Finite Group Databasebeta

← #34057548

Subgroups — #34057548

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
1#23750355C211yesmaximal, characteristic, complemented
2#23750290C211yesmaximal, characteristic
3#23750022C212yesmaximal, complemented
4#23749914C212yesmaximal
5#23749912C211yesmaximal, characteristic
6#23749909C212yesmaximal, complemented
7#23749907C211yesmaximal, characteristic, complemented
8C8×Dic298C211yesmaximal, characteristic, complemented
9C4×#14558725C211yesmaximal, characteristic
10#23749712C212yesmaximal
11#498×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149—21no—
13C2×#14558725C2211yescharacteristic
14#18011910—21no—
15#18011910C2212yes—
16#18011910C2212yes—
17#18011909—22no—
18#18011889—22no—
19#18011887—21no—
20C4×Dic298C2211yescharacteristic
21C4×Dic298C2211yescharacteristic
22#18011876—21no—
23#18011876C2212yes—
24C4×D596C2211yescharacteristic
25#18011873C2212yes—
26#18011843C2211yescharacteristic
27#18011842C2212yes—
28#18011842C2212yes—
29C4⋊#11924746C2211yescharacteristic
30#18011840C2212yes—
31#18011836C2211yescharacteristic
32#18011834C2212yes—
33#18011833C2211yescharacteristic
34#18011831C2212yes—
35C8×Dic149C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4⋊Q8×C149C2211yescharacteristic
38#106×C149C2212yes—
39#106×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41C2×Dic298—22no—
42C2×Dic298—22no—
43C2×Dic298D411yescharacteristic
44C2×Dic298C2311yescharacteristic
45Q8×D149—44no—
46C2×D596C2311yescharacteristic
47C2×C4×D149—22no—
48C2×C4×D149—21no—
49#14558725Q814yescomplemented
50C2×#11924746C2312yes—
51C4⋊Dic149D411yescharacteristic
52C4⋊Dic149C2311yescharacteristic
53C4⋊Dic149C2311yescharacteristic
54#14558715—22no—
55#14558715—22no—
56#14558715—22no—
57#14558715—21no—
58#14558715D412yes—
59#14558713—22no—
60#14558713—21no—
61C4×Dic149—22no—
62C4×Dic149—21no—
63C4×Dic149C2312yes—
64C2×Q8×C149—21no—
65C2×Q8×C149—21no—
66C2×C8×C149C2311yescharacteristic
67C2×C8×C149C2311yescharacteristic
68C4⋊C4×C149—21no—
69C4⋊C4×C149—21no—
70C4⋊C4×C149C2312yes—
71C4⋊C4×C149C2312yes—
72C42×C149C2311yescharacteristic
73C22×D149—21no—
74Dic298—21no—
75Dic298—44no—
76Dic298—82no—
77Dic298SD1614yescomplemented
78Dic298C2×Q812yescomplemented
79D596—21no—
80D596C2×Q812yes—
81C2×Dic149—21no—
82C2×Dic149—21no—
83C2×Dic149—21no—
84C2×Dic149C2×D412yes—
85C2×Dic149C2×D412yes—
86C4×D149—42no—
87C4×D149—44no—
88#11924746—22no—
89Q8×C149—42no—
90Q8×C149—81no—
91C2×C4×C149—22no—
92C2×C4×C149—22no—
93C2×C4×C149C2×D411yescharacteristic
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2411yescharacteristic
96C8×C149—21no—
97C8×C149C2×Q812yescomplemented
98C22×C149C22×D411yescharacteristic
99C2×D149—22no—
100C2×D149—41no—
101C2×D149—41no—
102C4×C149—21no—
103C4×C149—42no—
104C4×C149—42no—
105C4×C149C2×SD1612yes—
106C4×C149C22×Q811yescharacteristic
107C4×C1492−1+411yescharacteristic
108Dic149—22no—
109Dic149—22no—
110Dic149—41no—
111Dic149—41no—
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—1492no—
122#498D14911yescharacteristic, complemented
123#476—1491no—
124#476—1492no—
125#475—1491no—
126#475—1492no—
127C8×Q8—1491no—
128C4×SD16—1492no—
129C4×SD16—1491no—
130C22×Q8—2981no—
131C2×SD16—1491no—
132C4⋊Q8—2981no—
133C4⋊Q8—2982no—
134C4⋊Q8—1492no—
135C4⋊Q8—1492no—
136C4⋊Q8C2×D14911yescharacteristic
137C4⋊Q8C2×D14911yescharacteristic
138C22⋊Q8—2982no—
139C22⋊Q8—2981no—
140C4×Q8—2981no—
141C4×Q8—1492no—
142C4×Q8—1491no—
143C4×Q8—1491no—
144C4×D4—1492no—
145C4×D4—1491no—
146#106—1491no—
147#106—1492no—
148#106—1492no—
149#106C2×D14912yes—
150#106C2×D14912yes—
151C4⋊C8—1492no—
152C4⋊C8—1491no—
153#103—1491no—
154#103—1492no—
155#102—1491no—
156#102—1492no—
157C4×C8—1492no—
158C4×C8C2×D14911yescharacteristic
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—2982no—
162C2×Q8—2982no—
163C2×Q8—5964no—
164C2×Q8—1491no—
165C2×Q8—1491no—
166C2×D4—1491no—
167C22×C4—2982no—
168C22×C4—2981no—
169SD16—1494no—
170C2×C8—1492no—
171C2×C8C22×D14911yescharacteristic
172C2×C8C22×D14911yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4—2981no—
179C4⋊C4—1491no—
180C4⋊C4—1491no—
181C4⋊C4—1492no—
182C4⋊C4—1491no—
183C4⋊C4C22×D14912yes—
184C4⋊C4C22×D14912yes—
185C22⋊C4—2982no—
186C22⋊C4—2981no—
187C42—2982no—
188C42—2981no—
189C42—1492no—
190C42C22×D14911yescharacteristic
191C23—2981no—
192Q8—42no—
193Q8—81no—
194Q8—2981no—
195Q8—5964no—
196Q8—1492no—
197Q8—1494no—
198Q8—11922no—
199D4—2981no—
200D4—1492no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—2981no—
204C2×C4—2981no—
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—21no—
214C8—2982no—
215C8Q8×D14912yescomplemented
216C22—2982no—
217C22—5961no—
218C22—5961no—
219C22C2×D4×D14911yescharacteristic
220C4—21no—
221C4—42no—
222C4—42no—
223C4—2982no—
224C4—2982no—
225C4—5961no—
226C4—5961no—
227C4—1494no—
228C4SD16×D14912yes—
229C4C2×Q8×D14911yescharacteristic
230C4#1801195911yescharacteristic
231C2—5962no—
232C2#2375029011yescharacteristic
233C2C2×SD16×D14911yescharacteristic
234C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#498×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#498D149
hypocenterC149Q8×SD16
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #23750355C2semidirect
3. #23750022C2semidirect
6. #23749909C2semidirect
7. #23749907C2semidirect
8. C8×Dic298C2semidirect
11. #498×C149C2semidirect
49. #14558725Q8semidirect
77. Dic298SD16semidirect
78. Dic298C2×Q8semidirect
97. C8×C149C2×Q8semidirect
117. C149Q8×SD16semidirect
122. #498D149semidirect
215. C8Q8×D149semidirect

Maximal subgroups

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

subgroupindex
#237503552
#237502902
#237500222
#237499142
#237499122
#237499092
#237499072
C8×Dic2982
C4×#145587252
#237497122
#498×C1492
Q8×SD16149