Finite Group Databasebeta

← #37527126

Subgroups — #37527126

Order 22976.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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#26040010C211yesmaximal, characteristic, complemented
2#26040009C211yesmaximal, characteristic, complemented
3#26040007C211yesmaximal, characteristic, complemented
4#26040005C211yesmaximal, characteristic, complemented
5#26040005C211yesmaximal, characteristic, complemented
6#26039999C211yesmaximal, characteristic
7#26039998C211yesmaximal, characteristic
8#26039993C211yesmaximal, characteristic, complemented
9#26039992C211yesmaximal, characteristic, complemented
10C4×Dic718C211yesmaximal, characteristic, complemented
11(C22⋊D359)⋊C4C211yesmaximal, characteristic
12#26039972C211yesmaximal, characteristic, complemented
13#26039971C211yesmaximal, characteristic, complemented
14C2×Dic359⋊C4C211yesmaximal, characteristic, complemented
15#117×C359C211yesmaximal, characteristic, complemented
16C2×Dic718C2211yescharacteristic
17C2×C22⋊D359C2211yescharacteristic
18C22×Dic359C2211yescharacteristic
19C22×Dic359C2211yescharacteristic
20C2×C4×D359C2211yescharacteristic
21C4⋊Dic359C2211yescharacteristic
22C4⋊Dic359C2211yescharacteristic
23C4⋊Dic359C2211yescharacteristic
24Dic359⋊C4C2211yescharacteristic
25Dic359⋊C4C2211yescharacteristic
26Dic359⋊C4C2211yescharacteristic
27Dic359⋊C4C2211yescharacteristic
28Dic359⋊C4C2211yescharacteristic
29Dic359⋊C4C2211yescharacteristic
30Dic359⋊C4C2211yescharacteristic
31Dic359⋊C4C2211yescharacteristic
32Dic359⋊C4C2211yescharacteristic
33Dic359⋊C4C2211yescharacteristic
34Dic359⋊C4C2211yescharacteristic
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
44C22×C4×C359C2211yescharacteristic
45C4⋊C4×C359C2211yescharacteristic
46C4⋊C4×C359C2211yescharacteristic
47C22⋊C4×C359C2211yescharacteristic
48C22⋊C4×C359C2211yescharacteristic
49C42×C359C2211yescharacteristic
50C42×C359C2211yescharacteristic
51C22×D359C2311yescharacteristic
52Dic718—22no—
53C22⋊D359—22no—
54C2×Dic359—22no—
55C2×Dic359—22no—
56C2×Dic359C2311yescharacteristic
57C2×Dic359C2311yescharacteristic
58C2×Dic359C2311yescharacteristic
59C2×Dic359C2311yescharacteristic
60C2×Dic359C2311yescharacteristic
61C2×Dic359C2311yescharacteristic
62C2×Dic359C2311yescharacteristic
63C4×D359—22no—
64C23×C359C2311yescharacteristic
65C2×C4×C359—22no—
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#4112yes—
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#4112yes—
92C2×C359—22no—
93C2×C359C2×#4111yescharacteristic
94C2×C359C2×#4111yescharacteristic
95C2×C3592−1+411yescharacteristic
96D359—41no—
97C359#55211yescharacteristic, complemented, Sylow
98#552—3591nomaximal, Sylow
99#126—3591no—
100#126—3591no—
101#125—3591no—
102#125—3591no—
103#125—3591no—
104#123—3591no—
105#123—3591no—
106C22⋊Q8—3591no—
107C22⋊Q8—3591no—
108C4×Q8—3591no—
109C4×D4—3591no—
110#117—3591no—
111#117—3591no—
112#117D35911yescharacteristic, complemented
113C2×C4⋊C4—3591no—
114C2×Q8—3591no—
115C2×D4—3591no—
116C22×C4—3591no—
117C22×C4—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⋊C4—3591no—
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⋊C4C2×D35911yescharacteristic
143C22⋊C4C2×D35911yescharacteristic
144C42—3591no—
145C42—3591no—
146C42—3591no—
147C42C2×D35911yescharacteristic
148C42C2×D35911yescharacteristic
149C23—3591no—
150C23C22×D35911yescharacteristic
151Q8—7182no—
152D4—7182no—
153C2×C4—22no—
154C2×C4—3591no—
155C2×C4—3591no—
156C2×C4—3591no—
157C2×C4—3591no—
158C2×C4—3591no—
159C2×C4—3591no—
160C2×C4—3591no—
161C2×C4—7182no—
162C2×C4—7182no—
163C2×C4—7182no—
164C2×C4C22×D35911yescharacteristic
165C2×C4C22×D35911yescharacteristic
166C2×C4C22×D35911yescharacteristic
167C2×C4C22×D35911yescharacteristic
168C2×C4C22×D35911yescharacteristic
169C2×C4C22×D35911yescharacteristic
170C22—21no—
171C22—21no—
172C22—7181no—
173C22—7181no—
174C22—7181no—
175C22#1915334112yes—
176C22C23×D35911yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—3592no—
184C4—7181no—
185C4—7181no—
186C4—7181no—
187C4—7181no—
188C4—7181no—
189C4—7181no—
190C2—22no—
191C2—14361no—
192C2#41×D35911yescharacteristic
193C2C2×#1915334111yescharacteristic
194C2#2604006211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D359
commutator, socle, cosocleC22×C359C24
Fitting#117×C359C2
prime radicalC2×C22⋊D359C22
hypercenter#117D359
hypocenterC359#552
LCSG ▷ C22×C359 ▷ C359
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #26040010C2semidirect
2. #26040009C2semidirect
3. #26040007C2semidirect
4. #26040005C2semidirect
5. #26040005C2semidirect
8. #26039993C2semidirect
9. #26039992C2semidirect
10. C4×Dic718C2semidirect
12. #26039972C2semidirect
13. #26039971C2semidirect
14. C2×Dic359⋊C4C2semidirect
15. #117×C359C2semidirect
97. C359#552semidirect
112. #117D359semidirect

Maximal subgroups

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

subgroupindex
#260400102
#260400092
#260400072
#260400052
#260399992
#260399982
#260399932
#260399922
C4×Dic7182
(C22⋊D359)⋊C42
#260399722
#260399712
C2×Dic359⋊C42
#117×C3592
#552359