Finite Group Databasebeta

← #353×D101 #27172788

Subgroups — #353×D101 #27172788

Order 12928.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D101C211yesmaximal, characteristic
2#99×D101C212yesmaximal, complemented
3#20264970C211yesmaximal, characteristic, complemented
4#20264969C212yesmaximal, complemented
5#353×C101C211yesmaximal, characteristic, complemented
6C2×D4×D101—22no—
7C2×D4×D101C411yescharacteristic, complemented
8C2×D4×D101C2211yescharacteristic
9#15692252C411yescharacteristic, complemented
10#15692251—22no—
11#15692251C411yescharacteristic, complemented
12#15692251C2211yescharacteristic
13C22⋊C4×D101—22no—
14C42×D101C411yescharacteristic, complemented
15#15692170C2212yescomplemented
16#15692169—22no—
17#127×C101C2211yescharacteristic
18#99×C101C2212yescomplemented
19C23×D101—21no—
20C23×D101D412yes—
21C2×D404—22no—
22C2×D404C2×C411yescharacteristic
23C2×D404C2×C411yescharacteristic, complemented
24C2×C22⋊D101—22no—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—21no—
27C2×C22⋊D101D412yes—
28D4×D101—41no—
29D4×D101—41no—
30D4×D101—41no—
31D4×D101—41no—
32D4×D101—48no—
33C2×C4×D101—21no—
34C2×C4×D101—42no—
35C2×C4×D101C2×C411yescharacteristic
36C22⋊Dic101—22no—
37#13108051—42no—
38C4×Dic101—21no—
39C4×Dic101C2×C411yescharacteristic, complemented
40C2×D4×C101—22no—
41C2×D4×C101C2×C411yescharacteristic, complemented
42C2×D4×C101C2311yescharacteristic
43C22⋊C4×C101—22no—
44C42×C101C2×C411yescharacteristic, complemented
45C23×C101—21no—
46C23×C101C2×D412yes—
47C22×D101—22no—
48C22×D101—22no—
49C22×D101—21no—
50C22×D101—44no—
51C22×D101—44no—
52C22×D101—41no—
53C22×D101—41no—
54C22×D101C22⋊C411yescharacteristic
55C22×D101C22⋊C412yes—
56D4×C101—41no—
57D4×C101—41no—
58D4×C101—44no—
59C2×C4×C101—21no—
60C2×C4×C101—42no—
61C2×C4×C101C22×C411yescharacteristic
62C22⋊D101—42no—
63C22⋊D101—44no—
64C22⋊D101—44no—
65C22⋊D101—42no—
66C2×Dic101—21no—
67C2×Dic101—42no—
68C2×Dic101C22⋊C411yescharacteristic
69D404—41no—
70D404—41no—
71D404—44no—
72C4×D101—24no—
73C4×D101—21no—
74C4×D101—21no—
75C4×D101—84no—
76C22×C101—22no—
77C22×C101—42no—
78C22×C101—42no—
79C22×C101—41no—
80C22×C101C2×C22⋊C411yescharacteristic
81C2×D101—22no—
82C2×D101—22no—
83C2×D101—44no—
84C2×D101—42no—
85C2×D101—42no—
86C2×D101—41no—
87C2×D101—82no—
88C2×D101—82no—
89C2×D101—82no—
90C2×D101#9911yescharacteristic
91C2×D101#9911yescharacteristic
92C4×C101—22no—
93C4×C101—21no—
94C4×C101—82no—
95Dic101—22no—
96Dic101—21no—
97Dic101—82no—
98C2×C101—21no—
99C2×C101—42no—
100C2×C101—81no—
101C2×C101C2×#9911yescharacteristic
102D101—21no—
103D101—42no—
104D101—81no—
105D101#35312yesdirect product factor
106C2×#353—1011nomaximal, Sylow
107C101C2×#35311yescharacteristic, complemented, Sylow
108C2×#127—1011no—
109C2×#99—1012no—
110#353—1011no—
111#353—1012no—
112#353D10111yescharacteristic, direct product factor
113C22×D4—1011no—
114C22×D4—1011no—
115C22×D4—2022no—
116#127—1011no—
117#127—1011no—
118#127—1011no—
119#127—2022no—
120#127C2×D10111yescharacteristic
121C2×C22⋊C4—2022no—
122C2×C42—1011no—
123#99—1012no—
124#99—2022no—
125#99C2×D10112yescomplemented
126C24—1012no—
127C24—2021no—
128C2×D4—22no—
129C2×D4—1012no—
130C2×D4—1011no—
131C2×D4—1011no—
132C2×D4—4041no—
133C2×D4—4041no—
134C2×D4—4048no—
135C2×D4—4041no—
136C2×D4—4041no—
137C2×D4—2022no—
138C2×D4—2021no—
139C2×D4—2022no—
140C2×D4—2022no—
141C2×D4C4×D10111yescharacteristic, complemented
142C2×D4C22×D10111yescharacteristic
143C22×C4—1011no—
144C22×C4—4042no—
145C22×C4—2021no—
146C22⋊C4—22no—
147C22⋊C4—4042no—
148C22⋊C4—2022no—
149C42—1011no—
150C42—2021no—
151C42C4×D10111yescharacteristic, complemented
152C23—21no—
153C23—1011no—
154C23—1012no—
155C23—4041no—
156C23—4041no—
157C23—4044no—
158C23—4044no—
159C23—2022no—
160C23—2022no—
161C23—2021no—
162C23D4×D10112yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—4041no—
167D4—4044no—
168D4—4041no—
169D4—4042no—
170D4—4044no—
171D4—4044no—
172D4—4042no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—8084no—
176C2×C4—1011no—
177C2×C4—4042no—
178C2×C4—2021no—
179C2×C4—2021no—
180C2×C4—2024no—
181C2×C4—2021no—
182C2×C4C2×C4×D10111yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—8082no—
188C22—8082no—
189C22—8082no—
190C22—1011no—
191C22—1011no—
192C22—4042no—
193C22—4042no—
194C22—4044no—
195C22—4041no—
196C22—2022no—
197C22—2022no—
198C22C22⋊C4×D10111yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—8082no—
203C4—2022no—
204C4—2021no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—8081no—
209C2—1012no—
210C2—4042no—
211C2—2021no—
212C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#353×C101C2
socleC2×C101C2×#99
prime radical#127×D101C2
cosocleC2×D4×C101C23
hypercenter#353D101
hypocenterC101C2×#353
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D101C2semidirect
3. #20264970C2semidirect
4. #20264969C2semidirect
5. #353×C101C2semidirect
7. C2×D4×D101C4semidirect
9. #15692252C4semidirect
11. #15692251C4semidirect
14. C42×D101C4semidirect
15. #15692170C22semidirect
18. #99×C101C22semidirect
23. C2×D404C2×C4semidirect
39. C4×Dic101C2×C4semidirect
41. C2×D4×C101C2×C4semidirect
44. C42×C101C2×C4semidirect
105. D101#353direct (N × H)
107. C101C2×#353semidirect
112. #353D101direct (N × H)
125. #99C2×D101semidirect
141. C2×D4C4×D101semidirect
151. C42C4×D101semidirect

Maximal subgroups

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

subgroupindex
#127×D1012
#99×D1012
#202649702
#202649692
#353×C1012
C2×#353101