Finite Group Databasebeta

← #31797052

Subgroups — #31797052

Order 17536.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#22578390C211yesmaximal, characteristic
2#22578383C211yesmaximal, characteristic
3#22578306C212yesmaximal, complemented
4#22578305C212yesmaximal, complemented
5#22577975C211yesmaximal, characteristic
6C2×#14555519—21no—
7#17984167—21no—
8#17984149—21no—
9#17984148C2212yes—
10#17984146—21no—
11#17984146C2212yes—
12C2×#14555509C2211yescharacteristic
13C22×Dic274C411yescharacteristic, complemented
14#17984087—21no—
15#17984074—21no—
16#17984072C2211yescharacteristic
17#17984071C2211yescharacteristic
18#17984069C411yescharacteristic, complemented
19#17984066—21no—
20#17984065—21no—
21#17984044C411yescharacteristic, complemented
22#123×C137C411yescharacteristic, complemented
23#14555519—22no—
24#14555519—41no—
25C2×#11764047—41no—
26C4⋊#10587739—41no—
27C22⋊#10587739—22no—
28C4×#10587739—41no—
29#14555509—41no—
30#14555509—41no—
31C2×Dic274—21no—
32C2×Dic274—21no—
33C2×Dic274—41no—
34C2×Dic274—41no—
35C2×Dic274D411yescharacteristic
36C2×C22⋊D137D411yescharacteristic
37C2×C22⋊D137C2311yescharacteristic
38C22×Dic137C2×C411yescharacteristic
39C22×Dic137C2×C411yescharacteristic
40C22×Dic137D411yescharacteristic
41C2×C4×D137D411yescharacteristic
42C4⋊Dic137—21no—
43#14555489—21no—
44#14555489—21no—
45#14555489—21no—
46C22⋊Dic137—21no—
47C22⋊Dic137C2×C411yescharacteristic
48#14555487—21no—
49#14555487—21no—
50C4×Dic137—21no—
51C2×D4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic
53C4⋊C4×C137—21no—
54C22⋊C4×C137—21no—
55C22⋊C4×C137C2×C411yescharacteristic
56C2×#10587739—42no—
57#11764047—42no—
58C22×D137C2×D411yescharacteristic
59Dic274—41no—
60Dic274—42no—
61Dic274—41no—
62Dic274—41no—
63C22⋊D137—41no—
64C22⋊D137—41no—
65C2×Dic137—21no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137—21no—
69C2×Dic137—21no—
70C2×Dic137—21no—
71C2×Dic137—41no—
72C2×Dic137C22⋊C411yescharacteristic
73C2×Dic137C22⋊C411yescharacteristic
74C2×Dic137C2×D411yescharacteristic
75C4×D137—21no—
76C4×D137—21no—
77C23×C137C22⋊C411yescharacteristic
78C23×C137C22×C411yescharacteristic
79D4×C137—41no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—41no—
83C2×C4×C137C22⋊C411yescharacteristic
84C22×C137—21no—
85C22×C137—21no—
86C22×C137—41no—
87C22×C137—41no—
88C22×C137C2×C22⋊C411yescharacteristic
89C2×D137—21no—
90C2×D137—41no—
91#10587739—82no—
92C4×C137—21no—
93C4×C137—41no—
94C4×C137—41no—
95Dic137—21no—
96Dic137—21no—
97Dic137—41no—
98Dic137—41no—
99Dic137#9911yescharacteristic
100Dic137#9911yescharacteristic
101C2×C137—21no—
102C2×C137—41no—
103C2×C137—41no—
104C2×C137C2×#9911yescharacteristic
105D137—41no—
106C137#212311yescharacteristic, complemented, Sylow
107#2123—1371nomaximal, Sylow
108#536—1371no—
109#413—1371no—
110#410—1371no—
111#352—1372no—
112#352—1372no—
113C2×#41—1371no—
114C22×Q8—1371no—
115C2×M16—2741no—
116C4⋊Q8—2741no—
117#124—2741no—
118#123—1371no—
119#123—1371no—
120#123—1371no—
121#123#1058773911yescharacteristic, complemented
122C22⋊Q8—2741no—
123C22⋊Q8—2741no—
124#117—2741no—
125#117—1371no—
126#101—2741no—
127#100—1372no—
128#99—2741no—
129#99—1372no—
130#41—5481no—
131#41—5481no—
132C2×Q8—2741no—
133C2×Q8—2741no—
134C2×Q8—5481no—
135C2×Q8—5481no—
136C2×Q8—1371no—
137C2×D4—1371no—
138C2×D4—1371no—
139C2×D4C2×#1058773911yescharacteristic
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4—1371no—
143C22×C4—1371no—
144C22×C4C2×#1058773911yescharacteristic
145M16—2742no—
146M16—5481no—
147C2×C8—5481no—
148C4⋊C4—21no—
149C4⋊C4—2741no—
150C4⋊C4—2741no—
151C4⋊C4—2741no—
152C4⋊C4—2741no—
153C4⋊C4—5481no—
154C22⋊C4—21no—
155C22⋊C4—2741no—
156C22⋊C4—2741no—
157C22⋊C4—2741no—
158C22⋊C4—2742no—
159C22⋊C4—1371no—
160C22⋊C4C2×#1058773911yescharacteristic
161C42—2741no—
162C42—5481no—
163C23—1371no—
164C23C22⋊#1058773911yescharacteristic
165C23C22×#1058773911yescharacteristic
166Q8—5481no—
167Q8—5482no—
168Q8—5481no—
169Q8—5481no—
170D4—41no—
171D4—5481no—
172D4—5481no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—2741no—
177C2×C4—2741no—
178C2×C4—2741no—
179C2×C4—2741no—
180C2×C4—2741no—
181C2×C4—2741no—
182C2×C4—2741no—
183C2×C4—2741no—
184C2×C4—5481no—
185C2×C4—5482no—
186C2×C4—1371no—
187C2×C4—1371no—
188C2×C4—1371no—
189C2×C4C22⋊#1058773911yescharacteristic
190C8—5482no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—2741no—
196C22—5481no—
197C22C2×C22⋊#1058773911yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—2741no—
202C4—2741no—
203C4—5481no—
204C4—5481no—
205C4—10962no—
206C4—1371no—
207C4—1371no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—5481no—
212C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#123×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555509C22
cosocleC2×C22⋊D137C23
hypercenter#123#10587739
hypocenterC137#2123
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #22578306C2semidirect
4. #22578305C2semidirect
13. C22×Dic274C4semidirect
18. #17984069C4semidirect
21. #17984044C4semidirect
22. #123×C137C4semidirect
106. C137#2123semidirect
121. #123#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783832
#225783062
#225783052
#225779752
#2123137