Finite Group Databasebeta

← #31796106

Subgroups — #31796106

Order 17536.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×C4⋊#11764047C211yesmaximal, characteristic, complemented
2C4×#14555519C211yesmaximal, characteristic, complemented
3Q8⋊#11764047C212yesmaximal, complemented
4#22578251C212yesmaximal, complemented
5#22577939C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8C4⋊#11764047—21no—
9C4⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047C2212yes—
12#17984083—21no—
13#17984079C411yescharacteristic
14#17984077C411yescharacteristic
15#17984072—21no—
16#17984062C411yescharacteristic
17Q8×Dic137C2211yescharacteristic
18#17984050C2211yescharacteristic
19#17984048—21no—
20#17984046—21no—
21C2×C4×Dic137C2211yescharacteristic
22C22⋊Q8×C137C411yescharacteristic
23#14555519—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
27C2×D548C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C22×Dic137D411yescharacteristic
30C22×Dic137D411yescharacteristic
31C2×C4×D137—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C411yescharacteristic
35#14555489—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39#14555487—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×Q8×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50#11764047—24no—
51#11764047—42no—
52C22×D137—21no—
53D548—22no—
54C22⋊D137—42no—
55C2×Dic137—22no—
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C4×D137—41no—
64C4×D137—41no—
65C23×C137C22⋊C411yescharacteristic
66Q8×C137—22no—
67C2×C4×C137—22no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22×C411yescharacteristic
72C22×C137—21no—
73C22×C137—21no—
74C22×C137#10412yes—
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—41no—
77C2×D137—41no—
78C2×D137—41no—
79C4×C137—21no—
80C4×C137—41no—
81C4×C137—41no—
82C4×C137#13112yes—
83Dic137—21no—
84Dic137—21no—
85Dic137—21no—
86Dic137—41no—
87Dic137AGL(1,ℤ/8)11yescharacteristic
88Dic137#13711yescharacteristic
89C2×C137—22no—
90C2×C137#40811yescharacteristic
91C2×C137#41711yescharacteristic
92C2×C137C2×#10411yescharacteristic
93D137—81no—
94C137#148311yescharacteristic, complemented, Sylow
95#1483—1371nomaximal, Sylow
96#529—1371no—
97C2×C4⋊C8—1371no—
98C4×M16—1371no—
99Q8⋊C8—1372no—
100D4⋊C8—1372no—
101C2×M16—2741no—
102C22×C8—2741no—
103#126—2741no—
104#125—1371no—
105#124—1371no—
106#123—2741no—
107C22⋊Q8#1058773911yescharacteristic
108#121—1371no—
109C4×Q8—1371no—
110C4×D4—2741no—
111C4×D4—1371no—
112#117—2741no—
113C2×C42—1371no—
114C4⋊C8—2741no—
115C4⋊C8—1372no—
116#97—2741no—
117C4×C8—1372no—
118C2×Q8C2×#1058773911yescharacteristic
119C2×D4—2741no—
120C2×D4—1371no—
121C22×C4—2741no—
122C22×C4—1371no—
123C22×C4—1371no—
124C22×C4C2×#1058773911yescharacteristic
125M16—2744no—
126C2×C8—2742no—
127C2×C8—2742no—
128C2×C8—5482no—
129C4⋊C4—21no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—1371no—
134C4⋊C4C2×#1058773911yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—2741no—
137C22⋊C4—2741no—
138C22⋊C4—2741no—
139C22⋊C4—2741no—
140C42—2741no—
141C42—1371no—
142C42—1371no—
143C42—1371no—
144C42—1371no—
145C23—2741no—
146C23C22⋊#1058773911yescharacteristic
147Q8—22no—
148D4—2742no—
149D4—5482no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—2742no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—5481no—
158C2×C4—5481no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1371no—
163C2×C4C22⋊#1058773911yescharacteristic
164C2×C4C22×#1058773911yescharacteristic
165C8—2744no—
166C8—5482no—
167C22—21no—
168C22—21no—
169C22—5481no—
170C22—5481no—
171C22—5481no—
172C22#1798415312yes—
173C22C2×C22⋊#1058773911yescharacteristic
174C4—21no—
175C4—41no—
176C4—41no—
177C4—2741no—
178C4—2741no—
179C4—2741no—
180C4—5481no—
181C4—1371no—
182C4—1371no—
183C4#1798417512yes—
184C2—22no—
185C2—10961no—
186C2#2257837611yescharacteristic
187C2#2257839411yescharacteristic
188C2C2×#1798415311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984062C4
cosocleC4×Dic137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1483
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#11764047C2semidirect
2. C4×#14555519C2semidirect
3. Q8⋊#11764047C2semidirect
4. #22578251C2semidirect
94. C137#1483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#117640472
C4×#145555192
Q8⋊#117640472
#225782512
#225779392
#1483137