Finite Group Databasebeta

← #31796744

Subgroups — #31796744

Order 17536.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#22578470C211yesmaximal, characteristic
2#22578414C212yesmaximal, complemented
3#22578355C211yesmaximal, characteristic
4#22578275C212yesmaximal, complemented
5#100×D137C211yesmaximal, characteristic
6D4×#10587739—22no—
7D4×#10587739—22no—
8#17984167C2212yes—
9C2×C4⋊#10587739C2211yescharacteristic
10C2×C22⋊#10587739—22no—
11C2×C22⋊#10587739—22no—
12C2×C4×#10587739C2211yescharacteristic
13#17984157—22no—
14#17984156—22no—
15C2×D4×D137C2211yescharacteristic
16M16×D137C2212yes—
17#17984005C411yescharacteristic, complemented
18#17984004C412yescomplemented
19#100×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C22×#10587739—22no—
22C22×#10587739—21no—
23C22×#10587739—21no—
24C4⋊#10587739—22no—
25C4⋊#10587739D412yes—
26C4⋊#10587739D412yes—
27C22⋊#10587739—24no—
28C22⋊#10587739—22no—
29C22⋊#10587739—44no—
30C4×#10587739—22no—
31C4×#10587739—22no—
32C23×D137D412yes—
33C2×D548C2×C411yescharacteristic
34C2×C22⋊D137Q812yes—
35D4×D137—22no—
36D4×D137—22no—
37C2×C4×D137C2311yescharacteristic
38#14555494C2×C412yescomplemented
39#14555493—22no—
40C8×D137—22no—
41C2×D4×C137C2×C411yescharacteristic
42M16×C137C2×C412yescomplemented
43C2×#10587739—22no—
44C2×#10587739—22no—
45C2×#10587739—22no—
46C2×#10587739—41no—
47C2×#10587739—41no—
48C2×#10587739—42no—
49C2×#10587739—41no—
50C2×#10587739—82no—
51C2×#10587739—82no—
52C22×D137—22no—
53C22×D137—22no—
54C22×D137—42no—
55C22×D137—42no—
56C22×D137C4⋊C412yes—
57C22×D137C2×D411yescharacteristic
58D548—22no—
59C22⋊D137—24no—
60C2×Dic137C2×Q811yescharacteristic
61C4×D137C2×D412yes—
62C4×D137#4112yes—
63#11764040—22no—
64C23×C137C4⋊C412yes—
65D4×C137—22no—
66C2×C4×C137C22×C411yescharacteristic
67C8×C137—22no—
68C22×C137—22no—
69C22×C137—42no—
70C22×C137C2×C4⋊C411yescharacteristic
71C2×D137—21no—
72C2×D137—21no—
73C2×D137—22no—
74C2×D137—42no—
75C2×D137—42no—
76C2×D137—42no—
77C2×D137—82no—
78C2×D137#12011yescharacteristic
79C2×D137#12311yescharacteristic
80#10587739—42no—
81#10587739—42no—
82#10587739—42no—
83#10587739—81no—
84C4×C137C4×D412yes—
85Dic137C22⋊Q812yes—
86C2×C137—21no—
87C2×C137—42no—
88C2×C137#38511yescharacteristic
89D137—21no—
90D137—42no—
91D137#45311yescharacteristic
92D137#45511yescharacteristic
93C137#185611yescharacteristic, complemented, Sylow
94#1856—1371nomaximal, Sylow
95#518—1371no—
96#428—1372no—
97C2×#100—1371no—
98#390—1371no—
99#339—1372no—
100C22×D4—1371no—
101C2×M16—1372no—
102C4×D4—2742no—
103C4×D4—2742no—
104#117—1372no—
105C2×C4⋊C4—1371no—
106C2×C22⋊C4—2742no—
107C2×C22⋊C4—2742no—
108C2×C42—1371no—
109#107—2742no—
110#106—2742no—
111#100—1371no—
112#100—1372no—
113#100#1058773911yescharacteristic, complemented
114C24—1372no—
115C2×D4—2742no—
116C2×D4—2742no—
117C2×D4—1372no—
118C2×D4—1371no—
119C2×D4C2×#1058773911yescharacteristic
120C22×C4—2741no—
121C22×C4—2742no—
122C22×C4—2742no—
123C22×C4—2741no—
124C22×C4—1371no—
125M16—2742no—
126M16—1372no—
127M16C2×#1058773912yescomplemented
128C2×C8—2742no—
129C4⋊C4—2742no—
130C4⋊C4—1372no—
131C4⋊C4—1372no—
132C22⋊C4—2744no—
133C22⋊C4—2742no—
134C22⋊C4—5484no—
135C42—2742no—
136C42—2742no—
137C23—2742no—
138C23—2742no—
139C23—5482no—
140C23—5482no—
141C23—1371no—
142C23—1372no—
143C23C4⋊#1058773912yes—
144D4—22no—
145D4—2744no—
146D4—2742no—
147C2×C4—2742no—
148C2×C4—2742no—
149C2×C4—2742no—
150C2×C4—5481no—
151C2×C4—5481no—
152C2×C4—5482no—
153C2×C4—5481no—
154C2×C4—10962no—
155C2×C4—10962no—
156C2×C4—1371no—
157C2×C4—1372no—
158C2×C4—1372no—
159C2×C4C22×#1058773911yescharacteristic
160C8—22no—
161C8—2742no—
162C22—22no—
163C22—42no—
164C22—2741no—
165C22—2741no—
166C22—2742no—
167C22—5482no—
168C22—5482no—
169C22—5482no—
170C22—10962no—
171C22—1371no—
172C22—1371no—
173C22C2×C4⋊#1058773911yescharacteristic
174C4—5482no—
175C4—5482no—
176C4—5482no—
177C4—10961no—
178C4—1372no—
179C4D4×#1058773912yes—
180C2—21no—
181C2—42no—
182C2—2741no—
183C2—5482no—
184C2—1371no—
185C2—1371no—
186C2#2257835011yescharacteristic
subgroupG/H
centerC2#22578350
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#100×C137C4
socleC2×C137#385
prime radicalC2×D4×D137C22
cosocleC2×C4×D137C23
hypercenter#100#10587739
hypocenterC137#1856
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #22578414C2semidirect
4. #22578275C2semidirect
17. #17984005C4semidirect
18. #17984004C4semidirect
19. #100×C137C4semidirect
38. #14555494C2×C4semidirect
42. M16×C137C2×C4semidirect
93. C137#1856semidirect
113. #100#10587739semidirect
127. M16C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225784142
#225783552
#225782752
#100×D1372
#1856137