Finite Group Databasebeta

← #34057340

Subgroups — #34057340

Order 19072.

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

normal subgroups only

This group has 350 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#23750348C211yesmaximal, characteristic, complemented
2#23749982C212yesmaximal, complemented
3#23749977C211yesmaximal, characteristic, complemented
4#23749929C212yesmaximal, complemented
5#23749912C211yesmaximal, characteristic, complemented
6#23749840C212yesmaximal
7#23749830C211yesmaximal, characteristic, complemented
8Q8×#11924746C211yesmaximal, characteristic, complemented
9C4×#14558724C212yesmaximal, complemented
10C4×#14558721C211yesmaximal, characteristic
11#554×C149C211yesmaximal, characteristic, complemented
12C2×#14558724C2212yes—
13C2×#14558721C2211yescharacteristic
14#18011911C2211yescharacteristic
15#18011906—22no—
16#18011899C2212yes—
17#18011887—21no—
18C4×Dic298C2212yes—
19C4×D596C2211yescharacteristic
20#18011869—21no—
21#18011844C2211yescharacteristic
22#18011842C2212yes—
23C4⋊#11924746C2211yescharacteristic
24C4⋊#11924746C2212yes—
25#18011834C2212yes—
26#18011834C2212yes—
27#18011834C2212yes—
28#18011832C2211yescharacteristic
29#18011831C2212yes—
30#18011831C2211yescharacteristic
31C4×#11924746C2211yescharacteristic
32C4×#11924746C2212yes—
33C4⋊Q8×C149C2211yescharacteristic
34#125×C149C2212yes—
35C22⋊Q8×C149—21no—
36C22⋊Q8×C149—22no—
37C4×Q8×C149C2211yescharacteristic
38C4×D4×C149C2211yescharacteristic
39C4×D4×C149C2212yes—
40C2×C4⋊C4×C149—21no—
41C2×Dic298C2312yes—
42C2×D596C2311yescharacteristic
43C2×C4×D149—21no—
44#14558724—24no—
45#14558721—22no—
46C2×#11924746C2311yescharacteristic
47C2×#11924746C2312yes—
48C2×#11924746C2311yescharacteristic
49C4⋊Dic149—21no—
50C4⋊Dic149—21no—
51C4⋊Dic149C2311yescharacteristic
52#14558715—22no—
53#14558713—21no—
54#14558713—22no—
55#14558713—22no—
56C4×Dic149—22no—
57C2×Q8×C149—21no—
58C2×Q8×C149D411yescharacteristic
59C2×D4×C149C2311yescharacteristic
60C22×C4×C149—21no—
61C22×C4×C149—22no—
62C4⋊C4×C149—22no—
63C4⋊C4×C149—21no—
64C4⋊C4×C149—22no—
65C4⋊C4×C149D412yes—
66C4⋊C4×C149D411yescharacteristic
67C4⋊C4×C149C2311yescharacteristic
68C4⋊C4×C149C2312yes—
69C22⋊C4×C149—21no—
70C22⋊C4×C149—22no—
71C42×C149C2311yescharacteristic
72C42×C149C2312yes—
73C22×D149—21no—
74Dic298—22no—
75Dic298—22no—
76D596—21no—
77D596—21no—
78C2×Dic149—21no—
79C2×Dic149—22no—
80C4×D149—42no—
81#11924746—21no—
82#11924746—22no—
83#11924746—21no—
84C23×C149—21no—
85Q8×C149—22no—
86Q8×C149—81no—
87D4×C149—21no—
88D4×C149#4112yes—
89C2×C4×C149—21no—
90C2×C4×C149—22no—
91C2×C4×C149—42no—
92C2×C4×C149—82no—
93C2×C4×C149C2×D412yes—
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2×D412yes—
96C2×C4×C149C2×D411yescharacteristic
97C2×C4×C149C2411yescharacteristic
98C22×C149—22no—
99C22×C149—41no—
100C22×C149—41no—
101C22×C149C22×D411yescharacteristic
102C2×D149—41no—
103C2×D149—41no—
104C2×D149—41no—
105C4×C149—22no—
106C4×C149—22no—
107C4×C149—21no—
108C4×C149—41no—
109C4×C149—42no—
110C4×C149#13512yes—
111C4×C149C2×#4111yescharacteristic
112C4×C1492−1+411yescharacteristic
113Dic149—42no—
114Dic149—41no—
115C2×C149—42no—
116C2×C149#54811yescharacteristic
117C2×C149C2×#13511yescharacteristic
118C2×C149#57711yescharacteristic
119D149—81no—
120C149#336011yescharacteristic, complemented, Sylow
121#3360—1491nomaximal, Sylow
122#556—1491no—
123#554D14911yescharacteristic, complemented
124#488—1492no—
125#487—1491no—
126#479—1492no—
127#476—1491no—
128#463—1492no—
129#461—1491no—
130C8×Q8—1491no—
131C4×SD16—1492no—
132C4×D8—1491no—
133C2×SD16—1492no—
134C2×D8—1491no—
135C4⋊Q8—1491no—
136C4⋊Q8C2×D14911yescharacteristic
137#126—2982no—
138#125C2×D14912yes—
139#124—1492no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—2981no—
143C4×Q8—1492no—
144C4×Q8C2×D14911yescharacteristic
145C4×D4—1491no—
146C4×D4C2×D14911yescharacteristic
147C4×D4C2×D14912yes—
148#117—2981no—
149C2×C4⋊C4—21no—
150#107—1491no—
151#106—1492no—
152C4⋊C8—1491no—
153C4⋊C8—1492no—
154#103—1492no—
155#103—1492no—
156#103—1492no—
157#102—1491no—
158#102—1491no—
159#102—1492no—
160C4×C8—1491no—
161C4×C8—1492no—
162C2×Q8—21no—
163C2×Q8—1492no—
164C2×Q8C22⋊D14911yescharacteristic
165C2×D4—1491no—
166C2×D4C22×D14911yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—2981no—
170SD16—2984no—
171D8—2982no—
172C2×C8—1491no—
173C2×C8—1492no—
174C2×C8—1491no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—2981no—
179C4⋊C4—2982no—
180C4⋊C4—2981no—
181C4⋊C4—1491no—
182C4⋊C4C22⋊D14911yescharacteristic
183C4⋊C4C22⋊D14912yes—
184C4⋊C4C22×D14912yes—
185C4⋊C4C22×D14911yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—2982no—
189C22⋊C4—2981no—
190C22⋊C4—2982no—
191C42—2982no—
192C42C22×D14911yescharacteristic
193C42C22×D14912yes—
194C23—21no—
195C23—2981no—
196Q8—22no—
197Q8—81no—
198Q8—2982no—
199Q8—2982no—
200D4—21no—
201D4—2981no—
202D4—2981no—
203D4#1455873712yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—2981no—
209C2×C4—2982no—
210C2×C4—5962no—
211C2×C4C2×C22⋊D14912yes—
212C2×C4C2×C22⋊D14912yes—
213C2×C4C2×C22⋊D14911yescharacteristic
214C2×C4C2×C22⋊D14911yescharacteristic
215C2×C4C23×D14911yescharacteristic
216C8—2981no—
217C8—2982no—
218C8—2981no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—5961no—
223C22—5961no—
224C22—5961no—
225C22C22×C22⋊D14911yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—5961no—
232C4—5962no—
233C4#1801193112yes—
234C4C2×#1455873711yescharacteristic
235C4#1801195911yescharacteristic
236C2—42no—
237C2—11921no—
238C2#2375028611yescharacteristic
239C2C2×#1801193111yescharacteristic
240C2#2375043611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#554×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558721C22
hypercenter#554D149
hypocenterC149#3360
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #23750348C2semidirect
2. #23749982C2semidirect
3. #23749977C2semidirect
4. #23749929C2semidirect
5. #23749912C2semidirect
7. #23749830C2semidirect
8. Q8×#11924746C2semidirect
9. C4×#14558724C2semidirect
11. #554×C149C2semidirect
120. C149#3360semidirect
123. #554D149semidirect

Maximal subgroups

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

subgroupindex
#237503482
#237499822
#237499772
#237499292
#237499122
#237498402
#237498302
Q8×#119247462
C4×#145587242
C4×#145587212
#554×C1492
#3360149