Finite Group Databasebeta

← #27182573

Subgroups — #27182573

Order 12928.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265782C211yesmaximal, characteristic
2#20265484C212yesmaximal, complemented
3#20265474C212yesmaximal, complemented
4#20265425C212yesmaximal, complemented
5#20265408C212yesmaximal, complemented
6#20265379C212yesmaximal, complemented
7#20265377C212yesmaximal, complemented
8#20265204C211yesmaximal, characteristic, complemented
9#541×C101C211yesmaximal, characteristic, complemented
10C2×#13108061C2212yes—
11C2×#13108061C2212yes—
12C2×#13108058C2211yescharacteristic
13C22×#11717579C2211yescharacteristic
14#15692252C2211yescharacteristic
15#15692247C2211yescharacteristic
16#15692240—22no—
17#15692229C2212yes—
18#15692228—22no—
19C4×D404C2212yes—
20#15692188C2212yes—
21#15692186C2212yes—
22C4⋊#11717579C2212yes—
23C4⋊#11717579C2212yes—
24#15692179C2212yes—
25#15692179C2212yes—
26#15692175C2212yes—
27#15692175C2212yes—
28C4⋊Q8×C101C2211yescharacteristic
29#126×C101—22no—
30#125×C101—22no—
31#125×C101C2211yescharacteristic
32C22⋊Q8×C101C2212yes—
33C4×Q8×C101C2212yes—
34#117×C101C2211yescharacteristic
35C2×D404—22no—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C4×D101—22no—
39#13108061—42no—
40#13108061—42no—
41#13108058—41no—
42C2×#11717579—41no—
43C2×#11717579C2312yes—
44C2×#11717579C2312yes—
45C4⋊Dic101C2312yes—
46#13108053—22no—
47#13108051—22no—
48#13108051—22no—
49C2×Q8×C101C2312yes—
50C22×C4×C101C2311yescharacteristic
51C4⋊C4×C101—22no—
52C4⋊C4×C101—22no—
53C4⋊C4×C101—22no—
54C4⋊C4×C101—22no—
55C4⋊C4×C101D412yes—
56C4⋊C4×C101C2312yes—
57C22⋊C4×C101—22no—
58C22⋊C4×C101D412yes—
59C42×C101—22no—
60C42×C101C2311yescharacteristic
61C42×C101C2311yescharacteristic
62C23×C101C2×D411yescharacteristic
63C22×D101—22no—
64Q8×C101—22no—
65Q8×C101—22no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101C2×D412yes—
71C2×C4×C101C2×D412yes—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2411yescharacteristic
74C22⋊D101—82no—
75C2×Dic101—22no—
76D404—22no—
77D404—22no—
78D404—82no—
79C4×D101—82no—
80#11717579—22no—
81#11717579—22no—
82C22×C101—21no—
83C22×C101—22no—
84C22×C101C22×D411yescharacteristic
85C2×D101—42no—
86C2×D101—42no—
87C2×D101—42no—
88C4×C101—21no—
89C4×C101—22no—
90C4×C101—22no—
91C4×C101—42no—
92C4×C101—42no—
93C4×C101H(5,𝔽2)11yescharacteristic
94C4×C1012−1+411yescharacteristic
95Dic101—42no—
96C2×C101—41no—
97C2×C101#53711yescharacteristic
98C2×C101#57712yes—
99D101—82no—
100#3260—1011nomaximal, Sylow
101C101#326011yescharacteristic, complemented, Sylow
102#541D10111yescharacteristic, complemented
103#540—1011no—
104#479—1012no—
105#478—1012no—
106#468—1012no—
107#465—1012no—
108#461—1012no—
109#460—1012no—
110#424—1011no—
111C2×SD16—1012no—
112C2×SD16—1012no—
113C2×M16—1011no—
114C22×C8—1011no—
115C4⋊Q8C2×D10111yescharacteristic
116#127—1011no—
117#126—22no—
118#125—22no—
119#125—1011no—
120#125C2×D10111yescharacteristic
121#123—2022no—
122C22⋊Q8C2×D10112yes—
123#121—1012no—
124#121—2022no—
125C4×Q8C2×D10112yes—
126C4×D4—1012no—
127#117C2×D10111yescharacteristic
128#107—1012no—
129#106—1012no—
130C4⋊C8—1012no—
131C4⋊C8—1012no—
132#103—1012no—
133#103—1012no—
134#102—1012no—
135#102—1012no—
136C2×Q8C22×D10112yes—
137C2×D4—1012no—
138C2×D4—2022no—
139C2×D4—2022no—
140C22×C4—2022no—
141C22×C4C22×D10111yescharacteristic
142SD16—4042no—
143SD16—4042no—
144M16—4041no—
145C2×C8—1012no—
146C2×C8—1012no—
147C2×C8—4041no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1012no—
153C4⋊C4—2022no—
154C4⋊C4C22⋊D10112yes—
155C4⋊C4C22×D10112yes—
156C22⋊C4—22no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C22⋊C4C22⋊D10112yes—
160C42—22no—
161C42C22×D10111yescharacteristic
162C42C22×D10111yescharacteristic
163C23—2022no—
164C23C2×C22⋊D10111yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—8082no—
168D4—8082no—
169D4—2022no—
170D4—2022no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—8082no—
176C2×C4—2022no—
177C2×C4C2×C22⋊D10112yes—
178C2×C4C2×C22⋊D10112yes—
179C2×C4C2×C22⋊D10111yescharacteristic
180C2×C4C23×D10111yescharacteristic
181C8—2022no—
182C8—2022no—
183C22—22no—
184C22—21no—
185C22—4042no—
186C22—4042no—
187C22—4042no—
188C22C22×C22⋊D10111yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—4042no—
195C4#1569230111yescharacteristic
196C4#1569230211yescharacteristic
197C2—41no—
198C2—8082no—
199C2#2026575511yescharacteristic
200C2#2026598512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#541×C101C2
socleC22×C101C22×D4
prime radical#20265782C2
hypercenter#541D101
hypocenterC101#3260
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20265484C2semidirect
3. #20265474C2semidirect
4. #20265425C2semidirect
5. #20265408C2semidirect
6. #20265379C2semidirect
7. #20265377C2semidirect
8. #20265204C2semidirect
9. #541×C101C2semidirect
101. C101#3260semidirect
102. #541D101semidirect

Maximal subgroups

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

subgroupindex
#202657822
#202654842
#202654742
#202654252
#202654082
#202653792
#202653772
#202652042
#541×C1012
#3260101