Finite Group Databasebeta

← #34054218

Subgroups — #34054218

Order 19072.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#18011942C211yesmaximal, characteristic
2#23750199C211yesmaximal, characteristic
3#23750103C211yesmaximal, characteristic, complemented
4#23749849C214yesmaximal, complemented
5#23749833C214yesmaximal, complemented
6#23749651C211yesmaximal, characteristic, complemented
7#23749634C211yesmaximal, characteristic, complemented
8#23749622C211yesmaximal, characteristic
9C22×Q16×C149C211yesmaximal, characteristic, complemented
10C2×#14558736C2211yescharacteristic
11#18011942—24no—
12C2×#14558726C2212yes—
13C2×#14558723C2212yes—
14C2×#14558720C2211yescharacteristic
15#18011909C2212yes—
16#18011900C2212yes—
17#18011894—22no—
18#18011887—22no—
19Q8×Dic149—22no—
20Q8×Dic149C2214yes—
21#18011870—22no—
22#18011870C2211yescharacteristic
23#18011870C2211yescharacteristic
24#18011840C2214yes—
25#18011836C2212yes—
26#18011835C2214yes—
27#18011833C2212yes—
28C22×Q8×C149C2211yescharacteristic
29C22×Q8×C149C2211yescharacteristic
30C2×Q16×C149—24no—
31C2×Q16×C149C2214yes—
32C22×C8×C149C2211yescharacteristic
33#14558736—22no—
34#14558736—22no—
35C2×Dic298C2311yescharacteristic
36C2×D596C2311yescharacteristic
37C2×C22⋊D149—21no—
38C2×C4×D149—21no—
39#14558726—24no—
40#14558723—24no—
41#14558720—22no—
42C2×#11924746C2312yes—
43C4⋊Dic149—24no—
44C4⋊Dic149D412yes—
45C4⋊Dic149C2312yes—
46#14558715—24no—
47C22⋊Dic149—21no—
48C22⋊Dic149D412yes—
49#14558713—24no—
50C4×Dic149—24no—
51C4×Dic149—21no—
52C4×Dic149C2312yes—
53C2×Q8×C149—21no—
54C2×Q8×C149—21no—
55C2×Q8×C149—22no—
56C2×Q8×C149—22no—
57C2×Q8×C149—22no—
58C2×Q8×C149D414yes—
59C2×Q8×C149C2312yes—
60C2×Q8×C149C2312yes—
61C22×C4×C149—21no—
62C22×C4×C149—21no—
63C22×C4×C149C2311yescharacteristic
64Q16×C149—28no—
65C2×C8×C149—22no—
66C2×C8×C149C2312yes—
67C22×D149—21no—
68Dic298—21no—
69Dic298—21no—
70D596—21no—
71D596—21no—
72C22⋊D149—42no—
73C2×Dic149—21no—
74C2×Dic149—22no—
75C2×Dic149C2×D414yes—
76C4×D149—42no—
77#11924746—22no—
78C23×C149C2×D411yescharacteristic
79Q8×C149—24no—
80Q8×C149—22no—
81Q8×C149—22no—
82Q8×C149—22no—
83Q8×C149—42no—
84Q8×C149C2×D414yes—
85C2×C4×C149—24no—
86C2×C4×C149—22no—
87C2×C4×C149—22no—
88C2×C4×C149—42no—
89C2×C4×C149C2×D411yescharacteristic
90C2×C4×C149C2×D412yes—
91C2×C4×C149#4112yes—
92C2×C4×C149C2411yescharacteristic
93C8×C149—22no—
94C22×C149—22no—
95C22×C149#12112yes—
96C22×C149C22×D411yescharacteristic
97C2×D149—41no—
98C2×D149—42no—
99C4×C149—24no—
100C4×C149—42no—
101C4×C149#12112yes—
102C4×C149C22×D411yescharacteristic
103C4×C149C2×#4111yescharacteristic
104Dic149—24no—
105Dic149—41no—
106Dic149—42no—
107C2×C149—22no—
108C2×C149C2×#12111yescharacteristic
109C2×C149#57812yes—
110D149—81no—
111C149#304411yescharacteristic, complemented, Sylow
112#3044—1491nomaximal, Sylow
113C2×#137—1491no—
114C22×Q16D14911yescharacteristic, complemented
115#536—1491no—
116#519—1491no—
117#464—1494no—
118#462—1494no—
119#424—1491no—
120#419—1491no—
121#416—1491no—
122C2×#41—1491no—
123C22×Q8C2×D14911yescharacteristic
124C22×Q8C2×D14911yescharacteristic
125#137—2984no—
126C2×Q16—24no—
127C2×Q16—1492no—
128C2×Q16C2×D14914yes—
129C2×SD16—1492no—
130C2×M16—1491no—
131C22×C8C2×D14911yescharacteristic
132C4⋊Q8—1492no—
133#124—1492no—
134#123—2982no—
135C22⋊Q8—2982no—
136C4×Q8—2982no—
137C4×Q8—1494no—
138#117—2982no—
139#117—1491no—
140#117—1491no—
141C4⋊C8—1494no—
142#103—1494no—
143#103—1492no—
144#102—1492no—
145#41—2982no—
146#41—2982no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1491no—
153C2×Q8C22⋊D14914yes—
154C2×Q8C22×D14912yes—
155C2×Q8C22×D14912yes—
156C2×D4—2981no—
157C2×D4—1491no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2981no—
161C22×C4C22×D14911yescharacteristic
162Q16—28no—
163Q16—2984no—
164SD16—2984no—
165M16—2982no—
166C2×C8—22no—
167C2×C8—1492no—
168C2×C8C22×D14912yes—
169C4⋊C4—2984no—
170C4⋊C4—2984no—
171C4⋊C4—1492no—
172C4⋊C4—1492no—
173C22⋊C4—2984no—
174C22⋊C4—2981no—
175C22⋊C4—1492no—
176C42—2984no—
177C42—2981no—
178C42—1492no—
179C23—2981no—
180C23D4×D14911yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—2981no—
187Q8—2981no—
188Q8C2×C22⋊D14914yes—
189D4—2981no—
190D4—2981no—
191D4—5962no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—2982no—
197C2×C4—2981no—
198C2×C4—5962no—
199C2×C4—1494no—
200C2×C4D4×D14911yescharacteristic
201C2×C4C2×C22⋊D14912yes—
202C2×C4#1455873512yes—
203C2×C4C23×D14911yescharacteristic
204C8—22no—
205C8—2982no—
206C22—22no—
207C22—5962no—
208C22—5961no—
209C22#1801188312yes—
210C22C2×D4×D14911yescharacteristic
211C4—24no—
212C4—42no—
213C4—2984no—
214C4—5962no—
215C4—5961no—
216C4#1801188312yes—
217C4C22×C22⋊D14911yescharacteristic
218C4C2×#1455873511yescharacteristic
219C2—22no—
220C2—11921no—
221C2C2×#1801188311yescharacteristic
222C2#2375043812yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC22×Q16×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558736C22
hypercenterC22×Q16D149
hypocenterC149#3044
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #23750103C2semidirect
4. #23749849C2semidirect
5. #23749833C2semidirect
6. #23749651C2semidirect
7. #23749634C2semidirect
9. C22×Q16×C149C2semidirect
111. C149#3044semidirect
114. C22×Q16D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119422
#237501992
#237501032
#237498492
#237498332
#237496512
#237496342
#237496222
C22×Q16×C1492
#3044149