Finite Group Databasebeta

← #31779813

Subgroups — #31779813

Order 17536.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#22578029C211yesmaximal, characteristic
2#22577738C212yesmaximal, complemented
3#22577139C212yesmaximal, complemented
4#22577127C211yesmaximal, characteristic
5#399×C137C211yesmaximal, characteristic
6#17984067—21no—
7#17984067C2212yes—
8#17984065—22no—
9#17984065—22no—
10C2×C22⋊Dic137—21no—
11#17984021—22no—
12#17984011—22no—
13C22⋊#11764040C2212yes—
14C2×C4⋊C4×C137—21no—
15C2×C4⋊C4×C137C2211yescharacteristic
16C2×C22⋊C4×C137—21no—
17C2×C22⋊C4×C137C2211yescharacteristic
18#95×C137C2211yescharacteristic
19C2×Dic274—21no—
20C2×Dic274D412yes—
21C22×Dic137—21no—
22C22×Dic137—21no—
23C2×#11764040—22no—
24C4⋊Dic137—21no—
25C4⋊Dic137D412yes—
26#14555489—22no—
27#14555489—22no—
28C22⋊Dic137—22no—
29C22⋊Dic137—22no—
30C24×C137D411yescharacteristic
31C22×C4×C137—21no—
32C22×C4×C137—21no—
33C22×C4×C137D411yescharacteristic
34C22×C4×C137C2311yescharacteristic
35C4⋊C4×C137—22no—
36C4⋊C4×C137—42no—
37C22⋊C4×C137—22no—
38C22⋊C4×C137—42no—
39Dic274—42no—
40Dic274—81no—
41C2×Dic137—22no—
42C2×Dic137—22no—
43C2×Dic137—81no—
44C2×Dic137—81no—
45#11764040—42no—
46C23×C137—21no—
47C23×C137—21no—
48C23×C137—41no—
49C23×C137—41no—
50C23×C137Q1612yes—
51C23×C137C2×D411yescharacteristic
52C2×C4×C137—22no—
53C2×C4×C137—21no—
54C2×C4×C137—41no—
55C2×C4×C137—41no—
56C2×C4×C137—41no—
57C2×C4×C137—41no—
58C2×C4×C137—41no—
59C2×C4×C137—41no—
60C2×C4×C137—41no—
61C2×C4×C137C2×D412yes—
62C22×C137—21no—
63C22×C137—21no—
64C22×C137—22no—
65C22×C137—42no—
66C22×C137—41no—
67C22×C137—41no—
68C22×C137—81no—
69C22×C137#12011yescharacteristic
70C22×C137C2×Q1611yescharacteristic
71C22×C137#13711yescharacteristic
72C4×C137—22no—
73C4×C137—42no—
74C4×C137—41no—
75C4×C137—81no—
76Dic137—42no—
77Dic137—42no—
78C2×C137—21no—
79C2×C137—21no—
80C2×C137—42no—
81C2×C137#45111yescharacteristic
82C2×C137#45311yescharacteristic
83C2×C137#45811yescharacteristic
84C137#161411yescharacteristic, complemented, Sylow
85#1614—1371nomaximal, Sylow
86#543—1371no—
87#484—1372no—
88#399D13711yescharacteristic
89#328—1372no—
90#323—1371no—
91C22⋊Q8—2742no—
92C22⋊Q8—2742no—
93C22⋊Q8—2741no—
94C22⋊Q8—1372no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×D13711yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—2741no—
99C2×C22⋊C4C2×D13711yescharacteristic
100#107—2742no—
101#103—2742no—
102C22⋊C8—1372no—
103#95C2×D13711yescharacteristic
104C24C22⋊D13711yescharacteristic
105C2×Q8—2741no—
106C2×Q8—1372no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—2741no—
110C22×C4—2741no—
111C22×C4C22⋊D13711yescharacteristic
112C22×C4C22×D13711yescharacteristic
113C2×C8—2742no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4—2742no—
117C4⋊C4—2741no—
118C4⋊C4—2742no—
119C4⋊C4—1372no—
120C22⋊C4—22no—
121C22⋊C4—42no—
122C22⋊C4—2742no—
123C22⋊C4—2742no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#1455550012yes—
129C23C2×C22⋊D13711yescharacteristic
130Q8—5482no—
131Q8—10961no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—2742no—
142C2×C4—2742no—
143C2×C4—10961no—
144C2×C4—10961no—
145C2×C4D4×D13712yes—
146C8—5482no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1798405611yescharacteristic
155C22C2×#1455550011yescharacteristic
156C22#1798411811yescharacteristic
157C4—22no—
158C4—42no—
159C4—41no—
160C4—81no—
161C4—5482no—
162C4—5482no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2257755811yescharacteristic
167C2#2257757011yescharacteristic
168C2#2257759211yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC22×C4×C137C23
FrattiniC22×C4C22×D137
Fitting#399×C137C2
socleC22×C137#120
prime radicalC24×C137D4
hypercenter#399D137
hypocenterC137#1614
LCSG ▷ C22×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #22577738C2semidirect
3. #22577139C2semidirect
84. C137#1614semidirect

Maximal subgroups

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

subgroupindex
#225780292
#225777382
#225771392
#225771272
#399×C1372
#1614137