Finite Group Databasebeta

← #34057550

Subgroups — #34057550

Order 19072.

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

normal subgroups only

This group has 370 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
1Q8×Dic298C211yesmaximal, characteristic, complemented
2#23750290C211yesmaximal, characteristic
3#23750022C212yesmaximal, complemented
4#23750021C211yesmaximal, characteristic, complemented
5#23749916C211yesmaximal, characteristic
6#23749914C212yesmaximal
7#23749908C212yesmaximal, complemented
8#23749749C211yesmaximal, characteristic, complemented
9C4×#14558723C211yesmaximal, characteristic
10#23749710C212yesmaximal
11#475×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149—21no—
13C2×#14558723C2211yescharacteristic
14#18011911—21no—
15#18011911C2211yescharacteristic
16#18011910—22no—
17#18011910C2212yes—
18#18011910C2212yes—
19#18011889—22no—
20#18011887—21no—
21Q8×Dic149C2212yes—
22C4×Dic298—21no—
23C4×Dic298C2211yescharacteristic
24C4×D596C2211yescharacteristic
25#18011873C2212yes—
26#18011843C2212yes—
27#18011842C2212yes—
28#18011842C2212yes—
29#18011842C2211yescharacteristic
30#18011840C2212yes—
31#18011835C2211yescharacteristic
32#18011833C2212yes—
33#18011831C2211yescharacteristic
34C4×#11924746C2211yescharacteristic
35C8×Dic149C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4×Q8×C149C2211yescharacteristic
38#106×C149C2212yes—
39C4⋊C8×C149C2211yescharacteristic
40#103×C149C2212yes—
41C2×Dic298—22no—
42C2×Dic298—22no—
43C2×Dic298—21no—
44C2×Dic298D411yescharacteristic
45Q8×D149—44no—
46C2×D596C2311yescharacteristic
47C2×C4×D149—22no—
48C2×C4×D149—21no—
49#14558723Q814yescomplemented
50C2×#11924746C2311yescharacteristic
51C2×#11924746C2311yescharacteristic
52C4⋊Dic149—22no—
53C4⋊Dic149—21no—
54C4⋊Dic149D411yescharacteristic
55C4⋊Dic149C2311yescharacteristic
56C4⋊Dic149C2312yes—
57#14558715—22no—
58#14558715—22no—
59#14558715D412yes—
60#14558713—22no—
61#14558713—21no—
62C4×Dic149—22no—
63C4×Dic149C2312yes—
64C2×Q8×C149—21no—
65C2×Q8×C149C2311yescharacteristic
66C2×C8×C149C2312yes—
67C4⋊C4×C149—21no—
68C4⋊C4×C149—21no—
69C4⋊C4×C149C2311yescharacteristic
70C4⋊C4×C149C2312yes—
71C42×C149—21no—
72C42×C149C2311yescharacteristic
73C22×D149—21no—
74Dic298—44no—
75Dic298—82no—
76Dic298—81no—
77Dic298SD1614yescomplemented
78D596—21no—
79D596C2×Q812yes—
80C2×Dic149—21no—
81C2×Dic149—22no—
82C2×Dic149C2×D412yes—
83C2×Dic149C2×D412yes—
84C4×D149—42no—
85C4×D149—44no—
86#11924746—21no—
87#11924746C2×Q812yescomplemented
88Q8×C149—21no—
89Q8×C149—42no—
90Q8×C149C2×Q812yescomplemented
91C2×C4×C149—21no—
92C2×C4×C149—22no—
93C2×C4×C149—21no—
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2×D411yescharacteristic
96C2×C4×C149C2411yescharacteristic
97C8×C149—22no—
98C22×C149C22×D411yescharacteristic
99C2×D149—22no—
100C2×D149—41no—
101C2×D149—41no—
102C4×C149—22no—
103C4×C149—21no—
104C4×C149—42no—
105C4×C149—41no—
106C4×C149C2×SD1612yes—
107C4×C149C22×Q811yescharacteristic
108C4×C1492−1+411yescharacteristic
109Dic149—22no—
110Dic149—41no—
111Dic149—42no—
112Dic149C2×SD1614yes—
113C2×C149D4×Q811yescharacteristic
114C2×C149C22×SD1611yescharacteristic
115C2×C149#57711yescharacteristic
116D149—42no—
117C149Q8×SD1611yescharacteristic, complemented, Sylow
118Q8×SD16—1491nomaximal, Sylow
119Q82—1491no—
120D4×Q8—1491no—
121#498—1491no—
122#498—1492no—
123#476—1492no—
124#476—1491no—
125#475—1492no—
126#475D14911yescharacteristic, complemented
127C8×Q8—1491no—
128C4×SD16—1491no—
129C4×SD16—1492no—
130C22×Q8—2981no—
131C2×SD16—1491no—
132C4⋊Q8—2981no—
133C4⋊Q8—2982no—
134C4⋊Q8—1492no—
135C4⋊Q8—1492no—
136C4⋊Q8—1491no—
137C4⋊Q8C2×D14911yescharacteristic
138C22⋊Q8—2982no—
139C22⋊Q8—2981no—
140C4×Q8—2981no—
141C4×Q8—1492no—
142C4×Q8—1491no—
143C4×Q8C2×D14911yescharacteristic
144C4×D4—1492no—
145C4×D4—1491no—
146#106—1492no—
147#106—1492no—
148#106—1491no—
149#106—1492no—
150#106C2×D14912yes—
151C4⋊C8—1492no—
152C4⋊C8C2×D14911yescharacteristic
153#103—1491no—
154#103C2×D14912yes—
155#102—1492no—
156#102—1491no—
157C4×C8—1492no—
158C4×C8—1491no—
159C2×Q8—21no—
160C2×Q8—2982no—
161C2×Q8—2982no—
162C2×Q8—2981no—
163C2×Q8—5964no—
164C2×Q8—1491no—
165C2×Q8C22×D14911yescharacteristic
166C2×D4—1491no—
167C22×C4—2982no—
168C22×C4—2981no—
169SD16—1494no—
170C2×C8—1491no—
171C2×C8—1491no—
172C2×C8C22×D14912yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4—2981no—
179C4⋊C4—1492no—
180C4⋊C4—1491no—
181C4⋊C4—1492no—
182C4⋊C4—1491no—
183C4⋊C4C22×D14911yescharacteristic
184C4⋊C4C22×D14912yes—
185C22⋊C4—2982no—
186C22⋊C4—2981no—
187C42—21no—
188C42—2982no—
189C42—1492no—
190C42C22×D14911yescharacteristic
191C23—2981no—
192Q8—21no—
193Q8—42no—
194Q8—5964no—
195Q8—1494no—
196Q8—11921no—
197Q8—11922no—
198Q8Q8×D14912yescomplemented
199D4—2981no—
200D4—1492no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—2982no—
205C2×C4—2981no—
206C2×C4—5964no—
207C2×C4—5962no—
208C2×C4—1492no—
209C2×C4—1492no—
210C2×C4D4×D14911yescharacteristic
211C2×C4D4×D14911yescharacteristic
212C2×C4C23×D14911yescharacteristic
213C8—22no—
214C8—2981no—
215C8—1492no—
216C22—2982no—
217C22—5961no—
218C22—5961no—
219C22C2×D4×D14911yescharacteristic
220C4—21no—
221C4—22no—
222C4—41no—
223C4—42no—
224C4—2982no—
225C4—5962no—
226C4—5961no—
227C4—1494no—
228C4SD16×D14912yes—
229C4C2×Q8×D14911yescharacteristic
230C4#1801195811yescharacteristic
231C2—5962no—
232C2#2375028811yescharacteristic
233C2C2×SD16×D14911yescharacteristic
234C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#475×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#475D149
hypocenterC149Q8×SD16
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×Dic298C2semidirect
3. #23750022C2semidirect
4. #23750021C2semidirect
7. #23749908C2semidirect
8. #23749749C2semidirect
11. #475×C149C2semidirect
49. #14558723Q8semidirect
77. Dic298SD16semidirect
87. #11924746C2×Q8semidirect
90. Q8×C149C2×Q8semidirect
117. C149Q8×SD16semidirect
126. #475D149semidirect
198. Q8Q8×D149semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237502902
#237500222
#237500212
#237499162
#237499142
#237499082
#237497492
C4×#145587232
#237497102
#475×C1492
Q8×SD16149