Finite Group Databasebeta

← #31796951

Subgroups — #31796951

Order 17536.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D137⋊#108C211yesmaximal, characteristic
2#22578396C212yesmaximal
3D137⋊#101C212yesmaximal
4#22578366C211yesmaximal, characteristic
5C2×Q16×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7D137⋊M16C2211yescharacteristic
8#17984167C2211yescharacteristic
9#17984159—21no—
10#17984158—21no—
11#17984151—24no—
12#17984149—24no—
13#17984142—21no—
14C8×#10587739—21no—
15C2×Q8×D137C2212yes—
16C2×Dic548C411yescharacteristic, complemented
17C2×#14555500C412yescomplemented
18Q16×D137—21no—
19Q16×D137—21no—
20Q16×D137—21no—
21Q16×D137—21no—
22C2×C8×D137C2211yescharacteristic
23C2×Q16×C137C411yescharacteristic, complemented
24#14555520—22no—
25#14555520—21no—
26#14555519—22no—
27#14555519D412yes—
28#14555518—21no—
29#14555518—22no—
30C4⋊#10587739D412yes—
31C22⋊#10587739—21no—
32C4×#10587739—21no—
33C2×Dic274—22no—
34C2×Dic274C2×C412yes—
35Q8×D137—22no—
36Q8×D137—22no—
37Q8×D137—22no—
38Q8×D137—22no—
39C2×C4×D137—22no—
40C2×C4×D137C2311yescharacteristic
41Dic548—21no—
42Dic548—21no—
43#14555500—22no—
44#14555500—22no—
45C2×#11764040C2×C411yescharacteristic
46C8×D137D411yescharacteristic
47C8×D137D411yescharacteristic
48C8×D137D411yescharacteristic
49C8×D137D411yescharacteristic
50C2×Q8×C137C2×C412yes—
51Q16×C137—21no—
52Q16×C137—21no—
53C2×C8×C137C2×C411yescharacteristic
54C2×#10587739—22no—
55#11764047—22no—
56#11764047—42no—
57C22×D137#4111yescharacteristic
58Dic274—22no—
59Dic274—22no—
60Dic274—42no—
61Dic274—42no—
62C2×Dic137—22no—
63C2×Dic137C2×D411yescharacteristic
64C4×D137—42no—
65C4×D137—42no—
66C4×D137C2×D411yescharacteristic
67C4×D137C2×D411yescharacteristic
68C4×D137C2×D411yescharacteristic
69C4×D137#4111yescharacteristic
70#11764040C22⋊C411yescharacteristic
71#11764040C22⋊C411yescharacteristic
72Q8×C137—22no—
73Q8×C137—22no—
74C2×C4×C137—22no—
75C2×C4×C137C22×C411yescharacteristic
76C8×C137C22⋊C411yescharacteristic
77C8×C137C22⋊C411yescharacteristic
78C22×C137C4×D411yescharacteristic
79C2×D137—22no—
80C2×D137#12111yescharacteristic
81C2×D137#12411yescharacteristic
82#10587739—42no—
83C4×C137—42no—
84C4×C137C2×C22⋊C411yescharacteristic
85C4×C137C4×D411yescharacteristic
86Dic137—42no—
87Dic137#12111yescharacteristic
88Dic137#12711yescharacteristic
89C2×C137—21no—
90C2×C137#39011yescharacteristic
91D137—21no—
92D137#47312yes—
93C137#197911yescharacteristic, complemented, Sylow
94#1979—1371nomaximal, Sylow
95C22×Q16—1371no—
96C2×#108—1371no—
97#419—1372no—
98C2×#101—1372no—
99#405—1371no—
100C22×Q8—1372no—
101C2×Q16—2741no—
102C2×Q16—2741no—
103C2×Q16—2741no—
104C2×Q16—2741no—
105C2×Q16—1372no—
106C2×Q16—1371no—
107C2×Q16#1058773911yescharacteristic, complemented
108C2×M16—1372no—
109C2×M16—1371no—
110C22×C8—1371no—
111#117—1371no—
112#108—2741no—
113#108—2741no—
114#103—2744no—
115#101—2744no—
116#97—2741no—
117C4×C8—2741no—
118C2×Q8—2742no—
119C2×Q8—2742no—
120C2×Q8—2742no—
121C2×Q8—2742no—
122C2×Q8—2742no—
123C2×Q8—1372no—
124C2×Q8C2×#1058773912yes—
125C22×C4—2742no—
126C22×C4—1371no—
127Q16—21no—
128Q16—21no—
129Q16—2742no—
130Q16—2742no—
131Q16—2741no—
132Q16—2741no—
133M16—2741no—
134M16—2742no—
135M16—2742no—
136M16—1372no—
137C2×C8—2741no—
138C2×C8—2742no—
139C2×C8—1371no—
140C2×C8—1371no—
141C2×C8—1371no—
142C2×C8—1371no—
143C2×C8—1371no—
144C2×C8C2×#1058773911yescharacteristic
145C4⋊C4—1372no—
146C22⋊C4—2741no—
147C42—2741no—
148C23—1371no—
149Q8—22no—
150Q8—22no—
151Q8—2742no—
152Q8—2742no—
153Q8—5482no—
154Q8—5482no—
155C2×C4—22no—
156C2×C4—2742no—
157C2×C4—2742no—
158C2×C4—5482no—
159C2×C4—5482no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1371no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4C22×#1058773911yescharacteristic
166C8—2742no—
167C8—5482no—
168C8—1371no—
169C8—1371no—
170C8C22⋊#1058773911yescharacteristic
171C8C22⋊#1058773911yescharacteristic
172C22—2742no—
173C22—1371no—
174C22—1371no—
175C22D4×#1058773911yescharacteristic
176C4—42no—
177C4—5482no—
178C4—5482no—
179C4—1371no—
180C4—1371no—
181C4C2×C22⋊#1058773911yescharacteristic
182C4D4×#1058773911yescharacteristic
183C2—21no—
184C2—2741no—
185C2—1372no—
186C2#2257835511yescharacteristic
subgroupG/H
centerC2#22578355
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×Q16×C137C4
socleC2×C137#390
prime radicalC22×D137#41
cosocleC2×C4×D137C23
hypercenterC2×Q16#10587739
hypocenterC137#1979
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
16. C2×Dic548C4semidirect
17. C2×#14555500C4semidirect
23. C2×Q16×C137C4semidirect
93. C137#1979semidirect
107. C2×Q16#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1082
#225783962
D137⋊#1012
#225783662
C2×Q16×D1372
#1979137