Finite Group Databasebeta

← #27140349

Subgroups — #27140349

Order 12416.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#20261291C211yesmaximal, characteristic
2#20260790C211yesmaximal, characteristic
3#97×D97C211yesmaximal, characteristic
4#20260477C211yesmaximal, characteristic, complemented
5#20260476C211yesmaximal, characteristic, complemented
6#20260476C211yesmaximal, characteristic, complemented
7#330×C97C211yesmaximal, characteristic, complemented
8C2×C8×D97—21no—
9#15690809C411yescharacteristic
10#15690809C411yescharacteristic
11#15690802C2211yescharacteristic
12#15690788—21no—
13#15690788—21no—
14#15690775—21no—
15#15690775—21no—
16#15690768C411yescharacteristic
17#15690768C411yescharacteristic
18C42×D97C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690721C2211yescharacteristic
22#15690721C2211yescharacteristic
23#15690720—21no—
24#125×C97C2211yescharacteristic
25#97×C97C2211yescharacteristic
26#97×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×#11715895—21no—
33C2×#11715895—21no—
34C8×D97—24no—
35C4⋊Dic97C2×C411yescharacteristic
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97—21no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97C2×C411yescharacteristic
53C4⋊C4×C97C2×C411yescharacteristic
54C42×C97C2311yescharacteristic
55C22×D97C22⋊C411yescharacteristic
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22×C411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×Dic97—21no—
62C2×Dic97—21no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C22⋊C411yescharacteristic
66C4×D97—22no—
67C4×D97—22no—
68C4×D97—41no—
69Dic194—42no—
70Dic194—42no—
71C8×C97—22no—
72C8×C97—41no—
73#11715895—22no—
74#11715895—41no—
75C22×C97C2×C22⋊C411yescharacteristic
76C2×D97—21no—
77C2×D97—21no—
78C2×D97#10112yes—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97—41no—
84Dic97—21no—
85Dic97—21no—
86Dic97—41no—
87Dic97—41no—
88Dic97#10412yes—
89C2×C97C2×#10111yescharacteristic
90C2×C97C2×#10411yescharacteristic
91C2×C97#42111yescharacteristic
92D97—22no—
93#1531—971nomaximal, Sylow
94C97#153111yescharacteristic, complemented, Sylow
95#533—971no—
96#432—971no—
97C2×#97—971no—
98#330—971no—
99#330—971no—
100#330—971no—
101#330D9711yescharacteristic, complemented
102C22×C8—1941no—
103C4⋊Q8—971no—
104C4⋊Q8—971no—
105#125—971no—
106#125C2×D9711yescharacteristic
107C22⋊Q8—1941no—
108C22⋊Q8—1941no—
109C4×Q8—1941no—
110C4×Q8—1941no—
111#117—971no—
112#117—971no—
113C2×C42—971no—
114C4⋊C8—1941no—
115C22⋊C8—1941no—
116#97—971no—
117#97—971no—
118#97—1941no—
119#97C2×D9711yescharacteristic
120#97C2×D9711yescharacteristic
121C2×Q8—1941no—
122C2×Q8—1941no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—971no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—1944no—
129C2×C8—1941no—
130C2×C8—1941no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—971no—
134C4⋊C4—971no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C4⋊C4C4×D9711yescharacteristic
140C4⋊C4C4×D9711yescharacteristic
141C22⋊C4—1941no—
142C22⋊C4—1941no—
143C42—971no—
144C42—971no—
145C42—971no—
146C42—1941no—
147C42—1941no—
148C42C22×D9711yescharacteristic
149C23—971no—
150Q8—3882no—
151Q8—3882no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—3881no—
158C2×C4—1942no—
159C2×C4—1942no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4C2×C4×D9711yescharacteristic
163C2×C4D4×D9711yescharacteristic
164C2×C4D4×D9711yescharacteristic
165C8—22no—
166C8—41no—
167C8—3881no—
168C8—1942no—
169C22—972no—
170C22—1941no—
171C22—1941no—
172C22C22⋊C4×D9711yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—972no—
179C4—3881no—
180C4—3881no—
181C4—1941no—
182C4—1941no—
183C2—1942no—
184C2#101×D9711yescharacteristic
185C2#104×D9711yescharacteristic
186C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#330×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#330D97
hypocenterC97#1531
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. #20260477C2semidirect
5. #20260476C2semidirect
6. #20260476C2semidirect
7. #330×C97C2semidirect
94. C97#1531semidirect
101. #330D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
#202607902
#97×D972
#202604772
#202604762
#330×C972
#153197