Finite Group Databasebeta

← #34056044

Subgroups — #34056044

Order 19072.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#23750161C212yesmaximal, complemented
2#23749934C212yesmaximal, complemented
3#23749928C212yesmaximal, complemented
4#23749861C211yesmaximal, characteristic
5#23749856C212yesmaximal, complemented
6#23749849C212yesmaximal, complemented
7#23749843C212yesmaximal, complemented
8C2×#18011840C211yesmaximal, characteristic, complemented
9#467×C149C211yesmaximal, characteristic, complemented
10C2×#14558723C2212yes—
11C2×D1192C2211yescharacteristic
12C22×#11924746C2211yescharacteristic
13#18011906—22no—
14#18011902C2212yes—
15#18011900C2212yes—
16#18011895—22no—
17#18011885C2212yes—
18Q8×Dic149C2212yes—
19#18011873C2212yes—
20#18011871—22no—
21#18011869—22no—
22C2×C4×Dic149C2211yescharacteristic
23#18011845C2211yescharacteristic
24#18011842C2212yes—
25#18011840C2212yes—
26#18011840C2212yes—
27#18011835C2212yes—
28#18011833C2212yes—
29#18011831C2212yes—
30C2×Q16×C149C2211yescharacteristic
31C22×C8×C149C2211yescharacteristic
32C22⋊Q8×C149C2212yes—
33#107×C149C2211yescharacteristic
34#103×C149C2212yes—
35C2×D596C2312yes—
36C2×C22⋊D149—22no—
37C22×Dic149D411yescharacteristic
38C22×Dic149D411yescharacteristic
39C2×C4×D149—22no—
40#14558723—42no—
41D1192—41no—
42C2×#11924746—41no—
43C2×#11924746C2312yes—
44C4⋊Dic149—22no—
45C4⋊Dic149—22no—
46C4⋊Dic149C2312yes—
47#14558715—22no—
48#14558713—22no—
49#14558713—22no—
50#14558713—22no—
51#14558713—22no—
52C4×Dic149—22no—
53C4×Dic149D412yes—
54C4×Dic149C2312yes—
55C2×Q8×C149C2312yes—
56C22×C4×C149C2311yescharacteristic
57Q16×C149—41no—
58C2×C8×C149—41no—
59C2×C8×C149C2311yescharacteristic
60C2×C8×C149C2311yescharacteristic
61C4⋊C4×C149—22no—
62C4⋊C4×C149C2312yes—
63C22⋊C4×C149—22no—
64C22×D149—22no—
65D596—22no—
66D596—22no—
67C22⋊D149—44no—
68C2×Dic149—22no—
69C2×Dic149—22no—
70C2×Dic149—41no—
71C2×Dic149C2×D412yes—
72C2×Dic149C2×D412yes—
73C4×D149—44no—
74#11924746—22no—
75C23×C149C2×D411yescharacteristic
76Q8×C149—22no—
77Q8×C149—22no—
78C2×C4×C149—21no—
79C2×C4×C149—21no—
80C2×C4×C149—22no—
81C2×C4×C149—22no—
82C2×C4×C149C2×D411yescharacteristic
83C2×C4×C149C2411yescharacteristic
84C8×C149—21no—
85C8×C149—21no—
86C22×C149—21no—
87C22×C149—21no—
88C22×C149#13512yes—
89C22×C149C22×D411yescharacteristic
90C2×D149—42no—
91C2×D149—42no—
92C2×D149—42no—
93C4×C149—21no—
94C4×C149—42no—
95C4×C149—42no—
96C4×C149H(5,𝔽2)11yescharacteristic
97C4×C1492−1+411yescharacteristic
98Dic149—22no—
99Dic149—22no—
100Dic149—42no—
101C2×C149—22no—
102C2×C149#53711yescharacteristic
103C2×C149C2×#13511yescharacteristic
104C2×C149#57511yescharacteristic
105D149—82no—
106C149#325011yescharacteristic, complemented, Sylow
107#3250—1491nomaximal, Sylow
108#529—1492no—
109#479—1492no—
110#478—1492no—
111#467D14911yescharacteristic, complemented
112#466—1491no—
113#465—1492no—
114#464—1492no—
115#463—1492no—
116C2×C4⋊C8—1491no—
117C2×Q16C2×D14911yescharacteristic
118C2×SD16—1492no—
119C2×D8—1491no—
120C22×C8—1491no—
121C22×C8C2×D14911yescharacteristic
122#126—2982no—
123#125—1492no—
124#124—1492no—
125#123—2982no—
126C22⋊Q8C2×D14912yes—
127#121—1492no—
128C4×Q8—1492no—
129C4×D4—2982no—
130C4×D4—1492no—
131#117—2982no—
132C2×C42—1491no—
133#107—1491no—
134#107C2×D14911yescharacteristic
135#106—1492no—
136C4⋊C8—1492no—
137C4⋊C8—1492no—
138#103—1492no—
139#103C2×D14912yes—
140#102—1492no—
141#102—1492no—
142C2×Q8C22×D14912yes—
143C2×D4—2982no—
144C2×D4—1492no—
145C22×C4—2982no—
146C22×C4—1491no—
147C22×C4—1491no—
148C22×C4C22×D14911yescharacteristic
149Q16—41no—
150SD16—5962no—
151D8—5961no—
152C2×C8—41no—
153C2×C8—5961no—
154C2×C8—1492no—
155C2×C8C22×D14911yescharacteristic
156C2×C8C22×D14911yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—2982no—
159C4⋊C4—2982no—
160C4⋊C4—2982no—
161C4⋊C4—1492no—
162C4⋊C4C22×D14912yes—
163C22⋊C4—22no—
164C22⋊C4—2982no—
165C22⋊C4—2982no—
166C22⋊C4—2982no—
167C22⋊C4—2982no—
168C42—2982no—
169C42—1492no—
170C42—1492no—
171C23—2982no—
172C23D4×D14911yescharacteristic
173Q8—22no—
174Q8—22no—
175D4—2982no—
176D4—2982no—
177D4—5964no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2982no—
183C2×C4—2982no—
184C2×C4—5964no—
185C2×C4—5961no—
186C2×C4—1492no—
187C2×C4—1492no—
188C2×C4D4×D14911yescharacteristic
189C2×C4C23×D14911yescharacteristic
190C8—21no—
191C8—21no—
192C8—2982no—
193C22—21no—
194C22—21no—
195C22—5962no—
196C22—5962no—
197C22—5962no—
198C22#1801193412yes—
199C22C2×D4×D14911yescharacteristic
200C4—21no—
201C4—42no—
202C4—42no—
203C4—2982no—
204C4—2982no—
205C4—5962no—
206C4#1801195611yescharacteristic
207C4#1801195811yescharacteristic
208C2—22no—
209C2—11922no—
210C2#2375020211yescharacteristic
211C2C2×#1801193411yescharacteristic
212C2#2375042211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#467×C149C2
socleC22×C149C22×D4
prime radical#23749861C2
hypercenter#467D149
hypocenterC149#3250
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23750161C2semidirect
2. #23749934C2semidirect
3. #23749928C2semidirect
5. #23749856C2semidirect
6. #23749849C2semidirect
7. #23749843C2semidirect
8. C2×#18011840C2semidirect
9. #467×C149C2semidirect
106. C149#3250semidirect
111. #467D149semidirect

Maximal subgroups

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

subgroupindex
#237501612
#237499342
#237499282
#237498612
#237498562
#237498492
#237498432
C2×#180118402
#467×C1492
#3250149