Finite Group Databasebeta

← #34055813

Subgroups — #34055813

Order 19072.

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

normal subgroups only

This group has 404 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#23750028C211yesmaximal, characteristic, complemented
4#23749973C211yesmaximal, characteristic, complemented
5#23749972C211yesmaximal, characteristic, complemented
6#23749940C214yesmaximal, complemented
7#23749904C214yesmaximal, complemented
8#23749676C211yesmaximal, characteristic, complemented
9#500×C149C211yesmaximal, characteristic, complemented
10C2×#14558737—21no—
11C2×Q8×D149—21no—
12#18011911C2211yescharacteristic
13#18011910—21no—
14#18011910C2212yes—
15#18011909—21no—
16#18011908C2211yescharacteristic
17#18011900—21no—
18#18011886—21no—
19#18011886C2212yes—
20#18011873—21no—
21#18011873C2212yes—
22C4⋊C4×D149—21no—
23C4⋊C4×D149C2212yes—
24C42×D149C2211yescharacteristic
25#18011844C2214yes—
26#18011840C2212yes—
27#18011833C2214yes—
28#18011831C2214yes—
29#18011823C2212yes—
30C4×#11924746C2211yescharacteristic
31C4⋊Q8×C149C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33#107×C149C2214yes—
34C4×C8×C149C2211yescharacteristic
35#14558737—44no—
36C2×Dic298—22no—
37C2×Dic298—21no—
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
48C2×#11924746C2312yes—
49C4⋊Dic149—21no—
50C4⋊Dic149C2312yes—
51#14558715—22no—
52#14558715—22no—
53#14558713—22no—
54#14558713—22no—
55C4×Dic149D411yescharacteristic
56C4×Dic149D411yescharacteristic
57C4×Dic149C2311yescharacteristic
58C2×Q8×C149—21no—
59C2×Q8×C149—21no—
60C2×C8×C149C2312yes—
61C4⋊C4×C149—21no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149C2312yes—
64C4⋊C4×C149C2312yes—
65C42×C149C2311yescharacteristic
66C22×D149—22no—
67C22×D149C2×D411yescharacteristic
68Dic298—42no—
69Dic298—44no—
70D596—24no—
71D596—44no—
72D596—42no—
73D596—42no—
74C2×Dic149—22no—
75C2×Dic149C2×D411yescharacteristic
76C2×Dic149C2×D411yescharacteristic
77C2×Dic149C2×D411yescharacteristic
78C4×D149—22no—
79C4×D149—22no—
80C4×D149—44no—
81C4×D149—44no—
82C4×D149D814yes—
83#11924746—22no—
84Q8×C149—42no—
85Q8×C149—42no—
86C2×C4×C149—22no—
87C2×C4×C149—22no—
88C2×C4×C149C2×D411yescharacteristic
89C2×C4×C149C2×D411yescharacteristic
90C2×C4×C149C2411yescharacteristic
91C8×C149—22no—
92C22×C149C22×D411yescharacteristic
93C2×D149—21no—
94C2×D149—21no—
95C2×D149—42no—
96C2×D149—42no—
97C2×D149—42no—
98C2×D149C2×D812yes—
99C4×C149—42no—
100C4×C149—42no—
101C4×C149C2×D812yes—
102C4×C149#13712yes—
103C4×C1492−1+412yes—
104Dic149—21no—
105Dic149—21no—
106Dic149—42no—
107Dic149C2×D812yes—
108C2×C149#53611yescharacteristic
109C2×C149C22×D811yescharacteristic
110C2×C149C2×#13711yescharacteristic
111D149—22no—
112D149—82no—
113C149#322611yescharacteristic, complemented, Sylow
114#3226—1491nomaximal, Sylow
115#532—1491no—
116C2×C4⋊Q8—1491no—
117#500—1491no—
118#500D14911yescharacteristic, complemented
119#486—1491no—
120#486—1491no—
121#480—1494no—
122#474—1494no—
123#431—1491no—
124C2×#41—2981no—
125C22×Q8—2981no—
126C4⋊Q8—2981no—
127C4⋊Q8—2981no—
128C4⋊Q8—1492no—
129C4⋊Q8—1491no—
130C4⋊Q8C2×D14911yescharacteristic
131C4⋊Q8C2×D14911yescharacteristic
132#127—1491no—
133#124—2981no—
134#121—2981no—
135#121—1492no—
136C4×D4—2981no—
137C4×D4—1492no—
138C2×C4⋊C4—2981no—
139C2×C4⋊C4—1492no—
140C2×C42—1491no—
141#107—1494no—
142#107C2×D14914yes—
143C4⋊C8—1492no—
144#102—1494no—
145#102—1494no—
146C22⋊C8—1492no—
147C4×C8—1491no—
148C4×C8C2×D14911yescharacteristic
149#41—5964no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—2982no—
153C2×Q8—2981no—
154C2×Q8—5964no—
155C2×D4—2982no—
156C2×D4—2981no—
157C2×D4—2981no—
158C2×D4—1492no—
159C22×C4—2982no—
160C22×C4—2982no—
161C22×C4—1491no—
162C22×C4—1491no—
163C22×C4—1491no—
164C2×C8—1492no—
165C2×C8C22×D14912yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2982no—
169C4⋊C4—2982no—
170C4⋊C4—2981no—
171C4⋊C4—1492no—
172C4⋊C4C22×D14912yes—
173C4⋊C4C22×D14912yes—
174C22⋊C4—2982no—
175C22⋊C4—2982no—
176C42—1491no—
177C42—1491no—
178C42—1491no—
179C42C22×D14911yescharacteristic
180C23—2982no—
181C23—1491no—
182Q8—42no—
183Q8—42no—
184Q8—5964no—
185Q8—5962no—
186D4—2984no—
187D4—5964no—
188D4—5962no—
189D4—5962no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2982no—
193C2×C4—2982no—
194C2×C4—2982no—
195C2×C4—5964no—
196C2×C4—5964no—
197C2×C4—1491no—
198C2×C4—1491no—
199C2×C4—1491no—
200C2×C4—1494no—
201C2×C4D4×D14911yescharacteristic
202C2×C4D4×D14911yescharacteristic
203C2×C4C23×D14911yescharacteristic
204C8—22no—
205C8—2982no—
206C22—2981no—
207C22—2981no—
208C22—5962no—
209C22—5962no—
210C22—5962no—
211C22—1492no—
212C22C2×D4×D14911yescharacteristic
213C4—42no—
214C4—42no—
215C4—2981no—
216C4—2981no—
217C4—5962no—
218C4—1492no—
219C4D8×D14912yes—
220C4#1801194312yes—
221C4#1801195912yes—
222C2—2982no—
223C2—11922no—
224C2#2375020011yescharacteristic
225C2C2×D8×D14911yescharacteristic
226C2C2×#1801194311yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#500×C149C2
socleC22×C149C22×D4
prime radical#23750176C2
hypercenter#500D149
hypocenterC149#3226
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D149C2semidirect
3. #23750028C2semidirect
4. #23749973C2semidirect
5. #23749972C2semidirect
6. #23749940C2semidirect
7. #23749904C2semidirect
8. #23749676C2semidirect
9. #500×C149C2semidirect
113. C149#3226semidirect
118. #500D149semidirect

Maximal subgroups

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

subgroupindex
#237501762
C4⋊Q8×D1492
#237500282
#237499732
#237499722
#237499402
#237499042
#237496762
#500×C1492
#3226149