Finite Group Databasebeta

← #34055617

Subgroups — #34055617

Order 19072.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750176C211yesmaximal, characteristic
2C4⋊Q8×D149C211yesmaximal, characteristic, complemented
3#23749999C211yesmaximal, characteristic, complemented
4#23749998C211yesmaximal, characteristic, complemented
5#23749977C211yesmaximal, characteristic, complemented
6#23749829C214yesmaximal, complemented
7#23749765C214yesmaximal, complemented
8#23749676C211yesmaximal, characteristic, complemented
9#487×C149C211yesmaximal, characteristic, complemented
10C2×#14558737—21no—
11C2×Q8×D149C2212yes—
12C2×#14558725C2214yes—
13C2×#14558723C2214yes—
14#18011911C2211yescharacteristic
15#18011910—21no—
16#18011910—21no—
17#18011909C2212yes—
18#18011908C2211yescharacteristic
19#18011900—21no—
20#18011886—21no—
21#18011886C2212yes—
22#18011873—21no—
23#18011873C2212yes—
24C4⋊C4×D149—21no—
25C4⋊C4×D149—21no—
26C42×D149C2211yescharacteristic
27#18011840C2212yes—
28#18011834C2214yes—
29#18011823C2212yes—
30C4×#11924746C2211yescharacteristic
31C4⋊Q8×C149C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33#103×C149C2214yes—
34C4×C8×C149C2211yescharacteristic
35#14558737—44no—
36C2×Dic298—22no—
37C2×Dic298C2312yes—
38Q8×D149—44no—
39C2×D596—22no—
40C2×D596—21no—
41C2×D596—21no—
42C2×D596C2312yes—
43C2×C4×D149—22no—
44C2×C4×D149—22no—
45C2×C4×D149D411yescharacteristic
46C2×C4×D149D411yescharacteristic
47C2×C4×D149C2311yescharacteristic
48#14558725—44no—
49#14558723—44no—
50C2×#11924746C2312yes—
51C4⋊Dic149—21no—
52C4⋊Dic149—21no—
53#14558715—22no—
54#14558715—22no—
55#14558713—22no—
56#14558713—22no—
57C4×Dic149D411yescharacteristic
58C4×Dic149D411yescharacteristic
59C4×Dic149C2311yescharacteristic
60C2×Q8×C149—21no—
61C2×Q8×C149C2312yes—
62C2×C8×C149C2312yes—
63C4⋊C4×C149—21no—
64C4⋊C4×C149—21no—
65C4⋊C4×C149—21no—
66C4⋊C4×C149C2312yes—
67C42×C149C2311yescharacteristic
68C22×D149—22no—
69C22×D149C2×D411yescharacteristic
70Dic298—24no—
71Dic298—44no—
72D596—24no—
73D596—44no—
74D596—42no—
75D596—42no—
76C2×Dic149—22no—
77C2×Dic149C2×D411yescharacteristic
78C2×Dic149C2×D411yescharacteristic
79C2×Dic149C2×D411yescharacteristic
80C4×D149—22no—
81C4×D149—22no—
82C4×D149—44no—
83C4×D149—44no—
84C4×D149SD1614yescomplemented
85#11924746—22no—
86Q8×C149—24no—
87Q8×C149—42no—
88C2×C4×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149C2×D411yescharacteristic
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2411yescharacteristic
93C8×C149—22no—
94C22×C149C22×D411yescharacteristic
95C2×D149—21no—
96C2×D149—21no—
97C2×D149—42no—
98C2×D149—42no—
99C2×D149—42no—
100C2×D149C2×SD1612yes—
101C4×C149—42no—
102C4×C149—42no—
103C4×C149C2×SD1612yes—
104C4×C149#13712yes—
105C4×C149H(5,𝔽2)12yes—
106Dic149—21no—
107Dic149—21no—
108Dic149—42no—
109Dic149C2×SD1612yes—
110C2×C149#53511yescharacteristic
111C2×C149C22×SD1611yescharacteristic
112C2×C149C2×#13711yescharacteristic
113D149—22no—
114D149—82no—
115C149#320611yescharacteristic, complemented, Sylow
116#3206—1491nomaximal, Sylow
117#532—1491no—
118C2×C4⋊Q8—1491no—
119#492—1491no—
120#492—1491no—
121#487—1491no—
122#487D14911yescharacteristic, complemented
123#461—1494no—
124#448—1494no—
125#431—1491no—
126C2×#41—2981no—
127C22×Q8—1492no—
128C2×SD16—1494no—
129C2×SD16—1494no—
130C4⋊Q8—2981no—
131C4⋊Q8—2981no—
132C4⋊Q8—1492no—
133C4⋊Q8—1491no—
134C4⋊Q8C2×D14911yescharacteristic
135C4⋊Q8C2×D14911yescharacteristic
136#127—1491no—
137#124—2981no—
138#121—2981no—
139#121—1492no—
140C4×D4—2981no—
141C4×D4—1492no—
142C2×C4⋊C4—2981no—
143C2×C4⋊C4—2981no—
144C2×C42—1491no—
145C4⋊C8—1492no—
146#103—1494no—
147#103C2×D14914yes—
148C22⋊C8—1492no—
149C4×C8—1491no—
150C4×C8C2×D14911yescharacteristic
151#41—5964no—
152C2×Q8—21no—
153C2×Q8—2982no—
154C2×Q8—5964no—
155C2×Q8—1492no—
156C2×Q8C22×D14912yes—
157C2×D4—2982no—
158C2×D4—2981no—
159C2×D4—2981no—
160C2×D4—1492no—
161C22×C4—2982no—
162C22×C4—2982no—
163C22×C4—1491no—
164C22×C4—1491no—
165C22×C4—1491no—
166SD16—5964no—
167SD16—5964no—
168C2×C8—1492no—
169C2×C8C22×D14912yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—2982no—
174C4⋊C4—2982no—
175C4⋊C4—2981no—
176C4⋊C4—2981no—
177C4⋊C4C22×D14912yes—
178C22⋊C4—2982no—
179C22⋊C4—2982no—
180C42—1491no—
181C42—1491no—
182C42—1491no—
183C42C22×D14911yescharacteristic
184C23—2982no—
185C23—1491no—
186Q8—24no—
187Q8—42no—
188Q8—2984no—
189Q8—5964no—
190D4—2984no—
191D4—5964no—
192D4—5962no—
193D4—5962no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—2982no—
197C2×C4—2982no—
198C2×C4—2982no—
199C2×C4—5964no—
200C2×C4—5964no—
201C2×C4—1491no—
202C2×C4—1491no—
203C2×C4—1491no—
204C2×C4—1494no—
205C2×C4D4×D14911yescharacteristic
206C2×C4D4×D14911yescharacteristic
207C2×C4C23×D14911yescharacteristic
208C8—22no—
209C8—2982no—
210C22—2981no—
211C22—2981no—
212C22—5962no—
213C22—5962no—
214C22—5962no—
215C22—1492no—
216C22C2×D4×D14911yescharacteristic
217C4—42no—
218C4—42no—
219C4—2981no—
220C4—2981no—
221C4—5962no—
222C4—1492no—
223C4SD16×D14912yes—
224C4#1801194312yes—
225C4#1801195712yes—
226C2—2982no—
227C2—11922no—
228C2#2375019311yescharacteristic
229C2C2×SD16×D14911yescharacteristic
230C2C2×#1801194311yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#487×C149C2
socleC22×C149C22×D4
prime radical#23750176C2
hypercenter#487D149
hypocenterC149#3206
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D149C2semidirect
3. #23749999C2semidirect
4. #23749998C2semidirect
5. #23749977C2semidirect
6. #23749829C2semidirect
7. #23749765C2semidirect
8. #23749676C2semidirect
9. #487×C149C2semidirect
84. C4×D149SD16semidirect
115. C149#3206semidirect
122. #487D149semidirect

Maximal subgroups

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

subgroupindex
#237501762
C4⋊Q8×D1492
#237499992
#237499982
#237499772
#237498292
#237497652
#237496762
#487×C1492
#3206149