Finite Group Databasebeta

← #27187294

Subgroups — #27187294

Order 12928.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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⋊#108C211yesmaximal, characteristic
2#20266170C212yesmaximal
3D101⋊#101C212yesmaximal
4#20266140C211yesmaximal, characteristic
5C2×Q16×D101C211yesmaximal, characteristic
6D101⋊M16C2212yes—
7D101⋊M16C2211yescharacteristic
8#15692334C2211yescharacteristic
9#15692326—21no—
10#15692325—21no—
11#15692318—24no—
12#15692316—24no—
13#15692309—21no—
14C8×#87938—21no—
15C2×Q8×D101C2212yes—
16C2×Dic404C411yescharacteristic, complemented
17C2×#13108064C412yescomplemented
18Q16×D101—21no—
19Q16×D101—21no—
20Q16×D101—21no—
21Q16×D101—21no—
22C2×C8×D101C2211yescharacteristic
23C2×Q16×C101C411yescharacteristic, complemented
24#13108084—21no—
25#13108084—22no—
26#13108083—22no—
27#13108083D412yes—
28D101⋊C8—21no—
29D101⋊C8—22no—
30C4⋊#87938D412yes—
31C22⋊#87938—21no—
32C4×#87938—21no—
33C2×Dic202—22no—
34C2×Dic202C2×C412yes—
35Q8×D101—22no—
36Q8×D101—22no—
37Q8×D101—22no—
38Q8×D101—22no—
39C2×C4×D101—22no—
40C2×C4×D101C2311yescharacteristic
41Dic404—21no—
42Dic404—21no—
43#13108064—22no—
44#13108064—22no—
45C2×#11717579C2×C411yescharacteristic
46C8×D101D411yescharacteristic
47C8×D101D411yescharacteristic
48C8×D101D411yescharacteristic
49C8×D101D411yescharacteristic
50C2×Q8×C101C2×C412yes—
51Q16×C101—21no—
52Q16×C101—21no—
53C2×C8×C101C2×C411yescharacteristic
54C22×D101#4111yescharacteristic
55C2×#87938—22no—
56Q8×C101—22no—
57Q8×C101—22no—
58C2×C4×C101—22no—
59C2×C4×C101C22×C411yescharacteristic
60C2×Dic101—22no—
61C2×Dic101C2×D411yescharacteristic
62C4×D101—42no—
63C4×D101—42no—
64C4×D101C2×D411yescharacteristic
65C4×D101C2×D411yescharacteristic
66C4×D101C2×D411yescharacteristic
67C4×D101#4111yescharacteristic
68Dic202—22no—
69Dic202—22no—
70Dic202—42no—
71Dic202—42no—
72#11717581—22no—
73#11717581—42no—
74C8×C101C22⋊C411yescharacteristic
75C8×C101C22⋊C411yescharacteristic
76#11717579C22⋊C411yescharacteristic
77#11717579C22⋊C411yescharacteristic
78C22×C101C4×D411yescharacteristic
79C2×D101—22no—
80C2×D101#12111yescharacteristic
81C2×D101#12411yescharacteristic
82#87938—42no—
83C4×C101—42no—
84C4×C101C2×C22⋊C411yescharacteristic
85C4×C101C4×D411yescharacteristic
86Dic101—42no—
87Dic101#12111yescharacteristic
88Dic101#12711yescharacteristic
89C2×C101—21no—
90C2×C101#39011yescharacteristic
91D101—21no—
92D101#47312yes—
93#1979—1011nomaximal, Sylow
94C101#197911yescharacteristic, complemented, Sylow
95C22×Q16—1011no—
96C2×#108—1011no—
97#419—1012no—
98C2×#101—1012no—
99#405—1011no—
100C22×Q8—1012no—
101C2×Q16—1012no—
102C2×Q16—1011no—
103C2×Q16—2021no—
104C2×Q16—2021no—
105C2×Q16—2021no—
106C2×Q16—2021no—
107C2×Q16#8793811yescharacteristic, complemented
108C2×M16—1011no—
109C2×M16—1012no—
110C22×C8—1011no—
111#117—1011no—
112#108—2021no—
113#108—2021no—
114#103—2024no—
115#101—2024no—
116#97—2021no—
117C4×C8—2021no—
118C2×Q8—1012no—
119C2×Q8—2022no—
120C2×Q8—2022no—
121C2×Q8—2022no—
122C2×Q8—2022no—
123C2×Q8—2022no—
124C2×Q8C2×#8793812yes—
125C22×C4—1011no—
126C22×C4—2022no—
127Q16—21no—
128Q16—21no—
129Q16—2022no—
130Q16—2022no—
131Q16—2021no—
132Q16—2021no—
133M16—1012no—
134M16—2022no—
135M16—2022no—
136M16—2021no—
137C2×C8—1011no—
138C2×C8—1011no—
139C2×C8—1011no—
140C2×C8—1011no—
141C2×C8—1011no—
142C2×C8—2022no—
143C2×C8—2021no—
144C2×C8C2×#8793811yescharacteristic
145C4⋊C4—1012no—
146C22⋊C4—2021no—
147C42—2021no—
148C23—1011no—
149Q8—22no—
150Q8—22no—
151Q8—4042no—
152Q8—4042no—
153Q8—2022no—
154Q8—2022no—
155C2×C4—22no—
156C2×C4—1011no—
157C2×C4—1011no—
158C2×C4—1011no—
159C2×C4—1011no—
160C2×C4—1011no—
161C2×C4—4042no—
162C2×C4—4042no—
163C2×C4—2022no—
164C2×C4—2022no—
165C2×C4C22×#8793811yescharacteristic
166C8—1011no—
167C8—1011no—
168C8—4042no—
169C8—2022no—
170C8C22⋊#8793811yescharacteristic
171C8C22⋊#8793811yescharacteristic
172C22—1011no—
173C22—1011no—
174C22—2022no—
175C22D4×#8793811yescharacteristic
176C4—42no—
177C4—1011no—
178C4—1011no—
179C4—4042no—
180C4—4042no—
181C4C2×C22⋊#8793811yescharacteristic
182C4D4×#8793811yescharacteristic
183C2—21no—
184C2—1012no—
185C2—2021no—
186C2#2026612911yescharacteristic
subgroupG/H
centerC2#20266129
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC2×Q16×C101C4
socleC2×C101#390
prime radicalC22×D101#41
cosocleC2×C4×D101C23
hypercenterC2×Q16#87938
hypocenterC101#1979
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
16. C2×Dic404C4semidirect
17. C2×#13108064C4semidirect
23. C2×Q16×C101C4semidirect
94. C101#1979semidirect
107. C2×Q16#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊#1082
#202661702
D101⋊#1012
#202661402
C2×Q16×D1012
#1979101