Finite Group Databasebeta

← #34057721

Subgroups — #34057721

Order 19072.

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

normal subgroups only

This group has 350 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
1#23750353C211yesmaximal, characteristic, complemented
2#23750349C211yesmaximal, characteristic
3#23750018C212yesmaximal
4#23750004C211yesmaximal, characteristic, complemented
5#23749850C212yesmaximal
6#23749833C212yesmaximal, complemented
7#23749831C211yesmaximal, characteristic
8#23749752C211yesmaximal, characteristic, complemented
9#23749729C212yesmaximal, complemented
10C4×#14558723C211yesmaximal, characteristic
11#462×C149C211yesmaximal, characteristic, complemented
12C2×#14558726C2212yes—
13C2×#14558726C2212yes—
14C2×#14558725C2212yes—
15C2×#14558723C2211yescharacteristic
16#18011911C2211yescharacteristic
17#18011909C2212yes—
18#18011906—22no—
19#18011903—21no—
20#18011901—22no—
21#18011900C2212yes—
22#18011891—21no—
23Q8×Dic149C2212yes—
24C4×Dic298—21no—
25C4×Dic298C2211yescharacteristic
26#18011876C2212yes—
27C4×D596C2211yescharacteristic
28#18011869—21no—
29#18011842C2211yescharacteristic
30#18011840C2212yes—
31#18011836C2212yes—
32#18011835C2211yescharacteristic
33#18011831C2211yescharacteristic
34#18011821C2212yes—
35C4×#11924746C2211yescharacteristic
36C2×Q16×C149C2212yes—
37C4⋊Q8×C149C2211yescharacteristic
38C4×Q8×C149C2211yescharacteristic
39C4⋊C8×C149C2211yescharacteristic
40#103×C149C2212yes—
41C2×Dic298D411yescharacteristic
42C2×Dic298C2312yes—
43C2×D596C2311yescharacteristic
44C2×C4×D149—21no—
45#14558726—24no—
46#14558726—42no—
47#14558725—42no—
48#14558723—22no—
49C2×#11924746C2311yescharacteristic
50C2×#11924746C2311yescharacteristic
51C4⋊Dic149—22no—
52C4⋊Dic149—21no—
53C4⋊Dic149—21no—
54C4⋊Dic149D411yescharacteristic
55C4⋊Dic149C2311yescharacteristic
56#14558715—22no—
57#14558715—22no—
58#14558715—22no—
59#14558715D412yes—
60#14558713—22no—
61#14558713—22no—
62#14558713—21no—
63C4×Dic149—22no—
64C4×Dic149—22no—
65C4×Dic149C2312yes—
66C2×Q8×C149C2311yescharacteristic
67C2×Q8×C149C2312yes—
68Q16×C149—42no—
69C2×C8×C149C2312yes—
70C4⋊C4×C149—21no—
71C4⋊C4×C149—21no—
72C4⋊C4×C149—21no—
73C4⋊C4×C149C2311yescharacteristic
74C42×C149—21no—
75C42×C149C2311yescharacteristic
76C22×D149—21no—
77Dic298—22no—
78Dic298—22no—
79Dic298—22no—
80D596—21no—
81D596—21no—
82C2×Dic149—21no—
83C2×Dic149—22no—
84C2×Dic149C2×D412yes—
85C2×Dic149C2×D412yes—
86C4×D149—42no—
87#11924746—21no—
88#11924746#4112yes—
89Q8×C149—21no—
90Q8×C149—22no—
91Q8×C149—22no—
92Q8×C149—21no—
93C2×C4×C149—21no—
94C2×C4×C149—21no—
95C2×C4×C149—22no—
96C2×C4×C149C2×D411yescharacteristic
97C2×C4×C149C2×D411yescharacteristic
98C2×C4×C149C2411yescharacteristic
99C8×C149—22no—
100C22×C149C22×D411yescharacteristic
101C2×D149—41no—
102C2×D149—41no—
103C2×D149—41no—
104C4×C149—21no—
105C4×C149—41no—
106C4×C149—42no—
107C4×C149—41no—
108C4×C149#13712yes—
109C4×C149C2×#4111yescharacteristic
110C4×C149H(5,𝔽2)11yescharacteristic
111Dic149—22no—
112Dic149—22no—
113Dic149—42no—
114Dic149—41no—
115C2×C149#55011yescharacteristic
116C2×C149C2×#13711yescharacteristic
117C2×C149#57811yescharacteristic
118D149—81no—
119C149#339811yescharacteristic, complemented, Sylow
120#3398—1491nomaximal, Sylow
121#557—1491no—
122#556—1491no—
123#497—1492no—
124#494—1491no—
125#464—1492no—
126#462—1492no—
127#462D14911yescharacteristic, complemented
128#461—1491no—
129#446—1491no—
130#441—1492no—
131C4×SD16—1491no—
132C2×Q16—1492no—
133C2×Q16—1492no—
134C2×Q16C2×D14912yes—
135C2×SD16—1491no—
136C2×SD16—1492no—
137C4⋊Q8—1492no—
138C4⋊Q8—1491no—
139C4⋊Q8C2×D14911yescharacteristic
140#126—2982no—
141#125—2981no—
142#125—2982no—
143#124—1492no—
144C22⋊Q8—2981no—
145C4×Q8—2981no—
146C4×Q8—1492no—
147C4×Q8—1492no—
148C4×Q8—1491no—
149C4×Q8C2×D14911yescharacteristic
150C4×D4—1491no—
151#117—2981no—
152#106—1491no—
153C4⋊C8—1492no—
154C4⋊C8C2×D14911yescharacteristic
155#103—1491no—
156#103—1492no—
157#103C2×D14912yes—
158#102—1491no—
159#97—1492no—
160C4×C8—1491no—
161C2×Q8—1492no—
162C2×Q8—1491no—
163C2×Q8C22×D14911yescharacteristic
164C2×Q8C22×D14912yes—
165C2×D4—1491no—
166C22×C4—2981no—
167Q16—42no—
168Q16—2984no—
169Q16—5962no—
170SD16—2982no—
171SD16—5962no—
172C2×C8—1491no—
173C2×C8—1491no—
174C2×C8C22×D14912yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—2982no—
179C4⋊C4—2982no—
180C4⋊C4—2982no—
181C4⋊C4—2982no—
182C4⋊C4—2981no—
183C4⋊C4—2981no—
184C4⋊C4—1491no—
185C4⋊C4—1491no—
186C4⋊C4—1492no—
187C4⋊C4C22×D14911yescharacteristic
188C22⋊C4—2982no—
189C22⋊C4—2982no—
190C22⋊C4—2981no—
191C42—21no—
192C42—2982no—
193C42—2982no—
194C42—1492no—
195C42C22×D14911yescharacteristic
196C23—2981no—
197Q8—21no—
198Q8—22no—
199Q8—21no—
200Q8—22no—
201Q8—2982no—
202Q8—2982no—
203Q8—2982no—
204D4—2981no—
205D4—2981no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—2981no—
210C2×C4—2982no—
211C2×C4—5962no—
212C2×C4—1492no—
213C2×C4—1492no—
214C2×C4D4×D14911yescharacteristic
215C2×C4D4×D14911yescharacteristic
216C2×C4C23×D14911yescharacteristic
217C8—22no—
218C8—2981no—
219C8—1492no—
220C22—5961no—
221C22—5961no—
222C22—5961no—
223C22C2×D4×D14911yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—2982no—
229C4—2982no—
230C4—5961no—
231C4—5962no—
232C4#1801194312yes—
233C4#41×D14911yescharacteristic
234C4#1801195611yescharacteristic
235C2—11921no—
236C2#2375029311yescharacteristic
237C2C2×#1801194311yescharacteristic
238C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#462×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#462D149
hypocenterC149#3398
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #23750353C2semidirect
4. #23750004C2semidirect
6. #23749833C2semidirect
8. #23749752C2semidirect
9. #23749729C2semidirect
11. #462×C149C2semidirect
119. C149#3398semidirect
127. #462D149semidirect

Maximal subgroups

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

subgroupindex
#237503532
#237503492
#237500182
#237500042
#237498502
#237498332
#237498312
#237497522
#237497292
C4×#145587232
#462×C1492
#3398149