Finite Group Databasebeta

← #31779952

Subgroups — #31779952

Order 17536.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D548C211yesmaximal, characteristic
2#22577753C211yesmaximal, characteristic, complemented
3#22577690C211yesmaximal, characteristic
4#22577150C211yesmaximal, characteristic, complemented
5Q8⋊#11764040C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic
7#477×C137C211yesmaximal, characteristic, complemented
8C2×Q8×D137—21no—
9#17984088—21no—
10#17984088C2211yescharacteristic
11#17984068—22no—
12#17984068—21no—
13C4×D548—21no—
14C4×D548C2211yescharacteristic
15#17984019—21no—
16C4⋊#11764040C2211yescharacteristic
17#17984012—22no—
18#17984008—21no—
19C4×#11764040C2211yescharacteristic
20C4⋊Q8×C137C2211yescharacteristic
21C4×Q8×C137C2211yescharacteristic
22#107×C137—21no—
23C4⋊C8×C137C2211yescharacteristic
24#103×C137—21no—
25C2×Dic274—22no—
26C2×Dic274—21no—
27Q8×D137—42no—
28Q8×D137—42no—
29C2×D548D411yescharacteristic
30C2×C4×D137—22no—
31C2×C4×D137—21no—
32C2×#11764040—21no—
33C2×#11764040—21no—
34C4⋊Dic137—22no—
35C4⋊Dic137—21no—
36C4⋊Dic137D412yes—
37C4⋊Dic137D411yescharacteristic
38#14555487—22no—
39#14555487—21no—
40C2×Q8×C137—21no—
41C2×Q8×C137D411yescharacteristic
42C2×C8×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137D411yescharacteristic
47C42×C137—21no—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50Dic274—44no—
51Dic274—41no—
52Dic274—41no—
53D548—21no—
54D548Q1612yes—
55C2×Dic137—22no—
56C2×Dic137—21no—
57C4×D137—44no—
58C4×D137—41no—
59C4×D137—41no—
60#11764040—41no—
61#11764040—41no—
62Q8×C137—21no—
63Q8×C137—42no—
64Q8×C137SD1612yes—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137C2×D411yescharacteristic
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C8×C137—41no—
72C22×C137#12011yescharacteristic
73C2×D137—22no—
74C2×D137—41no—
75C2×D137—41no—
76C4×C137—22no—
77C4×C137—21no—
78C4×C137—41no—
79C4×C137—81no—
80C4×C137C2×SD1611yescharacteristic
81C4×C137C2×Q1611yescharacteristic
82C4×C137AGL(1,ℤ/8)11yescharacteristic
83C4×C137#13711yescharacteristic
84Dic137—42no—
85Dic137—41no—
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#477D13711yescharacteristic, complemented
95#476—1371no—
96#332—1371no—
97Q8⋊C8—1371no—
98D4⋊C8—1371no—
99C22×Q8—2741no—
100C4⋊Q8—2741no—
101C4⋊Q8—1371no—
102C4⋊Q8C2×D13711yescharacteristic
103C22⋊Q8—2742no—
104C22⋊Q8—2741no—
105C4×Q8C2×D13711yescharacteristic
106C4×D4—2741no—
107C4×D4—1371no—
108#107—21no—
109#106—2741no—
110C4⋊C8—1371no—
111C4⋊C8C2×D13711yescharacteristic
112#103—21no—
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×Q8C22⋊D13711yescharacteristic
122C2×D4—1371no—
123C22×C4—2742no—
124C22×C4—2741no—
125C2×C8—21no—
126C2×C8—2741no—
127C2×C8—2741no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2742no—
132C4⋊C4—2741no—
133C4⋊C4—1372no—
134C4⋊C4—1371no—
135C4⋊C4C22⋊D13711yescharacteristic
136C22⋊C4—2742no—
137C22⋊C4—2741no—
138C42—21no—
139C42C22×D13711yescharacteristic
140C23—2741no—
141Q8—21no—
142Q8—42no—
143Q8—5484no—
144Q8—5481no—
145Q8—5481no—
146Q8#1455549712yes—
147D4—2741no—
148D4—1372no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—2742no—
153C2×C4—2741no—
154C2×C4—5484no—
155C2×C4—5481no—
156C2×C4—5481no—
157C2×C4D4×D13711yescharacteristic
158C2×C4D4×D13711yescharacteristic
159C2×C4C2×C22⋊D13711yescharacteristic
160C8—41no—
161C8—5481no—
162C8—5481no—
163C22—2742no—
164C22—5481no—
165C22—5481no—
166C22#1798405611yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—81no—
171C4—5482no—
172C4—5481no—
173C4C2×#1455549711yescharacteristic
174C4Q16×D13711yescharacteristic
175C4#1798411311yescharacteristic
176C4#1798412011yescharacteristic
177C2—5482no—
178C2#2257755311yescharacteristic
179C2#2257755711yescharacteristic
180C2#2257757411yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#477×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#477D137
hypocenterC137#1637
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #22577753C2semidirect
4. #22577150C2semidirect
5. Q8⋊#11764040C2semidirect
7. #477×C137C2semidirect
90. C137#1637semidirect
94. #477D137semidirect

Maximal subgroups

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

subgroupindex
Q8×D5482
#225777532
#225776902
#225771502
Q8⋊#117640402
#225771302
#477×C1372
#1637137