Finite Group Databasebeta

← #34045052

Subgroups — #34045052

Order 19072.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#23750425C211yesmaximal, characteristic
2#23750391C211yesmaximal, characteristic, complemented
3#23749643C211yesmaximal, characteristic, complemented
4#23749619C211yesmaximal, characteristic, complemented
5#23749593C211yesmaximal, characteristic, complemented
6#23749389C211yesmaximal, characteristic, complemented
7#344×C149C211yesmaximal, characteristic, complemented
8C2×#14558736C2211yescharacteristic
9C2×#14558736C2211yescharacteristic
10#18011944—21no—
11#18011944C412yescomplemented
12#18011939—21no—
13#18011939C412yescomplemented
14#18011933—22no—
15#18011919—22no—
16#18011917—22no—
17C2×#14558720C2211yescharacteristic
18M16×D149—22no—
19C22×#11924746C2211yescharacteristic
20#18011839—22no—
21#18011834—21no—
22#18011831—21no—
23#18011823—22no—
24C2×M16×C149C2211yescharacteristic
25C2×M16×C149C2211yescharacteristic
26#117×C149C2211yescharacteristic
27#14558736—21no—
28#14558736—21no—
29#14558736—21no—
30#14558736—21no—
31#14558736—21no—
32#14558736—21no—
33#14558736C2×C412yes—
34#14558736D412yes—
35C2×Dic298C2×C411yescharacteristic
36C2×Dic298D411yescharacteristic
37C2×D596C2×C411yescharacteristic
38C2×D596D411yescharacteristic
39C2×C22⋊D149—21no—
40C2×C22⋊D149—21no—
41C2×C4×D149—21no—
42C2×C4×D149—21no—
43Dic596—22no—
44#14558725—22no—
45#14558725—22no—
46D1192—22no—
47#14558720—21no—
48#14558720D412yes—
49#14558719—24no—
50C2×#11924746—21no—
51C2×#11924746—21no—
52C2×#11924746—21no—
53C2×#11924746—21no—
54C8×D149—24no—
55C22×C4×C149C2311yescharacteristic
56M16×C149—21no—
57M16×C149—21no—
58M16×C149—21no—
59M16×C149C2×C412yes—
60C2×C8×C149—21no—
61C2×C8×C149D412yes—
62C4⋊C4×C149—21no—
63C22⋊C4×C149—21no—
64C42×C149—21no—
65C22×D149—21no—
66C22×D149—21no—
67Dic298—21no—
68Dic298—21no—
69Dic298—21no—
70Dic298C22⋊C412yes—
71D596—21no—
72D596—21no—
73D596—21no—
74D596C22⋊C412yes—
75C22⋊D149—22no—
76C22⋊D149—22no—
77C22⋊D149—41no—
78C22⋊D149—41no—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C4×D149—22no—
82C4×D149—22no—
83C4×D149—41no—
84C4×D149—41no—
85#11924746—22no—
86#11924746—41no—
87C23×C149#4111yescharacteristic
88C2×C4×C149—41no—
89C2×C4×C149C22×C411yescharacteristic
90C2×C4×C149C2×D411yescharacteristic
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2×D411yescharacteristic
93C2×C4×C149C2×D411yescharacteristic
94C2×C4×C149#4111yescharacteristic
95C8×C149—22no—
96C8×C149—22no—
97C22×C149—41no—
98C22×C149C4×D411yescharacteristic
99C22×C149#12111yescharacteristic
100C22×C149#12311yescharacteristic
101C2×D149—22no—
102C2×D149—42no—
103C2×D149—41no—
104C2×D149—81no—
105C4×C149—81no—
106C4×C149C2×C22⋊C411yescharacteristic
107C4×C149C4×D411yescharacteristic
108C4×C149#12011yescharacteristic
109C4×C149#12111yescharacteristic
110Dic149—22no—
111Dic149—41no—
112C2×C149—21no—
113C2×C149—21no—
114C2×C149—21no—
115C2×C149#38611yescharacteristic
116D149—42no—
117D149—81no—
118C149#188111yescharacteristic, complemented, Sylow
119#1881—1491nomaximal, Sylow
120#575—1491no—
121#568—1491no—
122#421—1491no—
123#416—1491no—
124#408—1491no—
125#344D14911yescharacteristic, complemented
126#341—1491no—
127C2×#41—1491no—
128C2×#41—1491no—
129#137—2981no—
130#137—1492no—
131AGL(1,ℤ/8)—2981no—
132AGL(1,ℤ/8)—1492no—
133#135—2982no—
134#131—2982no—
135#131—2982no—
136C2×M16—2982no—
137C2×M16—1491no—
138C2×M16C2×D14911yescharacteristic
139C2×M16C2×D14911yescharacteristic
140C22×C8—1491no—
141#117C2×D14911yescharacteristic
142#104—2982no—
143#103—2981no—
144#102—2981no—
145C22⋊C8—2982no—
146#41—2981no—
147#41—2981no—
148#41—2981no—
149#41—2981no—
150#41—2981no—
151#41—2981no—
152#41—1492no—
153#41—1492no—
154C2×Q8—1491no—
155C2×Q8—1491no—
156C2×D4—2981no—
157C2×D4—2981no—
158C2×D4—1491no—
159C2×D4—1491no—
160C22×C4—2981no—
161C22×C4—2981no—
162C22×C4C22×D14911yescharacteristic
163Q16—2982no—
164SD16—2982no—
165SD16—2982no—
166D8—2982no—
167M16—21no—
168M16—21no—
169M16—21no—
170M16—2984no—
171M16—2981no—
172M16—1492no—
173M16C4×D14912yes—
174C2×C8—21no—
175C2×C8—2984no—
176C2×C8—2981no—
177C2×C8—2981no—
178C2×C8—2981no—
179C2×C8—2981no—
180C2×C8D59612yes—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—2981no—
185C23—2981no—
186C23#1455873611yescharacteristic
187Q8—2981no—
188Q8—2981no—
189Q8—2981no—
190Q8—1492no—
191D4—2982no—
192D4—2982no—
193D4—2981no—
194D4—2981no—
195D4—2981no—
196D4—5961no—
197D4—5961no—
198D4—1492no—
199C2×C4—41no—
200C2×C4—2981no—
201C2×C4—2982no—
202C2×C4—2982no—
203C2×C4—2981no—
204C2×C4—5961no—
205C2×C4—5961no—
206C2×C4C2×C4×D14911yescharacteristic
207C2×C4D4×D14911yescharacteristic
208C2×C4D4×D14911yescharacteristic
209C2×C4D4×D14911yescharacteristic
210C2×C4C2×D59611yescharacteristic
211C2×C4#1455873711yescharacteristic
212C8—22no—
213C8—22no—
214C8—2982no—
215C8—5961no—
216C22—41no—
217C22—2982no—
218C22—5962no—
219C22—5961no—
220C22—11921no—
221C22C4×D59611yescharacteristic
222C22#1801188411yescharacteristic
223C22#1801189611yescharacteristic
224C4—81no—
225C4—2982no—
226C4—5961no—
227C4C22⋊C4×D14911yescharacteristic
228C4#1801187311yescharacteristic
229C4#1801188111yescharacteristic
230C4#1801188611yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—5962no—
235C2—11921no—
236C2#2374950311yescharacteristic
subgroupG/H
centerC4#18011881
commutatorC2×C4×C149C22×C4
FrattiniC22×C4C22×D149
Fitting#344×C149C2
socleC2×C149#386
prime radical#23750425C2
cosocleC22×C4×C149C23
hypercenter#344D149
hypocenterC149#1881
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #23750391C2semidirect
3. #23749643C2semidirect
4. #23749619C2semidirect
5. #23749593C2semidirect
6. #23749389C2semidirect
7. #344×C149C2semidirect
11. #18011944C4semidirect
13. #18011939C4semidirect
118. C149#1881semidirect
125. #344D149semidirect

Maximal subgroups

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

subgroupindex
#237504252
#237503912
#237496432
#237496192
#237495932
#237493892
#344×C1492
#1881149