Finite Group Databasebeta

← #34055922

Subgroups — #34055922

Order 19072.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D149C211yesmaximal, characteristic, complemented
2#23750160C211yesmaximal, characteristic
3#23750035C211yesmaximal, characteristic, complemented
4#23749987C211yesmaximal, characteristic, complemented
5#23749985C211yesmaximal, characteristic, complemented
6#23749962C212yesmaximal, complemented
7#23749945C212yesmaximal
8#23749909C212yesmaximal, complemented
9#23749904C212yesmaximal, complemented
10#23749680C211yesmaximal, characteristic, complemented
11#501×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149—21no—
13#18011911C2211yescharacteristic
14#18011910—21no—
15#18011910C2212yes—
16#18011909—21no—
17#18011906—21no—
18#18011899C2211yescharacteristic
19#18011896—21no—
20#18011891C2211yescharacteristic
21#18011886C2211yescharacteristic
22#18011876C2211yescharacteristic
23#18011873—21no—
24#18011873C2211yescharacteristic
25#18011869—21no—
26C4⋊C4×D149—21no—
27C4⋊C4×D149C2212yes—
28C42×D149C2211yescharacteristic
29#18011844C2212yes—
30#18011842C2212yes—
31#18011840C2212yes—
32#18011836C2212yes—
33#18011834C2212yes—
34#18011833C2212yes—
35#18011831C2212yes—
36#18011823C2212yes—
37#18011822C2211yescharacteristic
38C4⋊Q8×C149C2211yescharacteristic
39#125×C149C2211yescharacteristic
40#107×C149C2212yes—
41#106×C149C2212yes—
42#97×C149C2211yescharacteristic
43C2×Dic298—22no—
44C2×Dic298—21no—
45C2×Dic298C2311yescharacteristic
46Q8×D149—44no—
47C2×D596—21no—
48C2×D596C2311yescharacteristic
49C2×C4×D149—22no—
50C2×C4×D149—21no—
51C2×C4×D149D411yescharacteristic
52C2×C4×D149D411yescharacteristic
53C2×C4×D149C2311yescharacteristic
54C2×#11924746C2312yes—
55C4⋊Dic149—21no—
56C4⋊Dic149—21no—
57C4⋊Dic149C2312yes—
58#14558715—21no—
59#14558715—22no—
60#14558715—22no—
61#14558713—21no—
62#14558713—21no—
63#14558713—21no—
64#14558713—21no—
65#14558713—21no—
66C4×Dic149—21no—
67C4×Dic149D411yescharacteristic
68C4×Dic149D411yescharacteristic
69C4×Dic149C2311yescharacteristic
70C2×Q8×C149—21no—
71C2×C8×C149C2312yes—
72C4⋊C4×C149—21no—
73C4⋊C4×C149—21no—
74C4⋊C4×C149—21no—
75C4⋊C4×C149C2312yes—
76C4⋊C4×C149C2311yescharacteristic
77C4⋊C4×C149C2311yescharacteristic
78C42×C149C2311yescharacteristic
79C22×D149—21no—
80C22×D149C2×D411yescharacteristic
81Dic298—22no—
82Dic298—42no—
83Dic298—44no—
84D596—22no—
85D596—42no—
86C2×Dic149—22no—
87C2×Dic149—21no—
88C2×Dic149C2×D411yescharacteristic
89C2×Dic149C2×D411yescharacteristic
90C2×Dic149C2×D411yescharacteristic
91C4×D149—22no—
92C4×D149—22no—
93C4×D149—22no—
94C4×D149—44no—
95C4×D149—42no—
96#11924746—22no—
97Q8×C149—42no—
98C2×C4×C149—22no—
99C2×C4×C149—21no—
100C2×C4×C149—21no—
101C2×C4×C149C2×D411yescharacteristic
102C2×C4×C149C2×D411yescharacteristic
103C2×C4×C149C2411yescharacteristic
104C8×C149—22no—
105C22×C149C22×D411yescharacteristic
106C2×D149—21no—
107C2×D149—21no—
108C2×D149—41no—
109C2×D149—41no—
110C2×D149—41no—
111C2×D149#13712yes—
112C4×C149—21no—
113C4×C149—21no—
114C4×C149—42no—
115C4×C149—41no—
116C4×C149—41no—
117C4×C1492−1+412yes—
118Dic149—21no—
119Dic149—21no—
120Dic149—41no—
121Dic149—42no—
122Dic149AGL(1,ℤ/8)12yes—
123C2×C149#53611yescharacteristic
124C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C149C2×#13711yescharacteristic
126D149—22no—
127D149—81no—
128C149#323911yescharacteristic, complemented, Sylow
129#3239—1491nomaximal, Sylow
130C2×C4⋊Q8—1491no—
131#529—1491no—
132#501—1491no—
133#501D14911yescharacteristic, complemented
134#489—1491no—
135#489—1491no—
136#484—1492no—
137#481—1492no—
138#475—1492no—
139#474—1492no—
140#432—1491no—
141C22×Q8—2981no—
142C4⋊Q8—2981no—
143C4⋊Q8—2981no—
144C4⋊Q8—1492no—
145C4⋊Q8—1491no—
146C4⋊Q8C2×D14911yescharacteristic
147#126—2981no—
148#125C2×D14911yescharacteristic
149#124—1491no—
150#123—2981no—
151C22⋊Q8—1491no—
152#121—1491no—
153C4×Q8—1491no—
154C4×D4—2981no—
155C4×D4—1491no—
156#117—2981no—
157C2×C4⋊C4—2981no—
158C2×C4⋊C4—1492no—
159C2×C42—1491no—
160#107—1492no—
161#107C2×D14912yes—
162#106—1492no—
163#106C2×D14912yes—
164C4⋊C8—1492no—
165#103—1492no—
166#103—1492no—
167#102—1492no—
168#102—1492no—
169C22⋊C8—1492no—
170#97—1491no—
171#97C2×D14911yescharacteristic
172C2×Q8—21no—
173C2×Q8—2982no—
174C2×Q8—2981no—
175C2×Q8—5964no—
176C2×Q8—1491no—
177C2×D4—2981no—
178C2×D4—1491no—
179C22×C4—2982no—
180C22×C4—2981no—
181C22×C4—1491no—
182C22×C4—1491no—
183C22×C4—1491no—
184C2×C8—1492no—
185C2×C8C22×D14912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—2982no—
190C4⋊C4—2982no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2981no—
194C4⋊C4—1492no—
195C4⋊C4C22×D14912yes—
196C4⋊C4C22×D14911yescharacteristic
197C4⋊C4C22×D14911yescharacteristic
198C22⋊C4—2981no—
199C22⋊C4—2981no—
200C22⋊C4—2981no—
201C22⋊C4—2981no—
202C22⋊C4—2981no—
203C42—2981no—
204C42—1491no—
205C42—1491no—
206C42—1491no—
207C42C22×D14911yescharacteristic
208C23—2981no—
209C23—1491no—
210Q8—42no—
211Q8—2982no—
212Q8—5962no—
213Q8—5964no—
214D4—2982no—
215D4—5962no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—2982no—
220C2×C4—2982no—
221C2×C4—2982no—
222C2×C4—2981no—
223C2×C4—2982no—
224C2×C4—5964no—
225C2×C4—5962no—
226C2×C4—1491no—
227C2×C4—1491no—
228C2×C4—1491no—
229C2×C4D4×D14911yescharacteristic
230C2×C4D4×D14911yescharacteristic
231C2×C4C23×D14911yescharacteristic
232C8—22no—
233C8—2982no—
234C22—2981no—
235C22—2981no—
236C22—5961no—
237C22—5961no—
238C22—5961no—
239C22—1492no—
240C22C2×D4×D14911yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—2981no—
247C4—2981no—
248C4—5962no—
249C4—5961no—
250C4—1492no—
251C4#1801195912yes—
252C2—2982no—
253C2—11921no—
254C2#2375020011yescharacteristic
255C2AGL(1,ℤ/8)×D14911yescharacteristic
256C2#137×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#501×C149C2
socleC22×C149C22×D4
prime radical#18011886C22
hypercenter#501D149
hypocenterC149#3239
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D149C2semidirect
3. #23750035C2semidirect
4. #23749987C2semidirect
5. #23749985C2semidirect
6. #23749962C2semidirect
8. #23749909C2semidirect
9. #23749904C2semidirect
10. #23749680C2semidirect
11. #501×C149C2semidirect
128. C149#3239semidirect
133. #501D149semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1492
#237501602
#237500352
#237499872
#237499852
#237499622
#237499452
#237499092
#237499042
#237496802
#501×C1492
#3239149