Finite Group Databasebeta

← #31779491

Subgroups — #31779491

Order 17536.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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
3#22577455C211yesmaximal, characteristic, complemented
4#22577150C211yesmaximal, characteristic, complemented
5#22577148C211yesmaximal, characteristic, complemented
6#22577147C211yesmaximal, characteristic, complemented
7#331×C137C211yesmaximal, characteristic, complemented
8C2×#14555509—21no—
9#17984088C2211yescharacteristic
10#17984075—21no—
11#17984059—21no—
12#17984059C412yes—
13D4×Dic137—21no—
14D4×Dic137C412yes—
15C42×D137C2211yescharacteristic
16C4⋊#11764040C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18#17984000—21no—
19#17984000—21no—
20#17983998—21no—
21C8×Dic137—21no—
22#127×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555509—42no—
26#14555509—42no—
27C2×Dic274—21no—
28C2×C22⋊D137—22no—
29C22×Dic137—22no—
30C2×C4×D137C2×C411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×#11764040—21no—
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137C2×C412yes—
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—41no—
50Dic274—41no—
51C22⋊D137—44no—
52C2×Dic137—22no—
53C2×Dic137—44no—
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C22⋊C411yescharacteristic
56C2×Dic137C22⋊C411yescharacteristic
57C4×D137—21no—
58C4×D137—21no—
59C4×D137—21no—
60C4×D137—21no—
61C4×D137—41no—
62#11764040—41no—
63#11764040—41no—
64C23×C137—22no—
65D4×C137—42no—
66D4×C137—41no—
67D4×C137—41no—
68D4×C137—41no—
69D4×C137—41no—
70C2×C4×C137C22×C411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—41no—
74C8×C137—41no—
75C22×C137—42no—
76C22×C137—42no—
77C22×C137—42no—
78C22×C137C2×C22⋊C411yescharacteristic
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—21no—
82C4×C137—21no—
83C4×C137#13511yescharacteristic
84C4×C137#13511yescharacteristic
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87Dic137—21no—
88Dic137—21no—
89Dic137—21no—
90Dic137—42no—
91C2×C137—82no—
92C2×C137#41311yescharacteristic
93C2×C137#41611yescharacteristic
94C2×C137#41811yescharacteristic
95D137—41no—
96C137#155311yescharacteristic, complemented, Sylow
97#1553—1371nomaximal, Sylow
98#532—1371no—
99#432—1371no—
100#431—1371no—
101#332—1371no—
102#332—1371no—
103#331—1371no—
104#331D13711yescharacteristic, complemented
105C2×#41—2741no—
106C4⋊Q8—1371no—
107#127C2×D13711yescharacteristic
108#124—2741no—
109#121—2741no—
110#121—1372no—
111C4×D4—2741no—
112C4×D4—1372no—
113C2×C42—1371no—
114C4⋊C8—1371no—
115C4⋊C8—1371no—
116C4⋊C8C2×D13711yescharacteristic
117C4⋊C8C2×D13711yescharacteristic
118C22⋊C8—2741no—
119C22⋊C8—2741no—
120#97—2741no—
121C4×C8—2741no—
122#41—5482no—
123#41—5482no—
124C2×Q8—2741no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2742no—
128C2×D4C4×D13712yes—
129C22×C4—2742no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2741no—
136C2×C8—2741no—
137C4⋊C4—2741no—
138C4⋊C4—1372no—
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C42—1371no—
142C42—1371no—
143C42—1371no—
144C42C22×D13711yescharacteristic
145C23—22no—
146C23—1371no—
147Q8—5481no—
148Q8—5481no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—5484no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2742no—
160C2×C4—5481no—
161C2×C4—5484no—
162C2×C4—1371no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4C2×C4×D13711yescharacteristic
166C2×C4D4×D13711yescharacteristic
167C2×C4D4×D13711yescharacteristic
168C8—41no—
169C8—41no—
170C8—5481no—
171C8—5481no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—2741no—
176C22—2741no—
177C22—2741no—
178C22C22⋊C4×D13711yescharacteristic
179C4—21no—
180C4—2741no—
181C4—2741no—
182C4—2741no—
183C4—5482no—
184C4#1798410711yescharacteristic
185C4#1798410911yescharacteristic
186C4#1798411211yescharacteristic
187C4#1798411511yescharacteristic
188C2—82no—
189C2—5481no—
190C2#2257738411yescharacteristic
191C2#2257739611yescharacteristic
192C2#2257740611yescharacteristic
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#1553
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. #22577455C2semidirect
4. #22577150C2semidirect
5. #22577148C2semidirect
6. #22577147C2semidirect
7. #331×C137C2semidirect
96. C137#1553semidirect
104. #331D137semidirect

Maximal subgroups

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

subgroupindex
#225779512
#225774592
#225774552
#225771502
#225771482
#225771472
#331×C1372
#1553137