Finite Group Databasebeta

← C2×#31774079 #46845705

Subgroups — C2×#31774079 #46845705

Order 34688.

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22576208C211yesmaximal, characteristic
2C2×#22576191C212yesmaximal, complemented
3#31774079C218yesmaximal, direct product factor
4C2×#22576088C212yesmaximal, complemented
5C2×#22576079C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C271C211yesmaximal, characteristic, complemented
7#22576208—24no—
8#22576208—22no—
9#22576208C2214yes—
10C2×#16864655—22no—
11C2×#16864654—22no—
12C2×#16864654C2212yes—
13C2×Q8×D271—22no—
14#22576191—24no—
15#22576191C2218yescomplemented
16C2×#16864644—22no—
17C2×#16864641—22no—
18C2×#16864638C2212yes—
19#22576088C2218yescomplemented
20#22576079C2214yescomplemented
21C22×Q8×C271C2211yescharacteristic
22C22×Q8×C271C2211yescharacteristic
23C4⋊Q8×C271—22no—
24C4⋊Q8×C271C2214yescomplemented
25C2×C4⋊C4×C271—22no—
26C2×C42×C271C2211yescharacteristic
27#16864655—28no—
28#16864655—28no—
29#16864654—28no—
30#16864654—28no—
31#16864654—22no—
32#16864654—22no—
33#16864654D414yes—
34#16864654C2314yes—
35C2×Dic542—22no—
36C2×Dic542D412yes—
37Q8×D271—28no—
38Q8×D271—28no—
39C2×D1084—22no—
40C2×D1084D412yes—
41C2×C22⋊D271—22no—
42C2×C4×D271—24no—
43C2×C4×D271—22no—
44#16864644—28no—
45#16864641—28no—
46#16864638—22no—
47#16864638C2314yes—
48C2×C271⋊C8—22no—
49C2×Q8×C271—22no—
50C2×Q8×C271—22no—
51C2×Q8×C271—21no—
52C2×Q8×C271—22no—
53C2×Q8×C271—22no—
54C2×Q8×C271D412yes—
55C2×Q8×C271C2312yes—
56C2×Q8×C271C2312yes—
57C22×C4×C271—21no—
58C22×C4×C271—21no—
59C22×C4×C271D412yes—
60C22×C4×C271C2311yescharacteristic
61C4⋊C4×C271—28no—
62C42×C271—21no—
63C42×C271C2312yes—
64C22×D271—22no—
65Dic542—28no—
66Dic542—22no—
67Dic542C2×D414yes—
68D1084—28no—
69D1084—22no—
70D1084C2×D414yes—
71C22⋊D271—24no—
72C22⋊D271—24no—
73C2×Dic271—22no—
74C4×D271—216no—
75C4×D271—24no—
76C4×D271—24no—
77C271⋊C8—24no—
78C23×C271C2×D411yescharacteristic
79Q8×C271—24no—
80Q8×C271—24no—
81Q8×C271—24no—
82Q8×C271—42no—
83C2×C4×C271—24no—
84C2×C4×C271—22no—
85C2×C4×C271—22no—
86C2×C4×C271—22no—
87C2×C4×C271—22no—
88C2×C4×C271—41no—
89C2×C4×C271—41no—
90C2×C4×C271C2×D412yes—
91C2×C4×C271C2×D412yes—
92C2×C4×C271C2×D414yes—
93C2×C4×C271C2×D411yescharacteristic
94C2×C4×C271C2411yescharacteristic
95C22×C271—22no—
96C22×C271#12011yescharacteristic
97C22×C271#12011yescharacteristic
98C22×C271C22×D411yescharacteristic
99C2×D271—24no—
100C2×D271—44no—
101C4×C271—24no—
102C4×C271—24no—
103C4×C271—42no—
104C4×C271#12012yes—
105C4×C271C22×D412yes—
106Dic271—24no—
107C2×C271—21no—
108C2×C271—21no—
109C2×C271#45612yescomplemented
110C2×C271C2×#12011yescharacteristic
111D271—44no—
112C271C2×#45611yescharacteristic, complemented, Sylow
113C2×#456—2711nomaximal, Sylow
114C2×2−1+4—2711no—
115C2×#137—2712no—
116C2×C4⋊Q8D27111yescharacteristic, complemented
117#456—2718no—
118C2×#104—2712no—
119C2×#101—2711no—
1202−1+4—2714no—
1212−1+4—5424no—
1222−1+4—5422no—
123C2×#41—2712no—
124C2×#41—5422no—
125C2×#41—5422no—
126C22×Q8—5422no—
127C22×Q8C2×D27111yescharacteristic
128C22×Q8C2×D27111yescharacteristic
129#137—2718no—
130#137—5424no—
131C2×Q16—5422no—
132C2×SD16—5422no—
133C2×M16—2712no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D27114yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D27111yescharacteristic
138#104—2718no—
139#101—2714no—
140#41—2714no—
141#41—2714no—
142#41—5428no—
143#41—5428no—
144#41—5428no—
145#41—5422no—
146#41—5422no—
147#41—5428no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—2712no—
154C2×Q8—5422no—
155C2×Q8—5428no—
156C2×Q8—5428no—
157C2×Q8C22⋊D27112yes—
158C2×Q8C22×D27112yes—
159C2×Q8C22×D27112yes—
160C2×D4—2712no—
161C2×D4—5422no—
162C2×D4—5422no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—5424no—
166C22×C4—5422no—
167C22×C4C22⋊D27112yes—
168C22×C4C22×D27111yescharacteristic
169Q16—5428no—
170SD16—5428no—
171M16—2714no—
172M16—5422no—
173C2×C8—5422no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D27112yes—
177C23—5422no—
178C23C2×C22⋊D27111yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—2714no—
184Q8—5428no—
185Q8—5422no—
186D4—2714no—
187D4—5428no—
188D4—5424no—
189D4—5424no—
190D4—5422no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—54216no—
199C2×C4—5422no—
200C2×C4—5424no—
201C2×C4—5424no—
202C2×C4D4×D27112yes—
203C2×C4D4×D27112yes—
204C2×C4C2×C22⋊D27114yes—
205C2×C4C2×C22⋊D27111yescharacteristic
206C2×C4C23×D27111yescharacteristic
207C8—5424no—
208C22—22no—
209C22—5424no—
210C22—10844no—
211C22#2257612811yescharacteristic
212C22#2257612811yescharacteristic
213C22C22×C22⋊D27111yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—5424no—
218C4#2257612812yes—
219C4C2×D4×D27112yes—
220C2—21no—
221C2—21no—
222C2—10844no—
223C2#3177407912yesdirect product factor
224C2C2×#2257612811yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×C4⋊Q8×C271C2
socleC22×C271#120
prime radicalC2×#22576208C2
hypercenterC2×C4⋊Q8D271
hypocenterC271C2×#456
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22576191C2semidirect
3. #31774079C2direct (N × H)
4. C2×#22576088C2semidirect
5. C2×#22576079C2semidirect
6. C2×C4⋊Q8×C271C2semidirect
15. #22576191C22semidirect
19. #22576088C22semidirect
20. #22576079C22semidirect
24. C4⋊Q8×C271C22semidirect
109. C2×C271#456semidirect
112. C271C2×#456semidirect
116. C2×C4⋊Q8D271semidirect
135. C4⋊Q8C2×D271semidirect
223. C2#31774079direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225762082
C2×#225761912
#317740792
C2×#225760882
C2×#225760792
C2×C4⋊Q8×C2712
C2×#456271