Finite Group Databasebeta

← C2×#22550219 #30648624

Subgroups — C2×#22550219 #30648624

Order 16256.

416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861189C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic
3#22550219C218yesmaximal, direct product factor
4C2×#16861140C211yesmaximal, characteristic, complemented
5C2×#16861134C211yesmaximal, characteristic, complemented
6C2×#16861132C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic
8C2×#107×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×C4×D127C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12#16861189—22no—
13#16861189C2214yes—
14#16861167—22no—
15#16861167C2214yes—
16C2×C4⋊Dic127—21no—
17C2×C4⋊Dic127C2211yescharacteristic
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×C4×Dic127—21no—
21#16861140C2214yescomplemented
22#16861134C2214yescomplemented
23#16861132C2214yescomplemented
24(C2×D127)⋊C8C2214yes—
25C22×C8×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#107×C127C2214yescomplemented
29C23×D127D411yescharacteristic
30C2×Dic254—21no—
31C2×Dic254—22no—
32C2×Dic254—22no—
33C2×Dic254C2312yes—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37C2×C4×D127—22no—
38C2×C4×D127—22no—
39C2×C4×D127D412yes—
40C2×C4×D127C2312yes—
41C2×C127⋊C8—22no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—24no—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46(C2×D127)⋊C4—24no—
47(C2×D127)⋊C4—24no—
48C4×Dic127—24no—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—22no—
53C2×C8×C127C2312yes—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127C2312yes—
57C4⋊C4×C127C2312yes—
58C22×D127—22no—
59C22×D127—22no—
60C22×D127—22no—
61C22×D127C2×D412yes—
62Dic254—22no—
63Dic254—22no—
64Dic254—42no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—42no—
69C2×Dic127—42no—
70C2×Dic127C2×D412yes—
71C4×D127—24no—
72C4×D127—22no—
73C4×D127—22no—
74C127⋊C8—22no—
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127—42no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4112yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C8×C127—22no—
85C22×C127#12312yes—
86C22×C127#13512yes—
87C22×C127#13712yes—
88C22×C127C22×D411yescharacteristic
89C2×D127—22no—
90C2×D127—22no—
91C2×D127—22no—
92C2×D127—44no—
93C4×C127—42no—
94C4×C127—42no—
95C4×C127#12312yes—
96C4×C127C2×#4111yescharacteristic
97C4×C127C2×#4111yescharacteristic
98Dic127—22no—
99Dic127—42no—
100Dic127—42no—
101C2×C127#48514yescomplemented
102C2×C127C2×#12311yescharacteristic
103C2×C127C2×#13511yescharacteristic
104C2×C127C2×#13711yescharacteristic
105D127—42no—
106C2×#485—1271nomaximal, Sylow
107C127C2×#48511yescharacteristic, complemented, Sylow
108C2×C22⋊Q8—1271no—
109C2×#117—1271no—
110#485—1278no—
111C2×#107D12711yescharacteristic, complemented
112C2×#106—1271no—
113C2×#103—1271no—
114C2×#103—1271no—
115C2×C22⋊C8—1271no—
116C22×Q8—1271no—
117C23×C4—1271no—
118C22×C8—1271no—
119C22×C8C2×D12711yescharacteristic
120C22⋊Q8—1274no—
121C22⋊Q8—2542no—
122#117—1274no—
123#117—2542no—
124C2×C4⋊C4—1271no—
125C2×C4⋊C4—2541no—
126C2×C4⋊C4C2×D12711yescharacteristic
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C22⋊C4—2541no—
129C2×C22⋊C4—2541no—
130C2×C42—2541no—
131#107C2×D12714yescomplemented
132#106—1274no—
133#103—1274no—
134#103—1274no—
135C22⋊C8—1274no—
136C24—1271no—
137C2×Q8—1272no—
138C2×Q8—2542no—
139C2×Q8—2542no—
140C2×Q8—2541no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—1271no—
144C22×C4—1272no—
145C22×C4—1272no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4—2542no—
149C22×C4—2542no—
150C22×C4C22×D12711yescharacteristic
151C2×C8—22no—
152C2×C8—1272no—
153C2×C8—2542no—
154C2×C8C22×D12712yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—1272no—
158C4⋊C4—2544no—
159C4⋊C4—2541no—
160C4⋊C4C22×D12712yes—
161C4⋊C4C22×D12712yes—
162C22⋊C4—2544no—
163C22⋊C4—2544no—
164C42—2544no—
165C23—1272no—
166C23—2542no—
167C23—2542no—
168C23—2542no—
169C23D4×D12711yescharacteristic
170Q8—5082no—
171Q8—2542no—
172Q8—2542no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—1272no—
178C2×C4—5082no—
179C2×C4—5082no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4—2542no—
183C2×C4—2544no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4D4×D12711yescharacteristic
187C2×C4#1452324812yes—
188C2×C4#1452324912yes—
189C2×C4C23×D12711yescharacteristic
190C8—22no—
191C8—2542no—
192C22—5084no—
193C22—2542no—
194C22—2542no—
195C22—2542no—
196C22#1686119412yes—
197C22#1686122812yes—
198C22#1686124112yes—
199C22C2×D4×D12711yescharacteristic
200C4—42no—
201C4—42no—
202C4—5082no—
203C4—5082no—
204C4—2542no—
205C4#1686119412yes—
206C4C2×#1452324811yescharacteristic
207C4C2×#1452324911yescharacteristic
208C2—5082no—
209C2#2255021914yesdirect product factor
210C2C2×#1686119411yescharacteristic
211C2C2×#1686122811yescharacteristic
212C2C2×#1686124111yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#107×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
hypercenterC2×#107D127
hypocenterC127C2×#485
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
3. #22550219C2direct (N × H)
4. C2×#16861140C2semidirect
5. C2×#16861134C2semidirect
6. C2×#16861132C2semidirect
8. C2×#107×C127C2semidirect
21. #16861140C22semidirect
22. #16861134C22semidirect
23. #16861132C22semidirect
28. #107×C127C22semidirect
101. C2×C127#485semidirect
107. C127C2×#485semidirect
111. C2×#107D127semidirect
131. #107C2×D127semidirect
209. C2#22550219direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611892
C2×#168611672
#225502192
C2×#168611402
C2×#168611342
C2×#168611322
C2×(C2×D127)⋊C82
C2×#107×C1272
C2×#485127