Finite Group Databasebeta

← #27142574

Subgroups — #27142574

Order 12416.

396 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#15690806C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic, complemented
3C2×#15690727C211yesmaximal, characteristic, complemented
4C2×#15690726C211yesmaximal, characteristic, complemented
5C2×#15690726C211yesmaximal, characteristic, complemented
6M16×Dic97C211yesmaximal, characteristic, complemented
7C2×#100×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×C22⋊D97C414yes—
10C2×#13095495C2211yescharacteristic
11C2×#13095495C2211yescharacteristic
12#15690806—21no—
13#15690806—21no—
14#15690806—22no—
15C2×C4×Dic97C2211yescharacteristic
16#15690739—22no—
17#15690727—22no—
18#15690726—22no—
19#15690726—22no—
20#15690720—21no—
21C8×Dic97—21no—
22C22×D4×C97C2211yescharacteristic
23C2×M16×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25#100×C97—22no—
26C23×D97C2×C412yes—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—21no—
33C2×C22⋊D97—24no—
34C2×C22⋊D97—24no—
35C2×C22⋊D97—48no—
36C22×Dic97C2×C412yes—
37#13095495—21no—
38#13095495—21no—
39#13095495—21no—
40#13095495D412yescomplemented
41C2×#11715895—21no—
42C2×#11715895—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D412yes—
46C24×C97C2×C412yes—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97—21no—
56M16×C97—21no—
57M16×C97D412yescomplemented
58C2×C8×C97—21no—
59C2×C8×C97—21no—
60C23×C97—22no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—44no—
64C23×C97C22×C411yescharacteristic
65C22×D97—22no—
66C22×D97—22no—
67C22×D97—22no—
68C22×D97—44no—
69D4×C97—41no—
70D4×C97—41no—
71D4×C97—41no—
72D4×C97—41no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97#4111yescharacteristic
78C2×C4×C97#4111yescharacteristic
79C22⋊D97—48no—
80C22⋊D97—48no—
81C2×Dic97—24no—
82C2×Dic97C22⋊C412yes—
83C2×Dic97C22⋊C412yes—
84D388—41no—
85D388—41no—
86D388—41no—
87D388—41no—
88C8×C97—22no—
89C8×C97—41no—
90#11715895—22no—
91#11715895—41no—
92C22×C97—22no—
93C22×C97—42no—
94C22×C97—44no—
95C22×C97—82no—
96C22×C97—82no—
97C22×C97C2×C22⋊C411yescharacteristic
98C22×C97C4×D411yescharacteristic
99C22×C97C4×D411yescharacteristic
100C2×D97—42no—
101C2×D97—44no—
102C2×D97—82no—
103C2×D97—82no—
104C4×C97#12111yescharacteristic
105C4×C97#12111yescharacteristic
106C4×C97#12411yescharacteristic
107C4×C97#12711yescharacteristic
108Dic97—22no—
109Dic97#10014yescomplemented
110C2×C97—21no—
111C2×C97—21no—
112C2×C97—82no—
113C2×C97#39011yescharacteristic
114C2×C97C2×#10012yes—
115D97—82no—
116#1965—971nomaximal, Sylow
117C97#196511yescharacteristic, complemented, Sylow
118C2×#127—971no—
119#423—971no—
120C2×#100—971no—
121C2×#100—971no—
122C2×#100—971no—
123C2×#100D9711yescharacteristic, complemented
124C4×M16—971no—
125C22×D4—974no—
126C22×D4—971no—
127C22×D4C2×D9711yescharacteristic
128C2×M16—971no—
129C2×M16—971no—
130C2×M16C2×D9711yescharacteristic
131C2×M16C2×D9711yescharacteristic
132#127—1941no—
133#127—1941no—
134#127—1942no—
135C2×C42—971no—
136C4⋊C8—1942no—
137#100—22no—
138#100—1942no—
139#100—1942no—
140#100—1942no—
141#97—1941no—
142C4×C8—1941no—
143C24—972no—
144C24C4×D9712yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—3888no—
152C2×D4—1944no—
153C2×D4—1944no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4—1941no—
157C2×D4—1941no—
158C2×D4—1941no—
159C2×D4—1941no—
160C22×C4—972no—
161C22×C4C22×D9711yescharacteristic
162M16—21no—
163M16—21no—
164M16—21no—
165M16—972no—
166M16—1941no—
167M16—1941no—
168M16—1941no—
169M16C22⋊D9712yescomplemented
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—1941no—
173C2×C8—1941no—
174C42—971no—
175C42—971no—
176C42—972no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—44no—
181C23—3884no—
182C23—1942no—
183C23—1942no—
184C23—1942no—
185C23C2×C4×D9711yescharacteristic
186D4—41no—
187D4—41no—
188D4—41no—
189D4—41no—
190D4—3888no—
191D4—3888no—
192D4—3881no—
193D4—3881no—
194D4—3881no—
195D4—3881no—
196C2×C4—972no—
197C2×C4—972no—
198C2×C4—1944no—
199C2×C4D4×D9711yescharacteristic
200C2×C4D4×D9711yescharacteristic
201C2×C4D4×D9711yescharacteristic
202C2×C4C2×C22⋊D9711yescharacteristic
203C2×C4#1309551011yescharacteristic
204C2×C4#1309551111yescharacteristic
205C8—22no—
206C8—41no—
207C8—3881no—
208C8—1942no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22—82no—
213C22—82no—
214C22—7762no—
215C22—7762no—
216C22—3882no—
217C22—3884no—
218C22C22⋊C4×D9711yescharacteristic
219C22#1569077011yescharacteristic
220C22C4×C22⋊D9711yescharacteristic
221C4—974no—
222C4—1942no—
223C4#1569078211yescharacteristic
224C4#1569078311yescharacteristic
225C4#1569079611yescharacteristic
226C4#1569080611yescharacteristic
227C2—21no—
228C2—21no—
229C2—82no—
230C2—7762no—
231C2#2026063511yescharacteristic
232C2#100×D9712yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#100×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690806C2
cosocleC22×C4×C97C23
hypercenterC2×#100D97
hypocenterC97#1965
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #20260758C2semidirect
3. C2×#15690727C2semidirect
4. C2×#15690726C2semidirect
5. C2×#15690726C2semidirect
6. M16×Dic97C2semidirect
7. C2×#100×C97C2semidirect
40. #13095495D4semidirect
57. M16×C97D4semidirect
109. Dic97#100semidirect
117. C97#1965semidirect
123. C2×#100D97semidirect
169. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908062
#202607582
C2×#156907272
C2×#156907262
M16×Dic972
C2×#100×C972
#196597