Finite Group Databasebeta

← #34052736

Subgroups — #34052736

Order 19072.

580 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18011899C211yesmaximal, characteristic, complemented
2#23749541C214yesmaximal, complemented
3#23749537C212yesmaximal, complemented
4#23749536C212yesmaximal, complemented
5#23749534C212yesmaximal, complemented
6#23749533C212yesmaximal, complemented
7#23749491C211yesmaximal, characteristic, complemented
8C2×#124×C149C211yesmaximal, characteristic, complemented
9C22×Dic298C2211yescharacteristic
10C22×D596C2211yescharacteristic
11C22×C22⋊D149C2214yes—
12#18011899—24no—
13C2×C4⋊Dic149C2212yes—
14C2×C22⋊Dic149C2214yes—
15C2×#14558713C2212yes—
16C2×#14558713C2212yes—
17C2×#14558713C2214yes—
18#18011818C2212yes—
19#18011818C2212yes—
20#18011818C2214yes—
21C22×Q8×C149C2211yescharacteristic
22C22×D4×C149C2211yescharacteristic
23#124×C149—24no—
24C2×C22⋊C4×C149C2212yes—
25C2×C22⋊C4×C149C2212yes—
26C2×C42×C149C2211yescharacteristic
27C23×D149C2312yes—
28C2×Dic298—22no—
29C2×Dic298—42no—
30C2×D596—22no—
31C2×D596—42no—
32C2×C22⋊D149—44no—
33C2×C22⋊D149—44no—
34C2×C22⋊D149—44no—
35C22×Dic149C2314yes—
36C22×Dic149C2312yes—
37C4⋊Dic149—42no—
38C22⋊Dic149—44no—
39#14558713—24no—
40#14558713—24no—
41#14558713—44no—
42C24×C149C2312yes—
43C2×Q8×C149—22no—
44C2×Q8×C149—42no—
45C2×D4×C149—22no—
46C2×D4×C149—42no—
47C22×C4×C149C2312yes—
48C22×C4×C149C2311yescharacteristic
49C22×C4×C149C2311yescharacteristic
50C22×C4×C149C2311yescharacteristic
51C22⋊C4×C149—24no—
52C22⋊C4×C149—24no—
53C42×C149D414yescomplemented
54C22×D149—22no—
55C22×D149—22no—
56C22×D149—24no—
57C22×D149—24no—
58C22×D149—22no—
59Dic298—44no—
60D596—44no—
61C22⋊D149—416no—
62C2×Dic149—22no—
63C2×Dic149—24no—
64C2×Dic149—24no—
65C2×Dic149—24no—
66C2×Dic149—24no—
67C23×C149—22no—
68C23×C149—22no—
69C23×C149—24no—
70C23×C149—24no—
71C23×C149—22no—
72C23×C149C2411yescharacteristic
73Q8×C149—44no—
74D4×C149—44no—
75C2×C4×C149—22no—
76C2×C4×C149—22no—
77C2×C4×C149—21no—
78C2×C4×C149—21no—
79C2×C4×C149—22no—
80C2×C4×C149—24no—
81C2×C4×C149C2×D412yes—
82C2×C4×C149C2×D412yes—
83C2×C4×C149C2×D412yes—
84C22×C149—42no—
85C22×C149—42no—
86C22×C149—42no—
87C22×C149—44no—
88C22×C149—44no—
89C22×C149C22×D411yescharacteristic
90C22×C149H(5,𝔽2)12yes—
91C22×C149H(5,𝔽2)12yes—
92C22×C149H(5,𝔽2)11yescharacteristic
93C22×C1492−1+411yescharacteristic
94C2×D149—42no—
95C2×D149—42no—
96C2×D149—42no—
97C2×D149—44no—
98C2×D149—44no—
99C4×C149—22no—
100C4×C149—22no—
101C4×C149—22no—
102C4×C149—42no—
103Dic149—42no—
104Dic149—44no—
105C2×C149—82no—
106C2×C149#53511yescharacteristic
107C2×C149#53511yescharacteristic
108C2×C149#53711yescharacteristic
109C2×C149H(3,𝔽4)12yes—
110C2×C149#56212yes—
111D149—82no—
112C149#281911yescharacteristic, complemented, Sylow
113#2819—1491nomaximal, Sylow
114C2×#124—1491no—
115C2×#124D14911yescharacteristic, complemented
116#394—1494no—
117#393—1492no—
118#393—1492no—
119#392—1492no—
120#392—1492no—
121#382—1491no—
122C22×Q8—1491no—
123C22×Q8C2×D14911yescharacteristic
124C22×D4—1494no—
125C22×D4—1491no—
126C22×D4C2×D14911yescharacteristic
127#124—24no—
128#124—2984no—
129C2×C4⋊C4—1492no—
130C2×C22⋊C4—1492no—
131C2×C22⋊C4—1492no—
132C2×C22⋊C4—1494no—
133C2×C22⋊C4—1494no—
134C2×C22⋊C4C2×D14912yes—
135C2×C22⋊C4C2×D14912yes—
136C2×C42C2×D14911yescharacteristic
137#95—1492no—
138#95—1492no—
139#95—1494no—
140C24—1492no—
141C24C22×D14912yes—
142C2×Q8—22no—
143C2×Q8—42no—
144C2×Q8—2982no—
145C2×Q8—5962no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—2982no—
149C2×D4—5962no—
150C2×D4—5964no—
151C2×D4—5964no—
152C2×D4—5964no—
153C22×C4—1494no—
154C22×C4—1492no—
155C22×C4C22×D14912yes—
156C22×C4C22×D14911yescharacteristic
157C22×C4C22×D14911yescharacteristic
158C22×C4C22×D14911yescharacteristic
159C4⋊C4—5962no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—2984no—
163C22⋊C4—2984no—
164C22⋊C4—5964no—
165C22⋊C4—5964no—
166C42C22⋊D14914yescomplemented
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—2982no—
173C23—2984no—
174C23—2984no—
175C23—2982no—
176C23—2982no—
177C23C23×D14911yescharacteristic
178Q8—44no—
179Q8—5964no—
180D4—44no—
181D4—5964no—
182D4—59616no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—2982no—
190C2×C4—2984no—
191C2×C4—2984no—
192C2×C4—2984no—
193C2×C4—2984no—
194C2×C4C2×C22⋊D14912yes—
195C2×C4C2×C22⋊D14912yes—
196C2×C4C2×C22⋊D14912yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—44no—
201C22—44no—
202C22—5962no—
203C22—5962no—
204C22—5962no—
205C22—5964no—
206C22—5964no—
207C22C22×C22⋊D14911yescharacteristic
208C22#1801195612yes—
209C22#1801195712yes—
210C22#1801195711yescharacteristic
211C22#1801195811yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—5962no—
217C4—5964no—
218C2—82no—
219C2—11922no—
220C2#2375018911yescharacteristic
221C2#2375019211yescharacteristic
222C2#2375020311yescharacteristic
223C2#2375036712yes—
224C2#2375036912yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
FittingC2×#124×C149C2
hypercenterC2×#124D149
hypocenterC149#2819
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#18011899C2semidirect
2. #23749541C2semidirect
3. #23749537C2semidirect
4. #23749536C2semidirect
5. #23749534C2semidirect
6. #23749533C2semidirect
7. #23749491C2semidirect
8. C2×#124×C149C2semidirect
53. C42×C149D4semidirect
112. C149#2819semidirect
115. C2×#124D149semidirect
166. C42C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118992
#237495412
#237495372
#237495362
#237495342
#237495332
#237494912
C2×#124×C1492
#2819149