Finite Group Databasebeta

← #31779321

Subgroups — #31779321

Order 17536.

274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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×#17984067C211yesmaximal, characteristic
2#22577367C211yesmaximal, characteristic
3#22577139C214yesmaximal, complemented
4#381×C137C211yesmaximal, characteristic
5C22×Dic274C411yescharacteristic, complemented
6C2×#14555494—21no—
7#17984067—21no—
8#17984067—21no—
9#17984067C412yescomplemented
10#17984067C2212yes—
11C2×C4⋊Dic137C411yescharacteristic, complemented
12C2×#14555489—21no—
13C2×C22⋊Dic137—21no—
14C22⋊#11764040—21no—
15C22⋊#11764040C2212yes—
16C23×C4×C137C2211yescharacteristic
17C2×C22⋊C4×C137—21no—
18#95×C137—21no—
19#95×C137C2212yes—
20C2×Dic274—21no—
21C2×Dic274—42no—
22C2×Dic274C2×C412yes—
23C22×Dic137—21no—
24C22×Dic137—21no—
25#14555494—42no—
26C2×#11764040—22no—
27C4⋊Dic137—21no—
28C4⋊Dic137C2×C412yes—
29#14555489—22no—
30#14555489—22no—
31C22⋊Dic137—22no—
32C22⋊Dic137—22no—
33C24×C137D411yescharacteristic
34C22×C4×C137—22no—
35C22×C4×C137—42no—
36C22×C4×C137C2×C411yescharacteristic
37C22×C4×C137C2×C411yescharacteristic
38C22×C4×C137D411yescharacteristic
39C22×C4×C137D411yescharacteristic
40C22×C4×C137D411yescharacteristic
41C22×C4×C137C2311yescharacteristic
42C22⋊C4×C137—42no—
43Dic274—42no—
44Dic274—42no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47C2×Dic137—42no—
48C2×Dic137—42no—
49#11764040—42no—
50C23×C137—22no—
51C23×C137—22no—
52C23×C137—41no—
53C23×C137C22⋊C411yescharacteristic
54C23×C137C22⋊C411yescharacteristic
55C23×C137C2×D411yescharacteristic
56C2×C4×C137—22no—
57C2×C4×C137—22no—
58C2×C4×C137—41no—
59C2×C4×C137—41no—
60C2×C4×C137—41no—
61C2×C4×C137—41no—
62C2×C4×C137—42no—
63C2×C4×C137—42no—
64C2×C4×C137—42no—
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22×C411yescharacteristic
68C2×C4×C137C2×D411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137—21no—
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—21no—
75C22×C137—21no—
76C22×C137—44no—
77C22×C137#9912yes—
78C22×C137C2×C22⋊C411yescharacteristic
79C22×C137#13712yes—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—21no—
83C4×C137—21no—
84C4×C137—82no—
85Dic137—42no—
86Dic137—42no—
87C2×C137—22no—
88C2×C137—22no—
89C2×C137—41no—
90C2×C137C2×#9911yescharacteristic
91C2×C137#41911yescharacteristic
92C2×C137#42111yescharacteristic
93C137#151111yescharacteristic, complemented, Sylow
94#1511—1371nomaximal, Sylow
95C2×C22⋊Q8—1371no—
96#407—1371no—
97#381D13711yescharacteristic
98#328—1374no—
99C22×Q8—1371no—
100C23×C4C2×D13711yescharacteristic
101C2×M16—2741no—
102C22⋊Q8—2741no—
103C22⋊Q8—2741no—
104C22⋊Q8—1372no—
105C22⋊Q8—1372no—
106C2×C4⋊C4—2741no—
107C2×C4⋊C4—1371no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—2741no—
110C22⋊C8—2741no—
111C22⋊C8—1372no—
112#95—21no—
113#95C2×D13712yes—
114C24C22⋊D13711yescharacteristic
115C2×Q8—2741no—
116C2×Q8—5482no—
117C2×Q8—1372no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—2741no—
121C22×C4—2741no—
122C22×C4C4×D13711yescharacteristic
123C22×C4C4×D13711yescharacteristic
124C22×C4C22⋊D13711yescharacteristic
125C22×C4D54811yescharacteristic
126C22×C4D54811yescharacteristic
127C22×C4C22×D13711yescharacteristic
128M16—5482no—
129C2×C8—2742no—
130C4⋊C4—2742no—
131C4⋊C4—2741no—
132C4⋊C4—2742no—
133C4⋊C4—1372no—
134C22⋊C4—42no—
135C22⋊C4—2742no—
136C22⋊C4—2742no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23#1455548711yescharacteristic
141C23#1455548711yescharacteristic
142C23C2×C22⋊D13711yescharacteristic
143Q8—5482no—
144Q8—5482no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—2742no—
155C2×C4—2742no—
156C2×C4—5482no—
157C2×C4—5482no—
158C2×C4#1455548711yescharacteristic
159C2×C4#1455548711yescharacteristic
160C2×C4C2×C4×D13711yescharacteristic
161C2×C4C2×D54811yescharacteristic
162C8—5482no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#1798400212yes—
172C22C2×#1455548711yescharacteristic
173C22#1798411812yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—82no—
179C4—5482no—
180C4—5482no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#1798400211yescharacteristic
185C2#2257741111yescharacteristic
186C2#2257742111yescharacteristic
subgroupG/H
centerC22C2×#14555487
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
Fitting#381×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC24×C137D4
cosocleC22×C4×C137C23
hypercenter#381D137
hypocenterC137#1511
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #22577139C2semidirect
5. C22×Dic274C4semidirect
9. #17984067C4semidirect
11. C2×C4⋊Dic137C4semidirect
93. C137#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#179840672
#225773672
#225771392
#381×C1372
#1511137