Finite Group Databasebeta

← #31779481

Subgroups — #31779481

Order 17536.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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#22577951C211yesmaximal, characteristic
2#22577459C211yesmaximal, characteristic, complemented
3C4⋊C8×D137C211yesmaximal, characteristic, complemented
4#22577151C211yesmaximal, characteristic, complemented
5#22577151C211yesmaximal, characteristic, complemented
6#22577147C211yesmaximal, characteristic, complemented
7#331×C137C211yesmaximal, characteristic, complemented
8C2×#14555509C412yes—
9C2×C8×D137—21no—
10#17984088C2211yescharacteristic
11#17984075C412yes—
12#17984059—21no—
13#17984059—21no—
14D4×Dic137—21no—
15D4×Dic137—21no—
16C42×D137C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984017—21no—
20#17984000—21no—
21#17983998—21no—
22#127×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555509—44no—
26C2×Dic274C2×C412yes—
27C2×C22⋊D137—22no—
28C22×Dic137—22no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×#11764040—21no—
33C2×#11764040—21no—
34C8×D137—42no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×D4×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137C2×C412yes—
45C2×C8×C137—21no—
46C2×C8×C137—21no—
47C42×C137C2311yescharacteristic
48C22×D137C22⋊C411yescharacteristic
49Dic274—42no—
50C22⋊D137—44no—
51C2×Dic137—22no—
52C2×Dic137—44no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C22⋊C411yescharacteristic
56C4×D137—22no—
57C4×D137—21no—
58C4×D137—21no—
59C4×D137—21no—
60C4×D137—21no—
61#11764040—41no—
62#11764040—41no—
63C23×C137—22no—
64D4×C137—42no—
65D4×C137—41no—
66D4×C137—41no—
67D4×C137—41no—
68D4×C137—41no—
69C2×C4×C137C22×C411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137—41no—
74C22×C137—42no—
75C22×C137—42no—
76C22×C137—42no—
77C22×C137C2×C22⋊C411yescharacteristic
78C2×D137—21no—
79C2×D137—21no—
80C2×D137#10012yes—
81C4×C137—21no—
82C4×C137#13511yescharacteristic
83C4×C137#13511yescharacteristic
84C4×C137AGL(1,ℤ/8)11yescharacteristic
85C4×C137#13711yescharacteristic
86Dic137—21no—
87Dic137—21no—
88Dic137—21no—
89Dic137—42no—
90C2×C137—82no—
91C2×C137C2×#10011yescharacteristic
92C2×C137#41611yescharacteristic
93C2×C137#41711yescharacteristic
94D137—22no—
95C137#155111yescharacteristic, complemented, Sylow
96#1551—1371nomaximal, Sylow
97#532—1371no—
98#432—1371no—
99C2×C4⋊C8—1371no—
100#333—1371no—
101#333—1371no—
102#331—1371no—
103#331D13711yescharacteristic, complemented
104C2×#41—1372no—
105C22×C8—2741no—
106C4⋊Q8—1371no—
107#127C2×D13711yescharacteristic
108#124—1372no—
109#121—2741no—
110#121—2741no—
111C4×D4—2741no—
112C4×D4—2741no—
113C2×C42—1371no—
114C4⋊C8—2741no—
115C4⋊C8—1371no—
116C4⋊C8—1371no—
117C4⋊C8C2×D13711yescharacteristic
118C4⋊C8C2×D13711yescharacteristic
119C22⋊C8—2741no—
120#97—2741no—
121#41—5484no—
122C2×Q8—1372no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—2742no—
126C2×D4C4×D13712yes—
127C22×C4—2742no—
128C22×C4—1371no—
129C22×C4—1371no—
130C22×C4—1371no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2741no—
134C2×C8—2741no—
135C2×C8—5482no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C22⋊C4—2742no—
139C22⋊C4—2742no—
140C42—1371no—
141C42—1371no—
142C42—1371no—
143C42C22×D13711yescharacteristic
144C23—22no—
145C23—1371no—
146Q8—5482no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—5484no—
153C2×C4—2742no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2742no—
159C2×C4—5484no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1371no—
163C2×C4C2×C4×D13711yescharacteristic
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C8—41no—
167C8—41no—
168C8—5481no—
169C8—5481no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—2741no—
174C22—2741no—
175C22—1372no—
176C22C22⋊C4×D13711yescharacteristic
177C4—21no—
178C4—2741no—
179C4—2741no—
180C4—2741no—
181C4—5482no—
182C4#1798410711yescharacteristic
183C4#1798410911yescharacteristic
184C4#1798411211yescharacteristic
185C4#1798411711yescharacteristic
186C2—82no—
187C2—2742no—
188C2#100×D13711yescharacteristic
189C2#2257739611yescharacteristic
190C2#2257740111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#331×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC42×C137C23
hypercenter#331D137
hypocenterC137#1551
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22577459C2semidirect
3. C4⋊C8×D137C2semidirect
4. #22577151C2semidirect
5. #22577151C2semidirect
6. #22577147C2semidirect
7. #331×C137C2semidirect
95. C137#1551semidirect
103. #331D137semidirect

Maximal subgroups

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

subgroupindex
#225779512
#225774592
C4⋊C8×D1372
#225771512
#225771472
#331×C1372
#1551137