Finite Group Databasebeta

← #27179394

Subgroups — #27179394

Order 12928.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265118C212yesmaximal, complemented
2#20265110C212yesmaximal, complemented
3#20265107C212yesmaximal, complemented
4#20265100C212yesmaximal
5#20265097C212yesmaximal, complemented
6#20265089C211yesmaximal, characteristic
7#20265038C211yesmaximal, characteristic
8#20265035C212yesmaximal
9#399×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11C23×Dic101C2211yescharacteristic
12C2×C4⋊Dic101C2212yes—
13C2×#13108053C2212yes—
14C2×#13108053C2212yes—
15C2×#13108053C2212yes—
16C2×#13108053C2212yes—
17C2×C22⋊Dic101C2212yes—
18C2×C22⋊Dic101C2211yescharacteristic
19#15692162C2212yes—
20#15692162C2212yes—
21#15692162C2212yes—
22#15692162C2212yes—
23#15692162C2211yescharacteristic
24#15692162C2212yes—
25#15692162C2212yes—
26C2×C4⋊C4×C101C2212yes—
27C2×C4⋊C4×C101C2211yescharacteristic
28C2×C22⋊C4×C101C2212yes—
29C2×C22⋊C4×C101C2211yescharacteristic
30#95×C101C2211yescharacteristic
31C2×Dic202—42no—
32C2×Dic202—41no—
33C22×Dic101—41no—
34C22×Dic101—41no—
35C22×Dic101Q814yescomplemented
36C22×Dic101C2312yes—
37C22×Dic101C2312yes—
38C22×Dic101C2312yes—
39C22×Dic101C2312yes—
40C4⋊Dic101—42no—
41#13108053—42no—
42#13108053—42no—
43#13108053—42no—
44#13108053—42no—
45C22⋊Dic101—41no—
46C22⋊Dic101—42no—
47C24×C101C2311yescharacteristic
48C22×C4×C101C2311yescharacteristic
49C22×C4×C101C2311yescharacteristic
50C22×C4×C101C2312yes—
51C22×C4×C101C2312yes—
52C4⋊C4×C101—41no—
53C4⋊C4×C101—42no—
54C22⋊C4×C101—42no—
55C22⋊C4×C101—41no—
56C23×C101—22no—
57C23×C101—21no—
58C23×C101—21no—
59C23×C101—22no—
60C23×C101C2×Q812yes—
61C23×C101C2411yescharacteristic
62C2×C4×C101—22no—
63C2×C4×C101—21no—
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—22no—
67C2×C4×C101—22no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×Dic101—22no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—22no—
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—22no—
83C2×Dic101—44no—
84C2×Dic101C2×Q814yes—
85Dic202—44no—
86C22×C101—22no—
87C22×C101—24no—
88C22×C101—41no—
89C22×C101—42no—
90C22×C101—41no—
91C22×C101C22×Q811yescharacteristic
92C22×C101H(5,𝔽2)11yescharacteristic
93C22×C101H(5,𝔽2)12yes—
94C22×C101H(5,𝔽2)11yescharacteristic
95C22×C1012−1+412yes—
96C4×C101—41no—
97C4×C101—42no—
98C4×C101—42no—
99C4×C101—41no—
100Dic101—24no—
101Dic101—42no—
102Dic101—42no—
103Dic101—42no—
104C2×C101—42no—
105C2×C101#54311yescharacteristic
106C2×C101#54412yes—
107C2×C101#56011yescharacteristic
108C2×C101#56212yes—
109C2×C101#56311yescharacteristic
110#2841—1011nomaximal, Sylow
111C101#284111yescharacteristic, complemented, Sylow
112#400—1012no—
113#399D10111yescharacteristic
114#398—1012no—
115#398—1012no—
116#396—1012no—
117#395—1012no—
118#393—1011no—
119#381—1011no—
120#380—1012no—
121C22×Q8—1011no—
122C23×C4—1011no—
123C2×C4⋊C4—1012no—
124C2×C4⋊C4—1012no—
125C2×C4⋊C4—1012no—
126C2×C4⋊C4—1012no—
127C2×C4⋊C4—1012no—
128C2×C4⋊C4C2×D10112yes—
129C2×C4⋊C4C2×D10111yescharacteristic
130C2×C22⋊C4—1012no—
131C2×C22⋊C4—1011no—
132C2×C22⋊C4C2×D10112yes—
133C2×C22⋊C4C2×D10111yescharacteristic
134#95—1012no—
135#95—1012no—
136#95—1012no—
137#95—1012no—
138#95—1011no—
139#95—1012no—
140#95—1012no—
141#95C2×D10111yescharacteristic
142C24C22×D10111yescharacteristic
143C2×Q8—4041no—
144C2×Q8—4042no—
145C22×C4—1014no—
146C22×C4—1012no—
147C22×C4—1012no—
148C22×C4—1012no—
149C22×C4—1012no—
150C22×C4—4041no—
151C22×C4—4041no—
152C22×C4C22×D10111yescharacteristic
153C22×C4C22×D10111yescharacteristic
154C22×C4C22×D10112yes—
155C22×C4C22×D10112yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—4042no—
159C4⋊C4—4042no—
160C4⋊C4—4042no—
161C4⋊C4—4042no—
162C4⋊C4—4042no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—4042no—
166C22⋊C4—4041no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D10112yes—
172C23C23×D10111yescharacteristic
173Q8—4044no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1014no—
185C2×C4—4044no—
186C2×C4—2022no—
187C2×C4—2022no—
188C2×C4—2022no—
189C2×C4—2022no—
190C2×C4—2022no—
191C2×C4—2022no—
192C2×C4—2022no—
193C2×C4—2022no—
194C2×C4—2022no—
195C2×C4—2022no—
196C2×C4—2022no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D10111yescharacteristic
203C22#1569230011yescharacteristic
204C22#1569230012yes—
205C22#1569230111yescharacteristic
206C22#1569230212yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—4042no—
212C4—4042no—
213C4—4042no—
214C4—2024no—
215C2—42no—
216C2#2026580211yescharacteristic
217C2#2026580612yes—
218C2#2026591411yescharacteristic
219C2#2026592412yes—
220C2#2026593011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#399×C101C2
prime radicalC24×C101C23
hypercenter#399D101
hypocenterC101#2841
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #20265118C2semidirect
2. #20265110C2semidirect
3. #20265107C2semidirect
5. #20265097C2semidirect
35. C22×Dic101Q8semidirect
111. C101#2841semidirect

Maximal subgroups

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

subgroupindex
#202651182
#202651102
#202651072
#202651002
#202650972
#202650892
#202650382
#202650352
#399×C1012
#2841101