Finite Group Databasebeta

← #31782462

Subgroups — #31782462

Order 17536.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#22577973C211yesmaximal, characteristic
2#22577375C212yesmaximal
3#22577198C211yesmaximal, characteristic, complemented
4#22577197C212yesmaximal, complemented
5#354×C137C211yesmaximal, characteristic, complemented
6C2×#14555509C2211yescharacteristic
7C2×Q8×D137C411yescharacteristic, complemented
8#17984088C411yescharacteristic, complemented
9#17984086C411yescharacteristic, complemented
10#17984075C2211yescharacteristic
11#17984070—22no—
12#17984065—22no—
13#17984045C411yescharacteristic, complemented
14#17984044—22no—
15#17984003C2212yes—
16#17984002—22no—
17#124×C137C2211yescharacteristic
18#99×C137C2212yes—
19#14555509—42no—
20C2×Dic274—21no—
21C2×Dic274C2×C411yescharacteristic
22C2×Dic274C2×C411yescharacteristic
23Q8×D137—42no—
24C2×C22⋊D137D412yes—
25C22×Dic137D412yes—
26C2×C4×D137—21no—
27C2×C4×D137C2×C411yescharacteristic
28C4⋊Dic137—22no—
29#14555489—21no—
30#14555489—22no—
31#14555489—42no—
32C22⋊Dic137—22no—
33C22⋊Dic137—22no—
34#14555487—21no—
35C4×Dic137—42no—
36C4×Dic137C2×C411yescharacteristic
37C2×Q8×C137C2×C411yescharacteristic
38C2×D4×C137C2311yescharacteristic
39C22⋊C4×C137—22no—
40C22⋊C4×C137—22no—
41C42×C137C2×C411yescharacteristic
42C22×D137C22⋊C411yescharacteristic
43Dic274—41no—
44Dic274—42no—
45Dic274—41no—
46C22⋊D137—42no—
47C2×Dic137—22no—
48C2×Dic137—22no—
49C2×Dic137—21no—
50C2×Dic137—21no—
51C2×Dic137—42no—
52C2×Dic137C22⋊C412yes—
53C2×Dic137C22⋊C411yescharacteristic
54C4×D137—22no—
55C4×D137—42no—
56C23×C137C2×D412yes—
57Q8×C137—41no—
58D4×C137—41no—
59C2×C4×C137—21no—
60C2×C4×C137—21no—
61C2×C4×C137—42no—
62C2×C4×C137C22×C411yescharacteristic
63C22×C137—22no—
64C22×C137—42no—
65C22×C137C2×C22⋊C411yescharacteristic
66C2×D137—21no—
67C2×D137—41no—
68C4×C137—21no—
69C4×C137—41no—
70C4×C137—41no—
71C4×C137—82no—
72Dic137—22no—
73Dic137—41no—
74Dic137—41no—
75Dic137—82no—
76Dic137#9912yes—
77C2×C137—21no—
78C2×C137—42no—
79C2×C137C2×#9911yescharacteristic
80D137—41no—
81C137#212911yescharacteristic, complemented, Sylow
82#2129—1371nomaximal, Sylow
83#536—1371no—
84#410—1372no—
85#354—1371no—
86#354—1372no—
87#354D13711yescharacteristic, complemented
88C2×#41—1371no—
89C22×Q8—1371no—
90C4⋊Q8—1371no—
91C4⋊Q8—1371no—
92#124—1371no—
93#124C2×D13711yescharacteristic
94#123—2742no—
95C22⋊Q8—2742no—
96#117—2742no—
97#117—1371no—
98#99—2742no—
99#99—1372no—
100#99C2×D13712yes—
101#41—5482no—
102C2×Q8—2741no—
103C2×Q8—5482no—
104C2×Q8—1371no—
105C2×Q8—1371no—
106C2×Q8C4×D13711yescharacteristic
107C2×D4—1372no—
108C2×D4C22×D13711yescharacteristic
109C22×C4—2741no—
110C22×C4—1372no—
111C22×C4—1371no—
112C4⋊C4—2741no—
113C4⋊C4—2742no—
114C4⋊C4—2742no—
115C4⋊C4—5482no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—2741no—
119C22⋊C4—2742no—
120C22⋊C4—2742no—
121C42—5482no—
122C42—1371no—
123C42C4×D13711yescharacteristic
124C23—1371no—
125C23D4×D13712yes—
126Q8—41no—
127Q8—5481no—
128Q8—5481no—
129Q8—5482no—
130D4—41no—
131D4—5482no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2742no—
136C2×C4—2742no—
137C2×C4—2742no—
138C2×C4—2741no—
139C2×C4—2741no—
140C2×C4—5482no—
141C2×C4—5482no—
142C2×C4—1372no—
143C2×C4—1371no—
144C2×C4C2×C4×D13711yescharacteristic
145C22—22no—
146C22—42no—
147C22—2741no—
148C22—5481no—
149C22C22⋊C4×D13711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2742no—
155C4—5481no—
156C4—5481no—
157C4—10962no—
158C4—1372no—
159C2—21no—
160C2—42no—
161C2—5481no—
162C2#99×D13711yescharacteristic
subgroupG/H
centerC2#99×D137
commutatorC2×C4×C137C22×C4
FrattiniC2×D4C22×D137
Fitting#354×C137C2
socleC2×C137C2×#99
prime radicalC2×#14555509C22
cosocleC2×D4×C137C23
hypercenter#354D137
hypocenterC137#2129
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #22577198C2semidirect
4. #22577197C2semidirect
5. #354×C137C2semidirect
7. C2×Q8×D137C4semidirect
8. #17984088C4semidirect
9. #17984086C4semidirect
13. #17984045C4semidirect
81. C137#2129semidirect
87. #354D137semidirect

Maximal subgroups

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

subgroupindex
#225779732
#225773752
#225771982
#225771972
#354×C1372
#2129137