Finite Group Databasebeta

← #34055924

Subgroups — #34055924

Order 19072.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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×#18011910C211yesmaximal, characteristic, complemented
2#23750161C211yesmaximal, characteristic
3#23750033C212yesmaximal, complemented
4#23749986C212yesmaximal, complemented
5#23749958C211yesmaximal, characteristic, complemented
6#23749944C211yesmaximal, characteristic
7#23749943C211yesmaximal, characteristic
8#23749908C212yesmaximal, complemented
9#23749904C212yesmaximal, complemented
10#23749679C211yesmaximal, characteristic, complemented
11#484×C149C211yesmaximal, characteristic, complemented
12C22×Dic298—21no—
13#18011910—21no—
14#18011910—21no—
15#18011910C2212yes—
16#18011910C2212yes—
17#18011906—21no—
18#18011902C2211yescharacteristic
19#18011900C2211yescharacteristic
20#18011895—21no—
21#18011885C2211yescharacteristic
22Q8×Dic149C2211yescharacteristic
23#18011873C2211yescharacteristic
24#18011871—21no—
25#18011869—21no—
26C2×C4⋊Dic149C2211yescharacteristic
27C2×#14558715—21no—
28C2×C4×Dic149C2211yescharacteristic
29#18011844C2212yes—
30#18011843C2212yes—
31#18011842C2212yes—
32#18011840C2212yes—
33#18011835C2212yes—
34#18011833C2212yes—
35#18011831C2212yes—
36C22⋊#11924746C2211yescharacteristic
37#18011821C2212yes—
38C22⋊Q8×C149C2211yescharacteristic
39C2×C4⋊C4×C149C2211yescharacteristic
40#107×C149C2212yes—
41#103×C149C2212yes—
42C22⋊C8×C149C2211yescharacteristic
43C2×Dic298—22no—
44C2×Dic298—22no—
45C2×Dic298—44no—
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—21no—
48C22×Dic149—21no—
49C22×Dic149D411yescharacteristic
50C22×Dic149D411yescharacteristic
51C2×C4×D149—21no—
52C2×#11924746C2312yes—
53C4⋊Dic149—21no—
54C4⋊Dic149—21no—
55C4⋊Dic149—21no—
56C4⋊Dic149C2312yes—
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558715—22no—
60#14558715—22no—
61#14558713—21no—
62#14558713—21no—
63#14558713—21no—
64#14558713—21no—
65C4×Dic149—21no—
66C4×Dic149D411yescharacteristic
67C4×Dic149D411yescharacteristic
68C4×Dic149C2311yescharacteristic
69C4×Dic149C2311yescharacteristic
70C2×Q8×C149C2311yescharacteristic
71C22×C4×C149—21no—
72C22×C4×C149C2311yescharacteristic
73C2×C8×C149C2312yes—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—21no—
76C4⋊C4×C149C2312yes—
77C4⋊C4×C149C2311yescharacteristic
78C22⋊C4×C149—21no—
79C22×D149—21no—
80Dic298—44no—
81Dic298—44no—
82D596—22no—
83C22⋊D149—42no—
84C2×Dic149—22no—
85C2×Dic149—22no—
86C2×Dic149—22no—
87C2×Dic149—21no—
88C2×Dic149—42no—
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C2×Dic149C2×D411yescharacteristic
93C4×D149—42no—
94#11924746—22no—
95C23×C149C2×D411yescharacteristic
96Q8×C149—22no—
97C2×C4×C149—22no—
98C2×C4×C149—21no—
99C2×C4×C149—22no—
100C2×C4×C149—21no—
101C2×C4×C149—42no—
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2411yescharacteristic
104C8×C149—22no—
105C22×C149—21no—
106C22×C149—21no—
107C22×C149#13712yes—
108C22×C149C22×D411yescharacteristic
109C2×D149—41no—
110C2×D149—41no—
111C2×D149—41no—
112C4×C149—21no—
113C4×C149—41no—
114C4×C149—42no—
115C4×C149—41no—
116C4×C1492−1+412yes—
117Dic149—21no—
118Dic149—21no—
119Dic149—21no—
120Dic149—42no—
121Dic149—41no—
122Dic149AGL(1,ℤ/8)12yes—
123C2×C149—22no—
124C2×C149#53611yescharacteristic
125C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C149C2×#13711yescharacteristic
127D149—81no—
128C149#323911yescharacteristic, complemented, Sylow
129#3239—1491nomaximal, Sylow
130C2×C4⋊Q8—1491no—
131#529—1491no—
132#501—1492no—
133#489—1492no—
134#484—1491no—
135#484D14911yescharacteristic, complemented
136#481—1491no—
137#481—1491no—
138#475—1492no—
139#474—1492no—
140#432—1491no—
141C22×Q8—2981no—
142C4⋊Q8—2981no—
143C4⋊Q8—2981no—
144C4⋊Q8—1492no—
145C4⋊Q8—1492no—
146#126—2981no—
147#125—1491no—
148#124—1491no—
149#123—2981no—
150C22⋊Q8C2×D14911yescharacteristic
151#121—1491no—
152C4×Q8—1491no—
153C4×D4—2981no—
154C4×D4—1491no—
155#117—2981no—
156C2×C4⋊C4—2981no—
157C2×C4⋊C4—1491no—
158C2×C4⋊C4C2×D14911yescharacteristic
159C2×C42—1491no—
160#107—1492no—
161#107C2×D14912yes—
162#106—1492no—
163#106—1492no—
164C4⋊C8—1492no—
165#103—1492no—
166#103C2×D14912yes—
167#102—1492no—
168#102—1492no—
169C22⋊C8—1491no—
170C22⋊C8C2×D14911yescharacteristic
171#97—1492no—
172C2×Q8—2982no—
173C2×Q8—2982no—
174C2×Q8—5964no—
175C2×Q8C22×D14911yescharacteristic
176C2×D4—2981no—
177C2×D4—1491no—
178C22×C4—21no—
179C22×C4—2981no—
180C22×C4—2981no—
181C22×C4—1491no—
182C22×C4—1491no—
183C22×C4C22×D14911yescharacteristic
184C2×C8—1492no—
185C2×C8C22×D14912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2982no—
189C4⋊C4—2982no—
190C4⋊C4—2981no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2981no—
194C4⋊C4—1492no—
195C4⋊C4—1491no—
196C4⋊C4C22×D14912yes—
197C4⋊C4C22×D14911yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—2981no—
200C22⋊C4—2981no—
201C22⋊C4—2981no—
202C22⋊C4—2981no—
203C42—2981no—
204C42—1491no—
205C42—1491no—
206C42—1491no—
207C42—1491no—
208C23—2981no—
209C23D4×D14911yescharacteristic
210Q8—22no—
211Q8—5964no—
212Q8—5964no—
213D4—2982no—
214D4—5962no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—2982no—
221C2×C4—2982no—
222C2×C4—2981no—
223C2×C4—2982no—
224C2×C4—5962no—
225C2×C4—5962no—
226C2×C4—1491no—
227C2×C4—1491no—
228C2×C4—1491no—
229C2×C4—1491no—
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—22no—
233C8—2982no—
234C22—21no—
235C22—21no—
236C22—5961no—
237C22—5961no—
238C22—5961no—
239C22#1801194312yes—
240C22C2×D4×D14911yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—2981no—
246C4—2981no—
247C4—2981no—
248C4—5962no—
249C4—5961no—
250C4—1492no—
251C4#1801195812yes—
252C2—22no—
253C2—11921no—
254C2#2375019711yescharacteristic
255C2AGL(1,ℤ/8)×D14911yescharacteristic
256C2C2×#1801194311yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#484×C149C2
socleC22×C149C22×D4
prime radical#18011885C22
hypercenter#484D149
hypocenterC149#3239
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C2×#18011910C2semidirect
3. #23750033C2semidirect
4. #23749986C2semidirect
5. #23749958C2semidirect
8. #23749908C2semidirect
9. #23749904C2semidirect
10. #23749679C2semidirect
11. #484×C149C2semidirect
128. C149#3239semidirect
135. #484D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119102
#237501612
#237500332
#237499862
#237499582
#237499442
#237499432
#237499082
#237499042
#237496792
#484×C1492
#3239149