Finite Group Databasebeta

← #34055776

Subgroups — #34055776

Order 19072.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#23750237C211yesmaximal, characteristic, complemented
2#23750201C211yesmaximal, characteristic
3#23750019C211yesmaximal, characteristic, complemented
4#23750005C211yesmaximal, characteristic, complemented
5#23749994C211yesmaximal, characteristic, complemented
6#23749980C211yesmaximal, characteristic, complemented
7#23749848C212yesmaximal, complemented
8#23749836C212yesmaximal, complemented
9#23749789C212yesmaximal
10#23749783C212yesmaximal, complemented
11#433×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×Dic298C2211yescharacteristic
14C2×Dic596C2212yes—
15C2×Dic596C2212yes—
16C2×Dic596C2212yes—
17C2×#14558725C2212yes—
18#18011911C2211yescharacteristic
19#18011911C2211yescharacteristic
20#18011911C2211yescharacteristic
21#18011904—22no—
22#18011903C2211yescharacteristic
23#18011901—22no—
24#18011899C2211yescharacteristic
25#18011899C2211yescharacteristic
26#18011895—22no—
27#18011891—22no—
28#18011890C2212yes—
29C4×Dic298C2212yes—
30#18011836C2212yes—
31#18011836C2212yes—
32#18011836C2212yes—
33#18011833C2212yes—
34#117×C149C2211yescharacteristic
35C4⋊C8×C149C2212yes—
36C22⋊C8×C149C2212yes—
37#97×C149C2211yescharacteristic
38C4×C8×C149C2211yescharacteristic
39#14558736—42no—
40C2×Dic298—21no—
41C2×Dic298—42no—
42C2×Dic298C2311yescharacteristic
43C2×Dic298C2312yes—
44C2×Dic298C2311yescharacteristic
45C2×Dic298C2311yescharacteristic
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—21no—
48C22×Dic149—21no—
49C2×C4×D149—21no—
50Dic596—42no—
51Dic596—42no—
52Dic596—42no—
53#14558725—42no—
54C4⋊Dic149—22no—
55C4⋊Dic149—22no—
56C4⋊Dic149—22no—
57C4⋊Dic149C2312yes—
58#14558715—22no—
59#14558715—22no—
60#14558715—22no—
61C22⋊Dic149—22no—
62#14558713—22no—
63#14558713—22no—
64C4×Dic149—22no—
65C22×C4×C149C2311yescharacteristic
66C2×C8×C149C2312yes—
67C2×C8×C149C2312yes—
68C4⋊C4×C149D412yes—
69C22⋊C4×C149D412yes—
70C42×C149C2311yescharacteristic
71C42×C149C2311yescharacteristic
72C22×D149—21no—
73Dic298—22no—
74Dic298—22no—
75Dic298—22no—
76Dic298—22no—
77Dic298—22no—
78Dic298—42no—
79D596—22no—
80C22⋊D149—42no—
81C2×Dic149—21no—
82C2×Dic149—21no—
83C2×Dic149—21no—
84C2×Dic149—22no—
85C2×Dic149—22no—
86C2×Dic149—42no—
87C4×D149—42no—
88C23×C149C2×D411yescharacteristic
89C2×C4×C149—22no—
90C2×C4×C149C2×D411yescharacteristic
91C2×C4×C149C2×D412yes—
92C2×C4×C149C2×D412yes—
93C2×C4×C149C2411yescharacteristic
94C8×C149—22no—
95C8×C149—22no—
96C22×C149—21no—
97C22×C149—22no—
98C22×C149C22×D411yescharacteristic
99C2×D149—41no—
100C2×D149—42no—
101C4×C149—21no—
102C4×C149—22no—
103C4×C149—22no—
104C4×C149H(5,𝔽2)12yes—
105Dic149—41no—
106Dic149—41no—
107Dic149—41no—
108Dic149—42no—
109Dic149—42no—
110C2×C149—41no—
111C2×C149#53511yescharacteristic
112C2×C149#57812yes—
113D149—81no—
114C149#322311yescharacteristic, complemented, Sylow
115#3223—1491nomaximal, Sylow
116#541—1491no—
117#536—1491no—
118#497—1491no—
119#494—1491no—
120#491—1491no—
121#488—1491no—
122#464—1492no—
123#462—1492no—
124#452—1492no—
125#451—1492no—
126#433D14911yescharacteristic, complemented
127C2×#41—1491no—
128C22×Q8—1491no—
129C2×Q16—1492no—
130C2×Q16—1492no—
131C2×Q16—1492no—
132C2×SD16—1492no—
133C4⋊Q8—1491no—
134C4⋊Q8—1491no—
135C4⋊Q8—1491no—
136#126—2982no—
137#125—2982no—
138#125—1491no—
139#124—1491no—
140#124—1491no—
141#123—2982no—
142C22⋊Q8—2982no—
143C22⋊Q8—1492no—
144C4×Q8—1492no—
145#117C2×D14911yescharacteristic
146C4⋊C8C2×D14912yes—
147#103—1492no—
148#103—1492no—
149#103—1492no—
150#102—1492no—
151C22⋊C8C2×D14912yes—
152#97C2×D14911yescharacteristic
153C4×C8C2×D14911yescharacteristic
154#41—5962no—
155C2×Q8—2981no—
156C2×Q8—5962no—
157C2×Q8—1491no—
158C2×Q8—1491no—
159C2×Q8—1491no—
160C2×Q8—1492no—
161C2×D4—2981no—
162C2×D4—1491no—
163C22×C4—2981no—
164C22×C4—2981no—
165C22×C4C22×D14911yescharacteristic
166Q16—5962no—
167Q16—5962no—
168Q16—5962no—
169SD16—5962no—
170C2×C8C22×D14912yes—
171C2×C8C22×D14912yes—
172C4⋊C4—2982no—
173C4⋊C4—2982no—
174C4⋊C4—2982no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4—1492no—
179C4⋊C4D59612yes—
180C22⋊C4—2982no—
181C22⋊C4—2982no—
182C22⋊C4—2982no—
183C22⋊C4D59612yes—
184C42—2982no—
185C42C22×D14911yescharacteristic
186C42C22×D14911yescharacteristic
187C23—2981no—
188C23C2×D59611yescharacteristic
189Q8—2982no—
190Q8—2982no—
191Q8—2982no—
192Q8—2982no—
193Q8—2982no—
194Q8—5962no—
195D4—2982no—
196D4—5962no—
197C2×C4—22no—
198C2×C4—2982no—
199C2×C4—2982no—
200C2×C4—2981no—
201C2×C4—2981no—
202C2×C4—2981no—
203C2×C4—5962no—
204C2×C4—5962no—
205C2×C4C2×D59612yes—
206C2×C4C2×D59612yes—
207C2×C4C2×D59611yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—22no—
210C8—22no—
211C22—22no—
212C22—21no—
213C22—5962no—
214C22—5961no—
215C22C22×D59611yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—5962no—
220C4—5961no—
221C4—5961no—
222C4—5961no—
223C4—5962no—
224C4#1801195712yes—
225C2—41no—
226C2—11921no—
227C2#2375019411yescharacteristic
228C2#2375044212yes—
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#433×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558736C22
hypercenter#433D149
hypocenterC149#3223
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #23750237C2semidirect
3. #23750019C2semidirect
4. #23750005C2semidirect
5. #23749994C2semidirect
6. #23749980C2semidirect
7. #23749848C2semidirect
8. #23749836C2semidirect
10. #23749783C2semidirect
11. #433×C149C2semidirect
114. C149#3223semidirect
126. #433D149semidirect

Maximal subgroups

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

subgroupindex
#237502372
#237502012
#237500192
#237500052
#237499942
#237499802
#237498482
#237498362
#237497892
#237497832
#433×C1492
#3223149