Finite Group Databasebeta

← #34055767

Subgroups — #34055767

Order 19072.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#23750236C211yesmaximal, characteristic, complemented
2#23750200C211yesmaximal, characteristic
3#23750017C211yesmaximal, characteristic, complemented
4#23749993C211yesmaximal, characteristic, complemented
5#23749980C211yesmaximal, characteristic, complemented
6#23749849C212yesmaximal, complemented
7#23749833C212yesmaximal, complemented
8#23749786C212yesmaximal
9#23749779C212yesmaximal, complemented
10#23749684C211yesmaximal, characteristic, complemented
11#494×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2211yescharacteristic
13C2×Q8×D149C2211yescharacteristic
14C2×#14558726C2212yes—
15C2×#14558723C2212yes—
16#18011909C2211yescharacteristic
17#18011906—22no—
18#18011903C2211yescharacteristic
19#18011901—22no—
20#18011900C2211yescharacteristic
21#18011899C2211yescharacteristic
22#18011896—22no—
23#18011889—22no—
24#18011887C2212yes—
25Q8×Dic149C2212yes—
26#18011868C2211yescharacteristic
27#18011840C2212yes—
28#18011836C2212yes—
29#18011835C2212yes—
30#18011835C2212yes—
31#18011833C2212yes—
32#18011823C2212yes—
33#18011822C2211yescharacteristic
34C2×Q16×C149C2212yes—
35C2×Q16×C149C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4⋊Q8×C149C2211yescharacteristic
38C4×C8×C149C2211yescharacteristic
39#14558737—42no—
40C2×Dic298—21no—
41C2×Dic298C2311yescharacteristic
42Q8×D149—42no—
43C2×D596—21no—
44C2×D596C2311yescharacteristic
45C2×C4×D149—21no—
46C2×C4×D149—21no—
47C2×C4×D149C2311yescharacteristic
48#14558726—42no—
49#14558723—42no—
50C2×#11924746C2312yes—
51C4⋊Dic149—22no—
52C4⋊Dic149C2312yes—
53#14558715—22no—
54#14558715—22no—
55#14558715—22no—
56#14558715D412yes—
57#14558713—22no—
58#14558713—22no—
59#14558713—22no—
60#14558713D412yes—
61C4×Dic149—22no—
62C4×Dic149C2311yescharacteristic
63C2×Q8×C149C2311yescharacteristic
64C2×Q8×C149C2312yes—
65C2×Q8×C149C2311yescharacteristic
66Q16×C149—42no—
67Q16×C149—42no—
68C2×C8×C149C2312yes—
69C4⋊C4×C149—22no—
70C4⋊C4×C149—22no—
71C42×C149C2311yescharacteristic
72C22×D149—21no—
73C22×D149C2×D411yescharacteristic
74Dic298—22no—
75Dic298—42no—
76D596—22no—
77D596—42no—
78C2×Dic149—21no—
79C2×Dic149—22no—
80C2×Dic149C2×D411yescharacteristic
81C2×Dic149C2×D412yes—
82C4×D149—22no—
83C4×D149—42no—
84C4×D149—42no—
85#11924746—22no—
86Q8×C149—22no—
87Q8×C149—22no—
88Q8×C149—22no—
89Q8×C149—22no—
90C2×C4×C149—21no—
91C2×C4×C149—21no—
92C2×C4×C149—22no—
93C2×C4×C149C2×D412yes—
94C2×C4×C149C2411yescharacteristic
95C8×C149—22no—
96C22×C149C22×D411yescharacteristic
97C2×D149—21no—
98C2×D149—22no—
99C2×D149—41no—
100C2×D149—42no—
101C4×C149—22no—
102C4×C149—42no—
103C4×C149—41no—
104C4×C149—41no—
105C4×C149H(5,𝔽2)12yes—
106Dic149—21no—
107Dic149—22no—
108Dic149—41no—
109Dic149—42no—
110C2×C149#53511yescharacteristic
111C2×C149#57812yes—
112D149—41no—
113D149—81no—
114C149#322311yescharacteristic, complemented, Sylow
115#3223—1491nomaximal, Sylow
116#541—1491no—
117#536—1491no—
118#497—1491no—
119#494D14911yescharacteristic, complemented
120#491—1491no—
121#488—1491no—
122#464—1492no—
123#462—1492no—
124#452—1492no—
125#451—1492no—
126#433—1491no—
127C2×#41—1491no—
128C22×Q8—1491no—
129C2×Q16—1492no—
130C2×Q16C2×D14912yes—
131C2×Q16C2×D14912yes—
132C2×SD16—1492no—
133C4⋊Q8—1491no—
134C4⋊Q8C2×D14911yescharacteristic
135C4⋊Q8C2×D14911yescharacteristic
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#117—1491no—
146C4⋊C8—1492no—
147#103—1492no—
148#103—1492no—
149#103—1492no—
150#102—1492no—
151C22⋊C8—1492no—
152#97—1491no—
153C4×C8C2×D14911yescharacteristic
154#41—5962no—
155C2×Q8—2981no—
156C2×Q8—5962no—
157C2×Q8—1491no—
158C2×Q8C22×D14911yescharacteristic
159C2×Q8C22×D14911yescharacteristic
160C2×Q8C22×D14912yes—
161C2×D4—2981no—
162C2×D4—1491no—
163C22×C4—2981no—
164C22×C4—2981no—
165C22×C4—1491no—
166Q16—42no—
167Q16—42no—
168Q16—5962no—
169SD16—5962no—
170C2×C8—1492no—
171C2×C8C22×D14912yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—2982no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4—1492no—
179C4⋊C4—1492no—
180C22⋊C4—2982no—
181C22⋊C4—2982no—
182C22⋊C4—2982no—
183C22⋊C4—1492no—
184C42—2982no—
185C42—1491no—
186C42C22×D14911yescharacteristic
187C23—2981no—
188C23—1491no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—2982no—
194Q8—5962no—
195D4—2982no—
196D4—5962no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—2982no—
201C2×C4—2982no—
202C2×C4—2981no—
203C2×C4—5962no—
204C2×C4—5962no—
205C2×C4—1492no—
206C2×C4—1491no—
207C2×C4D4×D14912yes—
208C2×C4C23×D14911yescharacteristic
209C8—22no—
210C8—2982no—
211C22—2982no—
212C22—2981no—
213C22—5962no—
214C22—5961no—
215C22C2×D4×D14911yescharacteristic
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—2982no—
221C4—2981no—
222C4—5962no—
223C4—5961no—
224C4#1801195612yes—
225C2—5961no—
226C2—11921no—
227C2#2375018811yescharacteristic
228C2#2375043812yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#494×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558737C22
hypercenter#494D149
hypocenterC149#3223
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #23750236C2semidirect
3. #23750017C2semidirect
4. #23749993C2semidirect
5. #23749980C2semidirect
6. #23749849C2semidirect
7. #23749833C2semidirect
9. #23749779C2semidirect
10. #23749684C2semidirect
11. #494×C149C2semidirect
114. C149#3223semidirect
119. #494D149semidirect

Maximal subgroups

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

subgroupindex
#237502362
#237502002
#237500172
#237499932
#237499802
#237498492
#237498332
#237497862
#237497792
#237496842
#494×C1492
#3223149