Finite Group Databasebeta

← #34052766

Subgroups — #34052766

Order 19072.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18011906C212yesmaximal, complemented
2#23749559C212yesmaximal, complemented
3#23749549C212yesmaximal, complemented
4#23749547C212yesmaximal, complemented
5#23749547C212yesmaximal, complemented
6#23749543C212yesmaximal
7#23749542C211yesmaximal, characteristic
8#23749494C211yesmaximal, characteristic, complemented
9#398×C149C211yesmaximal, characteristic, complemented
10C22×D596C2211yescharacteristic
11#18011906—28no—
12C2×C4⋊Dic149C2212yes—
13C2×C4⋊Dic149C2211yescharacteristic
14C2×#14558715C2212yes—
15C2×#14558715C2212yes—
16C2×#14558715C2212yes—
17C2×#14558715C2212yes—
18C2×#14558713C2212yes—
19C2×#14558713C2212yes—
20C2×#14558713C2212yes—
21C2×#14558713C2212yes—
22C2×#14558713C2212yes—
23C2×#14558713C2212yes—
24C2×C4×Dic149C2211yescharacteristic
25#18011818C2211yescharacteristic
26#18011818C2212yes—
27C2×C4⋊C4×C149C2212yes—
28C2×C4⋊C4×C149C2211yescharacteristic
29C2×C4⋊C4×C149C2211yescharacteristic
30#95×C149C2212yes—
31#95×C149C2211yescharacteristic
32C23×D149C2312yes—
33C2×D596—41no—
34C2×D596—42no—
35C22×Dic149C2312yes—
36C22×Dic149C2312yes—
37C22×Dic149C2312yes—
38C4⋊Dic149—24no—
39C4⋊Dic149—41no—
40#14558715—24no—
41#14558715—42no—
42#14558715—42no—
43#14558715—42no—
44#14558713—24no—
45#14558713—24no—
46#14558713—24no—
47#14558713—42no—
48#14558713—42no—
49#14558713—42no—
50C4×Dic149D414yescomplemented
51C22×C4×C149C2311yescharacteristic
52C22×C4×C149C2312yes—
53C22×C4×C149C2312yes—
54C22×C4×C149C2311yescharacteristic
55C22×C4×C149C2311yescharacteristic
56C4⋊C4×C149—24no—
57C4⋊C4×C149—41no—
58C4⋊C4×C149—41no—
59C22×D149—22no—
60C22×D149—22no—
61C22×D149—22no—
62C22×D149—22no—
63C22×D149—22no—
64C22×D149—22no—
65C22×D149—22no—
66D596—44no—
67C2×Dic149—22no—
68C2×Dic149—22no—
69C2×Dic149—22no—
70C2×Dic149—22no—
71C2×Dic149—22no—
72C2×Dic149—22no—
73C2×Dic149—22no—
74C2×Dic149—22no—
75C2×Dic149C2×D414yes—
76C23×C149C2411yescharacteristic
77C2×C4×C149—21no—
78C2×C4×C149—21no—
79C2×C4×C149—22no—
80C2×C4×C149—22no—
81C2×C4×C149—21no—
82C2×C4×C149—22no—
83C2×C4×C149—21no—
84C2×C4×C149—22no—
85C2×C4×C149—22no—
86C2×C4×C149—22no—
87C2×C4×C149—22no—
88C2×C4×C149—22no—
89C2×C4×C149C2×D412yes—
90C22×C149C22×D411yescharacteristic
91C22×C149H(5,𝔽2)11yescharacteristic
92C22×C149H(5,𝔽2)12yes—
93C22×C1492−1+411yescharacteristic
94C22×C1492−1+412yes—
95C2×D149—42no—
96C2×D149—42no—
97C2×D149—42no—
98C2×D149—42no—
99C2×D149—42no—
100C2×D149—42no—
101C2×D149—42no—
102C4×C149—22no—
103C4×C149—42no—
104C4×C149—42no—
105C4×C149—41no—
106C4×C149—41no—
107Dic149—24no—
108Dic149—42no—
109Dic149—42no—
110C2×C149#53711yescharacteristic
111C2×C149#53712yes—
112C2×C149#56011yescharacteristic
113C2×C149#56311yescharacteristic
114C2×C149#56312yes—
115D149—82no—
116C149#282311yescharacteristic, complemented, Sylow
117#2823—1491nomaximal, Sylow
118C2×#126—1492no—
119#398—1492no—
120#398D14911yescharacteristic, complemented
121#396—1492no—
122#396—1492no—
123#396—1492no—
124#394—1491no—
125#394—1492no—
126#384—1491no—
127C22×D4—1491no—
128#126—2988no—
129C2×C4⋊C4—1492no—
130C2×C4⋊C4—1492no—
131C2×C4⋊C4—1492no—
132C2×C4⋊C4—1492no—
133C2×C4⋊C4—1491no—
134C2×C4⋊C4—1492no—
135C2×C4⋊C4C2×D14912yes—
136C2×C4⋊C4C2×D14911yescharacteristic
137C2×C4⋊C4C2×D14911yescharacteristic
138C2×C22⋊C4—1492no—
139C2×C22⋊C4—1492no—
140C2×C22⋊C4—1492no—
141C2×C22⋊C4—1492no—
142C2×C22⋊C4—1492no—
143C2×C22⋊C4—1492no—
144C2×C42—1491no—
145#95—1491no—
146#95—1492no—
147#95C2×D14912yes—
148#95C2×D14911yescharacteristic
149C24—1492no—
150C2×D4—5961no—
151C2×D4—5962no—
152C22×C4—1492no—
153C22×C4—1492no—
154C22×C4—1492no—
155C22×C4C22×D14911yescharacteristic
156C22×C4C22×D14912yes—
157C22×C4C22×D14912yes—
158C22×C4C22×D14911yescharacteristic
159C22×C4C22×D14911yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—2984no—
164C4⋊C4—2984no—
165C4⋊C4—5962no—
166C4⋊C4—5962no—
167C4⋊C4—5962no—
168C4⋊C4—5961no—
169C22⋊C4—2984no—
170C22⋊C4—2984no—
171C22⋊C4—2984no—
172C22⋊C4—5962no—
173C22⋊C4—5962no—
174C22⋊C4—5962no—
175C42—1494no—
176C23—2982no—
177C23—2982no—
178C23—2982no—
179C23—2982no—
180C23—2982no—
181C23—2982no—
182C23—2982no—
183C23C23×D14911yescharacteristic
184D4—5964no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—2982no—
198C2×C4—2982no—
199C2×C4—2982no—
200C2×C4—2982no—
201C2×C4—2982no—
202C2×C4—2982no—
203C2×C4—2982no—
204C2×C4—2982no—
205C2×C4—1494no—
206C2×C4D4×D14912yes—
207C22—5962no—
208C22—5962no—
209C22—5962no—
210C22—5962no—
211C22—5962no—
212C22—5962no—
213C22—5962no—
214C22C2×D4×D14911yescharacteristic
215C22#1801195611yescharacteristic
216C22#1801195712yes—
217C22#1801195811yescharacteristic
218C22#1801195912yes—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—2984no—
225C4—5962no—
226C4—5962no—
227C2—11922no—
228C2#2375020211yescharacteristic
229C2#2375020512yes—
230C2#2375036511yescharacteristic
231C2#2375037411yescharacteristic
232C2#2375037912yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#398×C149C2
prime radicalC22×D596C22
hypercenter#398D149
hypocenterC149#2823
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#18011906C2semidirect
2. #23749559C2semidirect
3. #23749549C2semidirect
4. #23749547C2semidirect
5. #23749547C2semidirect
8. #23749494C2semidirect
9. #398×C149C2semidirect
50. C4×Dic149D4semidirect
116. C149#2823semidirect
120. #398D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119062
#237495592
#237495492
#237495472
#237495432
#237495422
#237494942
#398×C1492
#2823149