Finite Group Databasebeta

← #27171345

Subgroups — #27171345

Order 12928.

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

normal subgroups only

This group has 272 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
1#20265654C211yesmaximal, characteristic
2#20265217C211yesmaximal, characteristic
3C2×#15692173C211yesmaximal, characteristic
4#20265082C211yesmaximal, characteristic
5#20264938C211yesmaximal, characteristic
6#20264936C211yesmaximal, characteristic
7#428×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10C4×Dic202—21no—
11C4×Dic202—21no—
12#15692220—21no—
13#15692220—21no—
14#15692214C2211yescharacteristic
15#15692211—22no—
16C2×C4×Dic101C2211yescharacteristic
17#15692188—21no—
18#15692186—21no—
19#15692173C414yescomplemented
20#15692162—22no—
21C2×M16×C101C2211yescharacteristic
22#117×C101C2211yescharacteristic
23C2×C4⋊C4×C101C2211yescharacteristic
24#107×C101—21no—
25#106×C101—21no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202C2×C412yes—
31C2×Dic202Q812yes—
32C22×Dic101—21no—
33C22×Dic101D412yes—
34#13108058—21no—
35#13108058C2×C412yes—
36C2×#11717579—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D411yescharacteristic
39#13108053—22no—
40#13108053—22no—
41C22⋊Dic101—21no—
42C4×Dic101—22no—
43C4×Dic101—22no—
44C4×Dic101—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C22×C4×C101—21no—
48C22×C4×C101C2311yescharacteristic
49M16×C101—21no—
50M16×C101C2×C412yes—
51C2×C8×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101D411yescharacteristic
55C4⋊C4×C101D411yescharacteristic
56C22⋊C4×C101—21no—
57C42×C101—21no—
58C23×C101C2×D411yescharacteristic
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—42no—
62C2×C4×C101—41no—
63C2×C4×C101C22×C411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×Q811yescharacteristic
67C2×C4×C101#4111yescharacteristic
68C2×C4×C101#4111yescharacteristic
69C2×Dic101—22no—
70C2×Dic101—22no—
71C2×Dic101—22no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76C2×Dic101C4⋊C414yes—
77Dic202—22no—
78Dic202—22no—
79Dic202—22no—
80Dic202—22no—
81C8×C101—22no—
82#11717579—22no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101C2×C4⋊C411yescharacteristic
86C22×C101#12011yescharacteristic
87C22×C101#12311yescharacteristic
88C4×C101—41no—
89C4×C101—41no—
90C4×C101—81no—
91C4×C101C4×D411yescharacteristic
92C4×C101C4×D411yescharacteristic
93C4×C101C22⋊Q811yescharacteristic
94C4×C101C22⋊Q811yescharacteristic
95Dic101—24no—
96Dic101—42no—
97Dic101—41no—
98Dic101—41no—
99C2×C101—21no—
100C2×C101—21no—
101C2×C101#38511yescharacteristic
102C2×C101#45411yescharacteristic
103C2×C101#45611yescharacteristic
104#1857—1011nomaximal, Sylow
105C101#185711yescharacteristic, complemented, Sylow
106#519—1011no—
107#428—1011no—
108#428D10111yescharacteristic
109C2×#101—1011no—
110#391—1011no—
111#339—1011no—
112#339—1011no—
113C22×Q8—1011no—
114C2×M16—1011no—
115C2×M16C2×D10111yescharacteristic
116C4×Q8—2021no—
117C4×Q8—2021no—
118C4×Q8—2021no—
119C4×Q8—2021no—
120#117—1011no—
121#117—2022no—
122#117C2×D10111yescharacteristic
123C2×C4⋊C4C2×D10111yescharacteristic
124C2×C42—1011no—
125#107—21no—
126#107—2021no—
127#106—21no—
128#106—2021no—
129#101—1014no—
130#95—2022no—
131C2×Q8—1012no—
132C2×Q8—1012no—
133C2×Q8—2021no—
134C2×Q8—2021no—
135C2×Q8—2021no—
136C2×Q8—2021no—
137C22×C4—21no—
138C22×C4—1012no—
139C22×C4—2021no—
140C22×C4C22×D10111yescharacteristic
141M16—21no—
142M16—1012no—
143M16—2021no—
144M16C4×D10112yes—
145C2×C8—21no—
146C2×C8—2021no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—1011no—
150C4⋊C4—1011no—
151C4⋊C4—2022no—
152C4⋊C4—2022no—
153C4⋊C4C22⋊D10111yescharacteristic
154C4⋊C4C22⋊D10111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2021no—
157C42—21no—
158C42—2022no—
159C42—2022no—
160C42—2021no—
161C42—2021no—
162C42—2021no—
163C23D4×D10111yescharacteristic
164Q8—2022no—
165Q8—2022no—
166Q8—2022no—
167Q8—2022no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—1014no—
173C2×C4—4041no—
174C2×C4—4041no—
175C2×C4—2022no—
176C2×C4—2021no—
177C2×C4—2021no—
178C2×C4—2022no—
179C2×C4—2022no—
180C2×C4C2×C4×D10111yescharacteristic
181C2×C4D4×D10111yescharacteristic
182C2×C4C2×C22⋊D10111yescharacteristic
183C2×C4Q8×D10111yescharacteristic
184C2×C4#1310807411yescharacteristic
185C2×C4#1310807511yescharacteristic
186C8—22no—
187C8—2022no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D10111yescharacteristic
191C22#1569222311yescharacteristic
192C22#1569224011yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—4042no—
197C4—4041no—
198C4—4041no—
199C4—2024no—
200C4C4×C22⋊D10111yescharacteristic
201C4#1569221711yescharacteristic
202C4#1569223111yescharacteristic
203C4#1569223311yescharacteristic
204C2—21no—
205C2—21no—
206C2#2026504811yescharacteristic
207C2#2026534811yescharacteristic
208C2#2026535811yescharacteristic
subgroupG/H
centerC22#15692223
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#428×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
cosocleC22×C4×C101C23
hypercenter#428D101
hypocenterC101#1857
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #15692173C4semidirect
105. C101#1857semidirect

Maximal subgroups

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

subgroupindex
#202656542
#202652172
C2×#156921732
#202650822
#202649382
#202649362
#428×C1012
#1857101