Finite Group Databasebeta

← #37527194

Subgroups — #37527194

Order 22976.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#26040014C211yesmaximal, characteristic, complemented
2#26040013C211yesmaximal, characteristic, complemented
3#26040010C211yesmaximal, characteristic, complemented
4#26040010C211yesmaximal, characteristic, complemented
5#26040008C211yesmaximal, characteristic, complemented
6#26040008C211yesmaximal, characteristic, complemented
7#26040006C211yesmaximal, characteristic, complemented
8#26040005C211yesmaximal, characteristic, complemented
9#26040001C211yesmaximal, characteristic
10#26039998C211yesmaximal, characteristic
11#26039997C211yesmaximal, characteristic
12#26039994C211yesmaximal, characteristic, complemented
13#26039993C211yesmaximal, characteristic, complemented
14#26039991C211yesmaximal, characteristic, complemented
15C22⋊Q8×C359C211yesmaximal, characteristic, complemented
16C2×Dic718C2211yescharacteristic
17C2×Dic718C2211yescharacteristic
18C2×C22⋊D359C2211yescharacteristic
19C2×C4×D359C2211yescharacteristic
20C4⋊Dic359C2211yescharacteristic
21C4⋊Dic359C2211yescharacteristic
22C4⋊Dic359C2211yescharacteristic
23C4⋊Dic359C2211yescharacteristic
24Dic359⋊C4C2211yescharacteristic
25Dic359⋊C4C2211yescharacteristic
26Dic359⋊C4C2211yescharacteristic
27Dic359⋊C4C2211yescharacteristic
28Dic359⋊C4C2211yescharacteristic
29Dic359⋊C4C2211yescharacteristic
30Dic359⋊C4C2211yescharacteristic
31Dic359⋊C4C2211yescharacteristic
32Dic359⋊C4C2211yescharacteristic
33C22⋊Dic359C2211yescharacteristic
34C22⋊Dic359C2211yescharacteristic
35C22⋊Dic359C2211yescharacteristic
36(C2×D359)⋊C4C2211yescharacteristic
37(C2×D359)⋊C4C2211yescharacteristic
38(C2×D359)⋊C4C2211yescharacteristic
39(C2×D359)⋊C4C2211yescharacteristic
40(C2×D359)⋊C4C2211yescharacteristic
41C4×Dic359C2211yescharacteristic
42C4×Dic359C2211yescharacteristic
43C4×Dic359C2211yescharacteristic
44C2×Q8×C359C2211yescharacteristic
45C22×C4×C359C2211yescharacteristic
46C4⋊C4×C359C2211yescharacteristic
47C4⋊C4×C359C2211yescharacteristic
48C4⋊C4×C359C2211yescharacteristic
49C22⋊C4×C359C2211yescharacteristic
50C22⋊C4×C359C2211yescharacteristic
51C22×D359C2311yescharacteristic
52Dic718—41no—
53Dic718—41no—
54C22⋊D359—41no—
55C2×Dic359C2311yescharacteristic
56C2×Dic359C2311yescharacteristic
57C2×Dic359C2311yescharacteristic
58C2×Dic359C2311yescharacteristic
59C2×Dic359C2311yescharacteristic
60C2×Dic359C2311yescharacteristic
61C2×Dic359C2311yescharacteristic
62C4×D359—41no—
63C23×C359C2311yescharacteristic
64Q8×C359—41no—
65C2×C4×C359—41no—
66C2×C4×C359C2311yescharacteristic
67C2×C4×C359C2311yescharacteristic
68C2×C4×C359C2311yescharacteristic
69C2×C4×C359C2311yescharacteristic
70C2×C4×C359C2311yescharacteristic
71C2×C4×C359C2311yescharacteristic
72C22×C359—21no—
73C22×C359—21no—
74C22×C359—21no—
75C22×C359C2411yescharacteristic
76C2×D359—21no—
77C2×D359—21no—
78C2×D359—21no—
79C4×C359—21no—
80C4×C359—21no—
81C4×C359—21no—
82C4×C359—21no—
83C4×C359—21no—
84C4×C359—21no—
85Dic359—21no—
86Dic359—21no—
87Dic359—21no—
88Dic359—21no—
89Dic359—21no—
90Dic359—21no—
91Dic359—21no—
92C2×C359—41no—
93C2×C359H(5,𝔽2)11yescharacteristic
94C2×C3592−1+411yescharacteristic
95C2×C3592−1+411yescharacteristic
96D359—41no—
97C359#56311yescharacteristic, complemented, Sylow
98#563—3591nomaximal, Sylow
99C4⋊Q8—3591no—
100C4⋊Q8—3591no—
101#126—3591no—
102#126—3591no—
103#126—3591no—
104#126—3591no—
105#125—3591no—
106#125—3591no—
107#124—3591no—
108#123—3591no—
109#123—3591no—
110C22⋊Q8—3591no—
111C22⋊Q8—3591no—
112C22⋊Q8—3591no—
113C22⋊Q8D35911yescharacteristic, complemented
114C2×Q8—3591no—
115C2×Q8—3591no—
116C2×Q8C2×D35911yescharacteristic
117C2×D4—3591no—
118C22×C4—3591no—
119C22×C4C2×D35911yescharacteristic
120C4⋊C4—3591no—
121C4⋊C4—3591no—
122C4⋊C4—3591no—
123C4⋊C4—3591no—
124C4⋊C4—3591no—
125C4⋊C4—3591no—
126C4⋊C4—3591no—
127C4⋊C4—3591no—
128C4⋊C4—3591no—
129C4⋊C4—3591no—
130C4⋊C4—3591no—
131C4⋊C4—3591no—
132C4⋊C4—3591no—
133C4⋊C4C2×D35911yescharacteristic
134C4⋊C4C2×D35911yescharacteristic
135C4⋊C4C2×D35911yescharacteristic
136C22⋊C4—3591no—
137C22⋊C4—3591no—
138C22⋊C4—3591no—
139C22⋊C4—3591no—
140C22⋊C4—3591no—
141C22⋊C4—3591no—
142C22⋊C4—3591no—
143C22⋊C4—3591no—
144C22⋊C4C2×D35911yescharacteristic
145C22⋊C4C2×D35911yescharacteristic
146C42—3591no—
147C42—3591no—
148C42—3591no—
149C23—3591no—
150C23C22×D35911yescharacteristic
151Q8—41no—
152Q8—14361no—
153Q8—14361no—
154D4—14361no—
155C2×C4—41no—
156C2×C4—3591no—
157C2×C4—3591no—
158C2×C4—3591no—
159C2×C4—3591no—
160C2×C4—3591no—
161C2×C4—3591no—
162C2×C4—3591no—
163C2×C4—14361no—
164C2×C4C22×D35911yescharacteristic
165C2×C4C22×D35911yescharacteristic
166C2×C4C22×D35911yescharacteristic
167C2×C4C22×D35911yescharacteristic
168C2×C4C22×D35911yescharacteristic
169C2×C4C22×D35911yescharacteristic
170C22—21no—
171C22—21no—
172C22—21no—
173C22—7181no—
174C22—7181no—
175C22—7181no—
176C22C23×D35911yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—7181no—
184C4—7181no—
185C4—7181no—
186C4—7181no—
187C4—7181no—
188C4—7181no—
189C4—7181no—
190C2—41no—
191C2—14361no—
192C2#2604006011yescharacteristic
193C2#2604006211yescharacteristic
194C2#2604006311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
FittingC22⋊Q8×C359C2
prime radicalC2×C22⋊D359C22
hypercenterC22⋊Q8D359
hypocenterC359#563
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26040014C2semidirect
2. #26040013C2semidirect
3. #26040010C2semidirect
4. #26040010C2semidirect
5. #26040008C2semidirect
6. #26040008C2semidirect
7. #26040006C2semidirect
8. #26040005C2semidirect
12. #26039994C2semidirect
13. #26039993C2semidirect
14. #26039991C2semidirect
15. C22⋊Q8×C359C2semidirect
97. C359#563semidirect
113. C22⋊Q8D359semidirect

Maximal subgroups

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

subgroupindex
#260400142
#260400132
#260400102
#260400082
#260400062
#260400052
#260400012
#260399982
#260399972
#260399942
#260399932
#260399912
C22⋊Q8×C3592
#563359