Finite Group Databasebeta

← C2×#22558376 #30776156

Subgroups — C2×#22558376 #30776156

Order 17152.

450 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16863195C212yesmaximal, complemented
2#22558376C214yesmaximal, direct product factor
3C2×#492×C67C211yesmaximal, characteristic
4C2×#14523962—22no—
5#16863195—22no—
6#16863195C2214yescomplemented
7C22×SD16×C67—22no—
8C2×C4⋊Q8×C67C2211yescharacteristic
9C2×#127×C67C411yescharacteristic
10#492×C67—22no—
11#492×C67C412yes—
12#492×C67C2212yes—
13C2×C4×C8×C67C411yescharacteristic
14C2×#11761612—42no—
15#14523962—24no—
16#14523962—42no—
17C22×Q8×C67D412yes—
18C22×D4×C67—21no—
19C22×D4×C67—41no—
20C2×SD16×C67—24no—
21C2×SD16×C67—24no—
22C2×SD16×C67—44no—
23C2×SD16×C67—44no—
24C22×C8×C67—21no—
25C4⋊Q8×C67—41no—
26C4⋊Q8×C67C2×C411yescharacteristic
27C4⋊Q8×C67D412yes—
28C4⋊Q8×C67C2311yescharacteristic
29#127×C67—21no—
30#127×C67—41no—
31#127×C67C2×C412yes—
32C2×C4⋊C4×C67—41no—
33C2×C42×C67C2×C411yescharacteristic
34C4×C8×C67—21no—
35C4×C8×C67C2×C412yes—
36#11761612—44no—
37#11761612—82no—
38C2×C67⋊C8—82no—
39C24×C67—41no—
40C2×Q8×C67—22no—
41C2×Q8×C67—42no—
42C2×Q8×C67—42no—
43C2×Q8×C67C22⋊C412yes—
44C2×Q8×C67C2×D412yes—
45C2×D4×C67—22no—
46C2×D4×C67—42no—
47C2×D4×C67—42no—
48C2×D4×C67—42no—
49C2×D4×C67—42no—
50C2×D4×C67—41no—
51C2×D4×C67—84no—
52C22×C4×C67—21no—
53C22×C4×C67—22no—
54C22×C4×C67C22⋊C411yescharacteristic
55SD16×C67—48no—
56SD16×C67—48no—
57C2×C8×C67—22no—
58C2×C8×C67—22no—
59C2×C8×C67—22no—
60C4⋊C4×C67—41no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C42×C67—21no—
64C42×C67C22⋊C411yescharacteristic
65C42×C67C22×C411yescharacteristic
66C23×C67—42no—
67C23×C67—84no—
68C23×C67—81no—
69C23×C67—81no—
70C23×C67#9911yescharacteristic
71Q8×C67—42no—
72Q8×C67—42no—
73Q8×C67—82no—
74D4×C67—44no—
75D4×C67—82no—
76D4×C67—84no—
77D4×C67—84no—
78C2×C4×C67—24no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67#9911yescharacteristic
88C2×C4×C67C2×C22⋊C411yescharacteristic
89C8×C67—24no—
90C67⋊C8—84no—
91C22×C67—22no—
92C22×C67—84no—
93C22×C67—82no—
94C22×C67—162no—
95C22×C67—162no—
96C22×C67#35311yescharacteristic
97C22×C67#35311yescharacteristic
98C22×C67C2×#9911yescharacteristic
99C4×C67—22no—
100C4×C67—22no—
101C4×C67—21no—
102C4×C67—21no—
103C4×C67—42no—
104C4×C67—42no—
105C2×#1410—671nomaximal, Sylow
106C2×C67—21no—
107C2×C67—21no—
108C2×C67—162no—
109C2×C67#141012yescomplemented
110C2×C67C2×#35311yescharacteristic
111C2×#492D6711yescharacteristic
112C2×#356—672no—
113#1410—674no—
114C67C2×#141011yescharacteristic, complemented, Sylow
115C22×SD16—22no—
116C2×C4⋊Q8C2×D6711yescharacteristic
117C2×#127Dic6711yescharacteristic
118#492—22no—
119#492Dic6712yes—
120#492C2×D6712yes—
121C2×#101—1342no—
122C2×C4×C8Dic6711yescharacteristic
123#356—674no—
124#356—1342no—
125C22×Q8C22⋊D6712yes—
126C22×D4—21no—
127C22×D4—41no—
128C2×SD16—24no—
129C2×SD16—24no—
130C2×SD16—44no—
131C2×SD16—44no—
132C2×M16—2682no—
133C22×C8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×Dic6711yescharacteristic
136C4⋊Q8C22⋊D6712yes—
137C4⋊Q8C22×D6711yescharacteristic
138#127—21no—
139#127—41no—
140#127C2×Dic6712yes—
141C2×C4⋊C4—41no—
142C2×C42C2×Dic6711yescharacteristic
143#101—2682no—
144#101—1344no—
145C4×C8—21no—
146C4×C8C2×Dic6712yes—
147C24—41no—
148C2×Q8—22no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8C22⋊Dic6712yes—
152C2×Q8C2×C22⋊D6712yes—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—84no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4C22⋊Dic6711yescharacteristic
163SD16—48no—
164SD16—48no—
165M16—2684no—
166M16—5362no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—5362no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C42—21no—
175C42C22⋊Dic6711yescharacteristic
176C42C22×Dic6711yescharacteristic
177C23—42no—
178C23—84no—
179C23—81no—
180C23—81no—
181C23#1452395811yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—82no—
185D4—44no—
186D4—82no—
187D4—84no—
188D4—84no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4#1452395811yescharacteristic
199C2×C4C2×C22⋊Dic6711yescharacteristic
200C8—24no—
201C8—5364no—
202C22—22no—
203C22—84no—
204C22—82no—
205C22—162no—
206C22—162no—
207C22#1686318911yescharacteristic
208C22#1686318911yescharacteristic
209C22C2×#1452395811yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C2—21no—
217C2—21no—
218C2—162no—
219C2#2255837612yesdirect product factor
220C2C2×#1686318911yescharacteristic
subgroupG/H
centerC22#16863189
commutatorC42×C67C22×C4
FrattiniC4⋊Q8C22×D67
FittingC2×#492×C67C2
socleC22×C67#353
prime radicalC2×#127×C67C4
cosocleC4⋊Q8×C67C23
hypercenterC2×#492D67
hypocenterC67C2×#1410
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C2×#16863195C2semidirect
2. #22558376C2direct (N × H)
6. #16863195C22semidirect
109. C2×C67#1410semidirect
114. C67C2×#1410semidirect
219. C2#22558376direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168631952
#225583762
C2×#492×C672
C2×#141067