Finite Group Databasebeta

← #34052745

Subgroups — #34052745

Order 19072.

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

normal subgroups only

This group has 464 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×#18011902C211yesmaximal, characteristic, complemented
2C2×#18011897C211yesmaximal, characteristic
3#23749560C211yesmaximal, characteristic, complemented
4#23749552C211yesmaximal, characteristic, complemented
5#23749550C211yesmaximal, characteristic, complemented
6#23749548C212yesmaximal, complemented
7#23749547C212yesmaximal, complemented
8#23749545C212yesmaximal, complemented
9#23749540C212yesmaximal
10#23749489C211yesmaximal, characteristic, complemented
11#399×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C22×C22⋊D149C2211yescharacteristic
14#18011902—24no—
15#18011897—24no—
16C2×C4⋊Dic149C2212yes—
17C2×C4⋊Dic149C2211yescharacteristic
18C2×#14558715C2212yes—
19C2×#14558715C2212yes—
20C2×#14558715C2212yes—
21C2×C22⋊Dic149C2211yescharacteristic
22C2×C22⋊Dic149C2212yes—
23C2×#14558713C2211yescharacteristic
24C2×#14558713C2211yescharacteristic
25C2×#14558713C2212yes—
26C2×#14558713C2212yes—
27C2×C4×Dic149C2211yescharacteristic
28#18011818C2212yes—
29#18011818C2211yescharacteristic
30#18011818C2212yes—
31#18011818C2212yes—
32C2×C4⋊C4×C149C2211yescharacteristic
33C2×C4⋊C4×C149C2212yes—
34C2×C22⋊C4×C149C2211yescharacteristic
35C2×C22⋊C4×C149C2212yes—
36#95×C149C2211yescharacteristic
37C23×D149C2311yescharacteristic
38C2×Dic298—22no—
39C2×Dic298—42no—
40C2×C22⋊D149—22no—
41C2×C22⋊D149—42no—
42C22×Dic149C2311yescharacteristic
43C22×Dic149C2312yes—
44C22×Dic149C2312yes—
45C22×Dic149C2311yescharacteristic
46C22×Dic149C2311yescharacteristic
47C4⋊Dic149—24no—
48C4⋊Dic149—22no—
49#14558715—24no—
50#14558715—42no—
51#14558715—42no—
52C22⋊Dic149—22no—
53C22⋊Dic149—42no—
54#14558713—22no—
55#14558713—22no—
56#14558713—42no—
57#14558713—42no—
58C4×Dic149D414yes—
59C24×C149C2311yescharacteristic
60C22×C4×C149C2311yescharacteristic
61C22×C4×C149C2312yes—
62C22×C4×C149C2312yes—
63C22×C4×C149C2311yescharacteristic
64C4⋊C4×C149—22no—
65C4⋊C4×C149—42no—
66C22⋊C4×C149—22no—
67C22⋊C4×C149—42no—
68C22×D149—21no—
69C22×D149—22no—
70C22×D149—21no—
71C22×D149—22no—
72C22×D149—21no—
73Dic298—44no—
74C22⋊D149—44no—
75C2×Dic149—22no—
76C2×Dic149—21no—
77C2×Dic149—21no—
78C2×Dic149—21no—
79C2×Dic149—22no—
80C2×Dic149—22no—
81C2×Dic149—22no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149—22no—
86C2×Dic149C2×D412yes—
87C2×Dic149C2×D412yes—
88C23×C149—21no—
89C23×C149—22no—
90C23×C149—21no—
91C23×C149—22no—
92C23×C149—21no—
93C23×C149C2411yescharacteristic
94C2×C4×C149—22no—
95C2×C4×C149—21no—
96C2×C4×C149—22no—
97C2×C4×C149—22no—
98C2×C4×C149—22no—
99C2×C4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149C2×D412yes—
104C22×C149—41no—
105C22×C149—41no—
106C22×C149—41no—
107C22×C149—42no—
108C22×C149—42no—
109C22×C149C22×D411yescharacteristic
110C22×C149H(5,𝔽2)12yes—
111C22×C149H(5,𝔽2)11yescharacteristic
112C22×C1492−1+412yes—
113C22×C1492−1+411yescharacteristic
114C2×D149—41no—
115C2×D149—41no—
116C2×D149—41no—
117C2×D149—42no—
118C2×D149—42no—
119C4×C149—22no—
120C4×C149—42no—
121C4×C149—41no—
122C4×C149—42no—
123Dic149—22no—
124Dic149—22no—
125Dic149—41no—
126Dic149—42no—
127Dic149—42no—
128C2×C149—81no—
129C2×C149#53511yescharacteristic
130C2×C149#53611yescharacteristic
131C2×C149#53711yescharacteristic
132C2×C149#56212yes—
133C2×C149#56312yes—
134D149—81no—
135C149#282011yescharacteristic, complemented, Sylow
136#2820—1491nomaximal, Sylow
137C2×#125—1491no—
138C2×#124—1491no—
139#399—1491no—
140#399D14911yescharacteristic, complemented
141#397—1491no—
142#397—1491no—
143#396—1492no—
144#396—1492no—
145#395—1492no—
146#394—1492no—
147#382—1491no—
148C22×Q8—1491no—
149C22×D4—1491no—
150#125—2984no—
151#124—2984no—
152C2×C4⋊C4—1492no—
153C2×C4⋊C4—1492no—
154C2×C4⋊C4—1491no—
155C2×C4⋊C4—1492no—
156C2×C4⋊C4—1492no—
157C2×C4⋊C4C2×D14911yescharacteristic
158C2×C4⋊C4C2×D14912yes—
159C2×C22⋊C4—1491no—
160C2×C22⋊C4—1491no—
161C2×C22⋊C4—1491no—
162C2×C22⋊C4—1492no—
163C2×C22⋊C4—1492no—
164C2×C22⋊C4—1492no—
165C2×C22⋊C4C2×D14911yescharacteristic
166C2×C22⋊C4C2×D14912yes—
167C2×C42—1491no—
168#95—1492no—
169#95—1491no—
170#95—1492no—
171#95—1492no—
172#95C2×D14911yescharacteristic
173C24—1491no—
174C24C22×D14911yescharacteristic
175C2×Q8—2982no—
176C2×Q8—5962no—
177C2×D4—2982no—
178C2×D4—5962no—
179C22×C4—1492no—
180C22×C4—1492no—
181C22×C4—1491no—
182C22×C4—1491no—
183C22×C4—1491no—
184C22×C4C22×D14912yes—
185C22×C4C22×D14911yescharacteristic
186C22×C4C22×D14912yes—
187C22×C4C22×D14911yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—2984no—
191C4⋊C4—2984no—
192C4⋊C4—2982no—
193C4⋊C4—5962no—
194C4⋊C4—5962no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—2982no—
198C22⋊C4—2982no—
199C22⋊C4—2982no—
200C22⋊C4—5962no—
201C22⋊C4—5962no—
202C22⋊C4—5962no—
203C42—1494no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—2981no—
210C23—2982no—
211C23—2981no—
212C23—2982no—
213C23—2981no—
214C23C23×D14911yescharacteristic
215Q8—5964no—
216D4—5964no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—2982no—
227C2×C4—2981no—
228C2×C4—2981no—
229C2×C4—2981no—
230C2×C4—2982no—
231C2×C4—2982no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—2982no—
235C2×C4—2982no—
236C2×C4—2982no—
237C2×C4—1492no—
238C2×C4—1492no—
239C2×C4D4×D14912yes—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—5961no—
246C22—5961no—
247C22—5961no—
248C22—5962no—
249C22—5962no—
250C22C2×D4×D14911yescharacteristic
251C22#1801195612yes—
252C22#1801195711yescharacteristic
253C22#1801195812yes—
254C22#1801195911yescharacteristic
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—2982no—
260C4—2982no—
261C4—5961no—
262C4—5962no—
263C4—5962no—
264C2—81no—
265C2—11921no—
266C2#2375018811yescharacteristic
267C2#2375019711yescharacteristic
268C2#2375020511yescharacteristic
269C2#2375037212yes—
270C2#2375037612yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#399×C149C2
prime radicalC22×C22⋊D149C22
hypercenter#399D149
hypocenterC149#2820
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#18011902C2semidirect
3. #23749560C2semidirect
4. #23749552C2semidirect
5. #23749550C2semidirect
6. #23749548C2semidirect
7. #23749547C2semidirect
8. #23749545C2semidirect
10. #23749489C2semidirect
11. #399×C149C2semidirect
135. C149#2820semidirect
140. #399D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119022
C2×#180118972
#237495602
#237495522
#237495502
#237495482
#237495472
#237495452
#237495402
#237494892
#399×C1492
#2820149