Finite Group Databasebeta

← D101⋊#354 #27187401

Subgroups — D101⋊#354 #27187401

Order 12928.

284 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D101⋊#99C212yesmaximal
2#20266084C212yesmaximal, complemented
3#20266083C212yesmaximal, complemented
4#124×D101C211yesmaximal, characteristic
5C2×C22⋊#87938—22no—
6#15692314C2214yes—
7#15692313—22no—
8C2×Q8×D101C411yescharacteristic, complemented
9C2×D4×D101C2211yescharacteristic
10#15692244C2211yescharacteristic
11#15692243C2211yescharacteristic
12#15692242C411yescharacteristic, complemented
13#15692241—22no—
14C22⋊C4×D101—22no—
15C42×D101C411yescharacteristic, complemented
16#124×C101C411yescharacteristic, complemented
17C22×#87938—42no—
18C22⋊#87938—24no—
19C22⋊#87938—42no—
20C23×D101D412yes—
21C2×Dic202—21no—
22C2×Dic202C2×C411yescharacteristic, complemented
23Q8×D101—41no—
24Q8×D101—41no—
25C2×D404C2311yescharacteristic
26C2×C22⋊D101D412yes—
27D4×D101—41no—
28D4×D101—41no—
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101C2×C411yescharacteristic
32C22⋊Dic101—22no—
33#13108051—22no—
34#13108051—22no—
35C4×Dic101—21no—
36C4×Dic101C2×C411yescharacteristic, complemented
37C2×Q8×C101C2×C411yescharacteristic, complemented
38C2×D4×C101C2×C411yescharacteristic
39C22⋊C4×C101—22no—
40C42×C101C2×C411yescharacteristic, complemented
41C23×C101C22⋊C412yes—
42C22×D101—22no—
43C22×D101—22no—
44C22×D101—44no—
45C22×D101C22⋊C411yescharacteristic
46C22×D101C2×D412yes—
47C2×#87938—44no—
48C2×#87938—84no—
49Q8×C101—41no—
50D4×C101—41no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101C22×C411yescharacteristic
54C22⋊D101—42no—
55C2×Dic101—21no—
56C2×Dic101—21no—
57C2×Dic101C22⋊C411yescharacteristic
58D404—41no—
59C4×D101—21no—
60C4×D101—21no—
61C4×D101—42no—
62C4×D101—41no—
63C4×D101—41no—
64Dic202—42no—
65Dic202—41no—
66C22×C101—22no—
67C22×C101—42no—
68C22×C101C2×C22⋊C411yescharacteristic
69C2×D101—22no—
70C2×D101—22no—
71C2×D101—44no—
72C2×D101—42no—
73C2×D101—82no—
74C2×D101#9911yescharacteristic
75C2×D101#9911yescharacteristic
76#87938—84no—
77C4×C101—21no—
78C4×C101—41no—
79C4×C101—41no—
80Dic101—21no—
81Dic101—41no—
82Dic101—41no—
83C2×C101—21no—
84C2×C101—42no—
85C2×C101C2×#9911yescharacteristic
86D101—21no—
87D101—42no—
88D101#35412yescomplemented
89C2×#354—1011nomaximal, Sylow
90C101C2×#35411yescharacteristic, complemented, Sylow
91C2×#124—1011no—
92C2×#99—1012no—
93#354—1012no—
94#354—1012no—
95C22×Q8—1011no—
96C22×D4—1011no—
97#124—1011no—
98#124—1011no—
99#124—1011no—
100#124—2022no—
101#124#8793811yescharacteristic, complemented
102C2×C22⋊C4—2022no—
103C2×C22⋊C4—2022no—
104C2×C42—1011no—
105#99—1014no—
106#99—2022no—
107C24—1012no—
108C2×Q8—1011no—
109C2×Q8—4041no—
110C2×Q8—4041no—
111C2×Q8—2021no—
112C2×Q8C2×#8793811yescharacteristic, complemented
113C2×D4—1012no—
114C2×D4—1011no—
115C2×D4—4041no—
116C2×D4—4041no—
117C2×D4C2×#8793811yescharacteristic
118C22×C4—1011no—
119C22×C4—4042no—
120C22×C4—2021no—
121C22×C4—2021no—
122C22⋊C4—22no—
123C22⋊C4—4042no—
124C22⋊C4—2022no—
125C22⋊C4—2022no—
126C22⋊C4—2024no—
127C22⋊C4—2022no—
128C42—1011no—
129C42—2021no—
130C42C2×#8793811yescharacteristic, complemented
131C23—1011no—
132C23—1012no—
133C23—4044no—
134C23—2022no—
135C23—2022no—
136C23C22⋊#8793812yes—
137Q8—41no—
138Q8—4041no—
139Q8—4042no—
140D4—41no—
141D4—4041no—
142D4—4042no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—8084no—
146C2×C4—1011no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—4042no—
150C2×C4—4044no—
151C2×C4—2021no—
152C2×C4—2021no—
153C2×C4—2021no—
154C2×C4—2021no—
155C2×C4C22×#8793811yescharacteristic
156C22—22no—
157C22—42no—
158C22—8082no—
159C22—1011no—
160C22—1011no—
161C22—4042no—
162C22—4044no—
163C22—2022no—
164C22—2022no—
165C22C2×C22⋊#8793811yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—8084no—
170C4—4041no—
171C4—4041no—
172C4—2021no—
173C2—21no—
174C2—42no—
175C2—1012no—
176C2—4042no—
177C2—2021no—
178C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC2×C101C2×#99
prime radicalC2×D4×D101C22
cosocleC2×D404C23
hypercenter#124#87938
hypocenterC101C2×#354
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #20266084C2semidirect
3. #20266083C2semidirect
8. C2×Q8×D101C4semidirect
12. #15692242C4semidirect
15. C42×D101C4semidirect
16. #124×C101C4semidirect
22. C2×Dic202C2×C4semidirect
36. C4×Dic101C2×C4semidirect
37. C2×Q8×C101C2×C4semidirect
40. C42×C101C2×C4semidirect
88. D101#354semidirect
90. C101C2×#354semidirect
101. #124#87938semidirect
112. C2×Q8C2×#87938semidirect
130. C42C2×#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊#992
#202660842
#202660832
#124×D1012
C2×#354101