Finite Group Databasebeta

← C2×#20261029 #27151506

Subgroups — C2×#20261029 #27151506

Order 12416.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#15690809C211yesmaximal, characteristic
2C2×#15690775C211yesmaximal, characteristic
3#20261029C218yesmaximal, direct product factor
4C2×#15690743C211yesmaximal, characteristic
5C2×#15690739C211yesmaximal, characteristic
6C2×#15690735C211yesmaximal, characteristic
7C2×#15690733C211yesmaximal, characteristic
8C2×#107×C97C211yesmaximal, characteristic
9C22×Dic194—21no—
10C22×Dic194C2211yescharacteristic
11C22×#11715895C2211yescharacteristic
12#15690809—22no—
13#15690809C2214yes—
14#15690775—22no—
15#15690775C2214yes—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490—21no—
18C2×#13095490—21no—
19C2×C4×Dic97—21no—
20C2×C4×Dic97C2211yescharacteristic
21#15690743C2214yes—
22#15690739C2214yes—
23#15690735C2214yes—
24#15690733C2214yes—
25C22×C8×C97C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28#107×C97C2214yes—
29C2×Dic194—21no—
30C2×Dic194—22no—
31C2×Dic194—24no—
32C2×Dic194—44no—
33C2×Dic194Q814yescomplemented
34C2×Dic194C2312yes—
35C22×Dic97—21no—
36C22×Dic97—21no—
37C22×Dic97D411yescharacteristic
38C22×Dic97D411yescharacteristic
39C2×#11715895—22no—
40C2×#11715895C2312yes—
41C4⋊Dic97—21no—
42C4⋊Dic97C2312yes—
43#13095490—24no—
44#13095490—24no—
45C4×Dic97—24no—
46C4×Dic97D412yes—
47C4×Dic97C2312yes—
48C22×C4×C97—21no—
49C22×C4×C97—21no—
50C22×C4×C97C2311yescharacteristic
51C2×C8×C97—22no—
52C2×C8×C97C2312yes—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C4⋊C4×C97C2312yes—
56C4⋊C4×C97C2312yes—
57C23×C97C2×D411yescharacteristic
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—42no—
61C2×C4×C97—42no—
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×Q812yes—
64C2×C4×C97#4112yes—
65C2×C4×C97C2411yescharacteristic
66C2×Dic97—24no—
67C2×Dic97—22no—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—42no—
71C2×Dic97Q1614yes—
72C2×Dic97C2×D412yes—
73C2×Dic97C2×D412yes—
74Dic194—24no—
75Dic194—22no—
76Dic194—48no—
77Dic194C2×Q814yescomplemented
78C8×C97—22no—
79#11715895—22no—
80C22×C97C22⋊Q812yes—
81C22×C97C2×Q1612yes—
82C22×C97AGL(1,ℤ/8)12yes—
83C22×C97C22×D411yescharacteristic
84C4×C97—42no—
85C4×C97—42no—
86C4×C97C22⋊Q812yes—
87C4×C97C22×Q811yescharacteristic
88C4×C97C2×#4111yescharacteristic
89Dic97—24no—
90Dic97—22no—
91Dic97—42no—
92Dic97C2×Q1614yes—
93C2×C97#47714yescomplemented
94C2×C97C2×C22⋊Q811yescharacteristic
95C2×C97C22×Q1611yescharacteristic
96C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
97C2×#477—971nomaximal, Sylow
98C97C2×#47711yescharacteristic, complemented, Sylow
99C2×C4⋊Q8—971no—
100C2×C4×Q8—971no—
101#477—978no—
102C2×#107—971no—
103C2×#107D9711yescharacteristic
104C2×C4⋊C8—971no—
105C2×#103—971no—
106C2×#103—971no—
107C22×Q8—971no—
108C22×Q8—1941no—
109C22×C8—971no—
110C22×C8C2×D9711yescharacteristic
111C4⋊Q8—974no—
112C4⋊Q8—1942no—
113C4×Q8—974no—
114C4×Q8—1942no—
115C2×C4⋊C4—971no—
116C2×C4⋊C4—1941no—
117C2×C4⋊C4—1941no—
118C2×C4⋊C4C2×D9711yescharacteristic
119C2×C4⋊C4C2×D9711yescharacteristic
120C2×C42—971no—
121C2×C42—1941no—
122#107—974no—
123#107C2×D9714yes—
124C4⋊C8—974no—
125#103—974no—
126#103—974no—
127C2×Q8—974no—
128C2×Q8—972no—
129C2×Q8—3884no—
130C2×Q8—1944no—
131C2×Q8—1941no—
132C2×Q8—1942no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—1941no—
138C22×C4—1941no—
139C22×C4C22×D9711yescharacteristic
140C2×C8—22no—
141C2×C8—972no—
142C2×C8—1942no—
143C2×C8C22×D9712yes—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—972no—
147C4⋊C4—1944no—
148C4⋊C4—1944no—
149C4⋊C4—1941no—
150C4⋊C4C22×D9712yes—
151C4⋊C4C22×D9712yes—
152C42—972no—
153C42—972no—
154C42—1944no—
155C23D4×D9711yescharacteristic
156Q8—974no—
157Q8—3888no—
158Q8—1944no—
159Q8—1942no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—974no—
165C2×C4—972no—
166C2×C4—972no—
167C2×C4—3882no—
168C2×C4—1942no—
169C2×C4—1944no—
170C2×C4—1942no—
171C2×C4—1942no—
172C2×C4D4×D9711yescharacteristic
173C2×C4Q8×D9712yes—
174C2×C4#1309551112yes—
175C2×C4C23×D9711yescharacteristic
176C8—22no—
177C8—1942no—
178C22#1569078812yes—
179C22Q16×D9712yes—
180C22#1569083412yes—
181C22C2×D4×D9711yescharacteristic
182C4—42no—
183C4—42no—
184C4—974no—
185C4—3882no—
186C4—1944no—
187C4—1942no—
188C4#1569078812yes—
189C4C2×Q8×D9711yescharacteristic
190C4C2×#1309551111yescharacteristic
191C2#2026102914yesdirect product factor
192C2C2×#1569078811yescharacteristic
193C2C2×Q16×D9711yescharacteristic
194C2C2×#1569083411yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#107×C97C2
socle, prime radicalC23×C97C2×D4
hypercenterC2×#107D97
hypocenterC97C2×#477
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
3. #20261029C2direct (N × H)
33. C2×Dic194Q8semidirect
77. Dic194C2×Q8semidirect
93. C2×C97#477semidirect
98. C97C2×#477semidirect
191. C2#20261029direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908092
C2×#156907752
#202610292
C2×#156907432
C2×#156907392
C2×#156907352
C2×#156907332
C2×#107×C972
C2×#47797