Finite Group Databasebeta

← C2×#22578257 #31796159

Subgroups — C2×#22578257 #31796159

Order 17536.

336 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 82 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×#17984145C211yesmaximal, characteristic
2C2×(C2×#10587739)⋊C4C211yesmaximal, characteristic
3#22578257C214yesmaximal, direct product factor
4C2×#17984064C211yesmaximal, characteristic
5C23×#10587739—21no—
6C22×#11764047—21no—
7#17984145—21no—
8#17984145C2212yes—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11C22×C4×D137C2211yescharacteristic
12#17984064—21no—
13#17984064—21no—
14#17984064C412yescomplemented
15#17984064C2212yes—
16C2×C4⋊Dic137C411yescharacteristic, complemented
17C2×#14555489—21no—
18C2×#14555487—21no—
19C22×Q8×C137C411yescharacteristic, complemented
20C22×#10587739—24no—
21C22×#10587739—22no—
22C22×#10587739—42no—
23C22×#10587739—42no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C23×D137D411yescharacteristic
27C22×Dic137—21no—
28C22×Dic137D411yescharacteristic
29C2×C4×D137—42no—
30C2×C4×D137C2×C411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137C2311yescharacteristic
34C4⋊Dic137—21no—
35C4⋊Dic137C2×C412yescomplemented
36#14555489—22no—
37#14555489—22no—
38#14555487—22no—
39#14555487—22no—
40C2×Q8×C137—21no—
41C2×Q8×C137—42no—
42C2×Q8×C137C2×C412yescomplemented
43C22×C4×C137—21no—
44C22×C4×C137C2×C411yescharacteristic
45C2×#10587739—24no—
46C2×#10587739—48no—
47C2×#10587739—42no—
48C2×#10587739—42no—
49#11764047—42no—
50C22×D137—22no—
51C22×D137—22no—
52C22×D137C22⋊C411yescharacteristic
53C22×D137SD1612yes—
54C22×D137Q1612yes—
55C22×D137C2×D411yescharacteristic
56C2×Dic137—22no—
57C2×Dic137—22no—
58C2×Dic137—42no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—41no—
62C4×D137—41no—
63C4×D137—41no—
64C4×D137—41no—
65C23×C137C22⋊C411yescharacteristic
66Q8×C137—42no—
67Q8×C137—42no—
68C2×C4×C137—22no—
69C2×C4×C137—22no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22×C411yescharacteristic
73C22×C137#9912yes—
74C22×C137#10312yes—
75C22×C137#10412yes—
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—24no—
78C2×D137—24no—
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—21no—
82C2×D137—21no—
83C2×D137#10312yes—
84C2×D137C2×SD1611yescharacteristic
85C2×D137C2×Q1611yescharacteristic
86#10587739—44no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—42no—
90Dic137—21no—
91Dic137—21no—
92Dic137—42no—
93C2×C137#32814yescomplemented
94C2×C137C2×#9911yescharacteristic
95C2×C137C2×#10311yescharacteristic
96C2×C137C2×#10411yescharacteristic
97D137—21no—
98D137—22no—
99D137—21no—
100C137C2×#32811yescharacteristic, complemented, Sylow
101C2×#328—1371nomaximal, Sylow
102C2×C22⋊Q8—1371no—
103C2×C22⋊C8—1371no—
104C2×#95—1371no—
105#328—1374no—
106C22×Q8#1058773911yescharacteristic, complemented
107C23×C4—2741no—
108C23×C4—1371no—
109C22×C8—2741no—
110C22⋊Q8—2741no—
111C22⋊Q8—2741no—
112C22⋊Q8—1372no—
113C22⋊Q8—1372no—
114C2×C4⋊C4—2741no—
115C2×C4⋊C4—1371no—
116C2×C22⋊C4—2741no—
117C22⋊C8—2741no—
118C22⋊C8—1372no—
119#95—2741no—
120#95—1372no—
121C24—1371no—
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8C2×#1058773912yescomplemented
125C22×C4—21no—
126C22×C4—2744no—
127C22×C4—2742no—
128C22×C4—2741no—
129C22×C4—5482no—
130C22×C4—5482no—
131C22×C4—5482no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4C2×#1058773911yescharacteristic
138C2×C8—2742no—
139C2×C8—5482no—
140C4⋊C4—2742no—
141C4⋊C4—2742no—
142C4⋊C4—2741no—
143C4⋊C4—1372no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C23—2742no—
147C23—2742no—
148C23—1372no—
149C23—1372no—
150C23—1371no—
151C23—1371no—
152C23C22⋊#1058773911yescharacteristic
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—2742no—
159C2×C4—2744no—
160C2×C4—2742no—
161C2×C4—5481no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4—5481no—
165C2×C4—5488no—
166C2×C4—5482no—
167C2×C4—5482no—
168C2×C4—5482no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4C22⋊#1058773911yescharacteristic
172C2×C4C22×#1058773911yescharacteristic
173C8—5482no—
174C22—2744no—
175C22—2744no—
176C22—2741no—
177C22—2741no—
178C22—2741no—
179C22—2741no—
180C22—1371no—
181C22—1372no—
182C22—1371no—
183C22#1798414612yes—
184C22#1798415112yes—
185C22#1798415312yes—
186C22C2×C22⋊#1058773911yescharacteristic
187C4—21no—
188C4—21no—
189C4—42no—
190C4—2741no—
191C4—2741no—
192C4—5484no—
193C4—5482no—
194C2—2741no—
195C2—2742no—
196C2—2741no—
197C2#2257825714yesdirect product factor
198C2C2×#1798414611yescharacteristic
199C2C2×#1798415111yescharacteristic
200C2C2×#1798415311yescharacteristic
subgroupG/H
centerC23C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×Q8×C137C4
socleC23×C137C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22×Q8#10587739
hypocenterC137C2×#328
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #22578257C2direct (N × H)
14. #17984064C4semidirect
16. C2×C4⋊Dic137C4semidirect
19. C22×Q8×C137C4semidirect
35. C4⋊Dic137C2×C4semidirect
42. C2×Q8×C137C2×C4semidirect
93. C2×C137#328semidirect
100. C137C2×#328semidirect
106. C22×Q8#10587739semidirect
124. C2×Q8C2×#10587739semidirect
197. C2#22578257direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841452
C2×(C2×#10587739)⋊C42
#225782572
C2×#179840642
C2×#328137