Finite Group Databasebeta

← #34055774

Subgroups — #34055774

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#23750020C211yesmaximal, characteristic, complemented
3#23750004C211yesmaximal, characteristic, complemented
4#23749993C211yesmaximal, characteristic, complemented
5#23749979C211yesmaximal, characteristic, complemented
6#23749844C212yesmaximal, complemented
7#23749832C212yesmaximal, complemented
8#23749787C212yesmaximal
9#23749781C212yesmaximal, complemented
10#23749686C211yesmaximal, characteristic, complemented
11#536×C149C211yesmaximal, characteristic
12C2×#14558726C2212yes—
13C2×#14558726C2212yes—
14C2×#14558726C2212yes—
15C2×#14558724C2212yes—
16#18011911C2211yescharacteristic
17#18011904—22no—
18#18011903C2211yescharacteristic
19#18011901—22no—
20#18011890C2212yes—
21C4×Dic298C2212yes—
22C4⋊#11924746C2212yes—
23#18011835C2212yes—
24#18011835C2212yes—
25#18011835C2212yes—
26#18011832C2212yes—
27C22⋊#11924746C2212yes—
28#18011822C2211yescharacteristic
29C4×#11924746C2211yescharacteristic
30C2×#41×C149C2211yescharacteristic
31C22×Q8×C149C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33C4⋊Q8×C149C2211yescharacteristic
34#124×C149C2211yescharacteristic
35#124×C149C2211yescharacteristic
36#123×C149—22no—
37C22⋊Q8×C149—22no—
38#117×C149C2211yescharacteristic
39C2×Dic298C2312yes—
40#14558726—42no—
41#14558726—42no—
42#14558726—42no—
43#14558724—42no—
44C2×#11924746C2312yes—
45C2×#11924746C2312yes—
46C4⋊Dic149—22no—
47C4⋊Dic149C2312yes—
48#14558715—22no—
49#14558715—22no—
50#14558715—22no—
51C22⋊Dic149—22no—
52C4×Dic149—22no—
53#41×C149—42no—
54C2×Q8×C149—21no—
55C2×Q8×C149—42no—
56C2×Q8×C149C2311yescharacteristic
57C2×Q8×C149C2311yescharacteristic
58C2×Q8×C149C2311yescharacteristic
59C2×D4×C149—21no—
60C2×D4×C149C2311yescharacteristic
61C22×C4×C149—21no—
62C22×C4×C149—21no—
63C22×C4×C149C2311yescharacteristic
64C4⋊C4×C149—22no—
65C4⋊C4×C149—22no—
66C4⋊C4×C149D412yes—
67C22⋊C4×C149—22no—
68C22⋊C4×C149—22no—
69C22⋊C4×C149D412yes—
70C42×C149C2311yescharacteristic
71C42×C149C2311yescharacteristic
72Dic298—22no—
73Dic298—22no—
74C2×Dic149—22no—
75C2×Dic149—22no—
76#11924746—22no—
77#11924746—22no—
78C23×C149—21no—
79C23×C149C2×D411yescharacteristic
80Q8×C149—22no—
81Q8×C149—22no—
82Q8×C149—22no—
83Q8×C149—42no—
84D4×C149—22no—
85D4×C149—42no—
86C2×C4×C149—22no—
87C2×C4×C149—21no—
88C2×C4×C149—21no—
89C2×C4×C149—21no—
90C2×C4×C149—42no—
91C2×C4×C149—42no—
92C2×C4×C149C2×D412yes—
93C2×C4×C149C2×D412yes—
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2411yescharacteristic
96C22×C149—21no—
97C22×C149—22no—
98C22×C149—41no—
99C22×C149—42no—
100C22×C149C22×D411yescharacteristic
101C4×C149—21no—
102C4×C149—22no—
103C4×C149—22no—
104C4×C149—41no—
105C4×C149—41no—
106C4×C149—41no—
107C4×C149H(5,𝔽2)12yes—
108Dic149—42no—
109Dic149—42no—
110C2×C149—41no—
111C2×C149—81no—
112C2×C149#53511yescharacteristic
113C2×C149#57812yes—
114C149#322311yescharacteristic, complemented, Sylow
115#3223—1491nomaximal, Sylow
116#541—1491no—
117#536D14911yescharacteristic
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#433—1491no—
127C2×#41C2×D14911yescharacteristic
128C22×Q8C2×D14911yescharacteristic
129C2×Q16—1492no—
130C2×Q16—1492no—
131C2×Q16—1492no—
132C2×SD16—1492no—
133C4⋊Q8—1491no—
134C4⋊Q8C2×D14911yescharacteristic
135C4⋊Q8C2×D14911yescharacteristic
136#126—2982no—
137#125—2982no—
138#125—1491no—
139#124C2×D14911yescharacteristic
140#124C2×D14911yescharacteristic
141#123—22no—
142C22⋊Q8—22no—
143C22⋊Q8—1492no—
144C4×Q8—1492no—
145#117C2×D14911yescharacteristic
146C4⋊C8—1492no—
147#103—1492no—
148#103—1492no—
149#103—1492no—
150#102—1492no—
151C22⋊C8—1492no—
152#97—1491no—
153C4×C8—1491no—
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—1492no—
158C2×Q8C22×D14911yescharacteristic
159C2×Q8C22×D14911yescharacteristic
160C2×Q8C22×D14911yescharacteristic
161C2×D4—21no—
162C2×D4C22×D14911yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D14911yescharacteristic
166Q16—5962no—
167Q16—5962no—
168Q16—5962no—
169SD16—5962no—
170C2×C8—1492no—
171C2×C8—1492no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—2982no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4—1492no—
179C4⋊C4C22⋊D14912yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—2982no—
183C22⋊C4C22⋊D14912yes—
184C42—2982no—
185C42C22×D14911yescharacteristic
186C42C22×D14911yescharacteristic
187C23—21no—
188C23C2×C22⋊D14911yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—2982no—
194Q8—2982no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—2982no—
204C2×C4—2982no—
205C2×C4C2×C22⋊D14912yes—
206C2×C4C2×C22⋊D14912yes—
207C2×C4C2×C22⋊D14911yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—2982no—
210C8—2982no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D14911yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—5962no—
223C4—5962no—
224C4#1801195612yes—
225C2—41no—
226C2—81no—
227C2#2375019211yescharacteristic
228C2#2375044012yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#536×C149C2
socleC22×C149C22×D4
prime radicalC2×#41×C149C22
hypercenter#536D149
hypocenterC149#3223
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #23750237C2semidirect
2. #23750020C2semidirect
3. #23750004C2semidirect
4. #23749993C2semidirect
5. #23749979C2semidirect
6. #23749844C2semidirect
7. #23749832C2semidirect
9. #23749781C2semidirect
10. #23749686C2semidirect
114. C149#3223semidirect

Maximal subgroups

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

subgroupindex
#237502372
#237500202
#237500042
#237499932
#237499792
#237498442
#237498322
#237497872
#237497812
#237496862
#536×C1492
#3223149