal, characteristic
5#22577133C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic
7#332×C137C211yesmaximal, characteristic, complemented
8C2×Q8×D137—21no—
9#17984088C2211yescharacteristic
10#17984087—21no—
11#17984066—22no—
12#17984064—21no—
13C4×Dic274C2211yescharacteristic
14C4×D548C2211yescharacteristic
15#17984050—21no—
16#17984022—21no—
17#17984020—21no—
18#17984013—21no—
19#17984011—22no—
20#17984010—21no—
21C4×#11764040C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25C2×Dic274—22no—
26C2×Dic274—21no—
27C2×Dic274D411yescharacteristic
28Q8×D137—42no—
29Q8×D137—42no—
30C2×D548D411yescharacteristic
31C2×C4×D137—22no—
32C2×C4×D137—21no—
33C2×#11764040—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137D411yescharacteristic
37C4⋊Dic137D411yescharacteristic
38#14555489—22no—
39#14555489—21no—
40#14555487—22no—
41#14555487—21no—
42C4×Dic137—21no—
43C2×Q8×C137—21no—
44C2×C8×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137D412yes—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50Dic274—21no—
51Dic274—44no—
52Dic274—42no—
53Dic274SD1612yes—
54D548—21no—
55D548Q1612yes—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137—41no—
60C4×D137—44no—
61C4×D137—41no—
62C4×D137—41no—
63#11764040—41no—
64Q8×C137—41no—
65Q8×C137—41no—
66C2×C4×C137—22no—
67C2×C4×C137C2×D411yescharacteristic
68C2×C4×C137C2×D411yescharacteristic
69C2×C4×C137C2×D411yescharacteristic
70C8×C137—41no—
71C8×C137—41no—
72C22×C137#12011yescharacteristic
73C2×D137—22no—
74C2×D137—41no—
75C2×D137—41no—
76C4×C137—21no—
77C4×C137—42no—
78C4×C137C2×SD1611yescharacteristic
79C4×C137C2×Q1611yescharacteristic
80C4×C137AGL(1,ℤ/8)11yescharacteristic
81C4×C137#13711yescharacteristic
82Dic137—22no—
83Dic137—41no—
84Dic137—41no—
85Dic137—81no—
86C2×C137#45011yescharacteristic
87C2×C137#45111yescharacteristic
88C2×C137#45411yescharacteristic
89D137—42no—
90C137#163711yescharacteristic, complemented, Sylow
91#1637—1371nomaximal, Sylow
92D4×Q8—1371no—
93#487—1371no—
94#477—1371no—
95#476—1371no—
96#332D13711yescharacteristic, complemented
97Q8⋊C8—1371no—
98D4⋊C8—1371no—
99C22×Q8—2741no—
100C4⋊Q8—2741no—
101C4⋊Q8—1371no—
102C4⋊Q8C2×D13711yescharacteristic
103C22⋊Q8—2741no—
104C22⋊Q8—2742no—
105C4×Q8—1371no—
106C4×D4—2741no—
107C4×D4—1371no—
108#107—2741no—
109#106—2741no—
110C4⋊C8C2×D13711yescharacteristic
111C4⋊C8C2×D13711yescharacteristic
112#103—2741no—
113#103—2742no—
114#102—2741no—
115C4×C8—1371no—
116C2×Q8—21no—
117C2×Q8—2742no—
118C2×Q8—2741no—
119C2×Q8—5482no—
120C2×Q8—5482no—
121C2×Q8—1371no—
122C2×D4—1371no—
123C22×C4—2742no—
124C22×C4—2741no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—2741no—
128C4⋊C4—21no—
129C4⋊C4—2742no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—1371no—
134C4⋊C4—1371no—
135C4⋊C4D54812yes—
136C22⋊C4—2742no—
137C22⋊C4—2741no—
138C42—2741no—
139C42C22×D13711yescharacteristic
140C23—2741no—
141Q8—41no—
142Q8—41no—
143Q8—2741no—
144Q8—5482no—
145Q8—5484no—
146Q8—1372no—
147D4—2741no—
148D4—1372no—
149C2×C4—22no—
150C2×C4—2741no—
151C2×C4—2741no—
152C2×C4—2741no—
153C2×C4—5484no—
154C2×C4—5481no—
155C2×C4—5481no—
156C2×C4—5481no—
157C2×C4D4×D13711yescharacteristic
158C2×C4D4×D13711yescharacteristic
159C2×C4C2×D54811yescharacteristic
160C8—41no—
161C8—41no—
162C8—5481no—
163C22—2742no—
164C22—5481no—
165C22—5481no—
166C22#1798405811yescharacteristic
167C4—21no—
168C4—42no—
169C4—2742no—
170C4—5481no—
171C4—5481no—
172C4—10961no—
173C4SD16×D13711yescharacteristic
174C4Q16×D13711yescharacteristic
175C4#1798411211yescharacteristic
176C4#1798411711yescharacteristic
177C2—5482no—
178C2#2257755511yescharacteristic
179C2#2257756011yescharacteristic
180C2#2257757811yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#332D137
hypocenterC137#1637
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22577755C2semidirect
3. #22577699C2semidirect
5. #22577133C2semidirect
7. #332×C137C2semidirect
90. C137#1637semidirect
96. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225780682
#225777552
#225776992
#225776942
#225771332
#225771302
#332×C1372
#1637137