Finite Group Databasebeta

← C2×#22577193 #31782425

Subgroups — C2×#22577193 #31782425

Order 17536.

278 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 77 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×#17984005C211yesmaximal, characteristic
2C2×#17984003C211yesmaximal, characteristic
3#22577193C214yesmaximal, direct product factor
4C2×#123×C137C211yesmaximal, characteristic
5C2×#14555494—21no—
6C2×C22⋊Dic137—21no—
7#17984005—21no—
8#17984005C2212yes—
9#17984003—21no—
10#17984003C2212yes—
11C22×D4×C137C2211yescharacteristic
12C23×C4×C137C411yescharacteristic, complemented
13#123×C137—21no—
14#123×C137—21no—
15#123×C137C412yescomplemented
16#123×C137C2212yes—
17C2×C4⋊C4×C137—21no—
18C2×C22⋊C4×C137—21no—
19C2×C22⋊C4×C137C411yescharacteristic, complemented
20C22×Dic137—41no—
21#14555494—22no—
22#14555494—41no—
23C2×#11764040—41no—
24C22⋊Dic137—22no—
25C22⋊Dic137—41no—
26C24×C137C2×C411yescharacteristic
27C24×C137D411yescharacteristic
28C2×D4×C137—41no—
29C2×D4×C137—41no—
30C2×D4×C137C2×C411yescharacteristic
31C2×D4×C137D411yescharacteristic
32C2×D4×C137D411yescharacteristic
33C2×D4×C137C2311yescharacteristic
34C22×C4×C137—21no—
35C22×C4×C137—21no—
36C22×C4×C137—21no—
37C22×C4×C137—41no—
38C22×C4×C137—41no—
39C22×C4×C137C2×C412yescomplemented
40C22×C4×C137D411yescharacteristic
41C4⋊C4×C137—22no—
42C4⋊C4×C137—22no—
43C22⋊C4×C137—22no—
44C22⋊C4×C137—21no—
45C22⋊C4×C137—22no—
46C22⋊C4×C137C2×C412yescomplemented
47C2×Dic137—42no—
48C2×Dic137—82no—
49#11764040—42no—
50C23×C137—21no—
51C23×C137—21no—
52C23×C137—21no—
53C23×C137—21no—
54C23×C137—42no—
55C23×C137—42no—
56C23×C137C22⋊C411yescharacteristic
57C23×C137C22⋊C411yescharacteristic
58C23×C137C22⋊C411yescharacteristic
59C23×C137C22×C411yescharacteristic
60C23×C137C2×D411yescharacteristic
61D4×C137—41no—
62D4×C137—41no—
63D4×C137—41no—
64D4×C137—41no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—22no—
68C2×C4×C137—22no—
69C2×C4×C137—42no—
70C2×C4×C137—42no—
71C2×C4×C137—41no—
72C2×C4×C137—41no—
73C2×C4×C137—42no—
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C22×C137—22no—
77C22×C137—21no—
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137—41no—
85C22×C137—42no—
86C22×C137—42no—
87C22×C137—42no—
88C22×C137—81no—
89C22×C137#9911yescharacteristic
90C22×C137#9911yescharacteristic
91C22×C137C2×C22⋊C411yescharacteristic
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—42no—
95C4×C137—42no—
96Dic137—82no—
97C2×C137—21no—
98C2×C137—21no—
99C2×C137—41no—
100C2×C137—41no—
101C2×C137—41no—
102C2×C137—41no—
103C2×C137#35212yescomplemented
104C2×C137C2×#9911yescharacteristic
105C137C2×#35211yescharacteristic, complemented, Sylow
106C2×#352—1371nomaximal, Sylow
107C2×#123D13711yescharacteristic
108C2×#100—1371no—
109C2×#99—1371no—
110#352—1374no—
111C22×D4C2×D13711yescharacteristic
112C23×C4Dic13711yescharacteristic, complemented
113C2×M16—2741no—
114#123—21no—
115#123—21no—
116#123Dic13712yescomplemented
117#123C2×D13712yes—
118C2×C4⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—2741no—
121C2×C22⋊C4Dic13711yescharacteristic, complemented
122#100—2741no—
123#100—1372no—
124#99—2741no—
125#99—1372no—
126C24C2×Dic13711yescharacteristic
127C24C22⋊D13711yescharacteristic
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic13711yescharacteristic
131C2×D4C22⋊D13711yescharacteristic
132C2×D4C22⋊D13711yescharacteristic
133C2×D4C22×D13711yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—5481no—
140C22×C4C2×Dic13712yescomplemented
141C22×C4C22⋊D13711yescharacteristic
142M16—2742no—
143M16—5481no—
144C2×C8—5481no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—21no—
149C22⋊C4—22no—
150C22⋊C4—2742no—
151C22⋊C4—5481no—
152C22⋊C4C2×Dic13712yescomplemented
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—42no—
158C23—42no—
159C23C22⋊Dic13711yescharacteristic
160C23C22⋊Dic13711yescharacteristic
161C23C22⋊Dic13711yescharacteristic
162C23C22×Dic13711yescharacteristic
163C23C2×C22⋊D13711yescharacteristic
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—5482no—
178C2×C4—10962no—
179C2×C4C22⋊Dic13711yescharacteristic
180C2×C4C2×C22⋊D13711yescharacteristic
181C8—5482no—
182C22—22no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22#1798400311yescharacteristic
196C22#1798400311yescharacteristic
197C22C2×C22⋊Dic13711yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—10962no—
203C2—21no—
204C2—21no—
205C2—41no—
206C2—41no—
207C2—41no—
208C2—41no—
209C2#2257719312yesdirect product factor
210C2C2×#1798400311yescharacteristic
subgroupG/H
centerC22#17984003
commutatorC23×C137C22×C4
FrattiniC2×D4C22×D137
FittingC2×#123×C137C2
socleC22×C137#99
prime radicalC22×D4×C137C22
cosocleC2×D4×C137C23
hypercenterC2×#123D137
hypocenterC137C2×#352
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #22577193C2direct (N × H)
12. C23×C4×C137C4semidirect
15. #123×C137C4semidirect
19. C2×C22⋊C4×C137C4semidirect
39. C22×C4×C137C2×C4semidirect
46. C22⋊C4×C137C2×C4semidirect
103. C2×C137#352semidirect
105. C137C2×#352semidirect
112. C23×C4Dic137semidirect
116. #123Dic137semidirect
121. C2×C22⋊C4Dic137semidirect
140. C22×C4C2×Dic137semidirect
152. C22⋊C4C2×Dic137semidirect
209. C2#22577193direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179840052
C2×#179840032
#225771932
C2×#123×C1372
C2×#352137