Finite Group Databasebeta

← #34057303

Subgroups — #34057303

Order 19072.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#23750349C211yesmaximal, characteristic
2#23750335C211yesmaximal, characteristic
3#23750021C211yesmaximal, characteristic, complemented
4#23749965C212yesmaximal, complemented
5#23749957C212yesmaximal, complemented
6#23749921C211yesmaximal, characteristic, complemented
7#23749882C212yesmaximal, complemented
8C4×#14558726C211yesmaximal, characteristic, complemented
9#23749702C211yesmaximal, characteristic
10#23749664C212yesmaximal, complemented
11#475×C149C211yesmaximal, characteristic, complemented
12C2×#14558726C2211yescharacteristic
13C2×#14558719C2212yes—
14#18011911C2211yescharacteristic
15#18011906—22no—
16#18011903—21no—
17#18011900—21no—
18#18011891C2212yes—
19#18011889—22no—
20#18011887C2212yes—
21Q8×Dic149—21no—
22C4×Dic298C2211yescharacteristic
23C4×D596—21no—
24C4×D596C2211yescharacteristic
25#18011869C2212yes—
26C4⋊C4×D149C2212yes—
27#18011845C2212yes—
28#18011844C2211yescharacteristic
29#18011842C2212yes—
30#18011842C2212yes—
31#18011836C2212yes—
32#18011835C2211yescharacteristic
33#18011834C2211yescharacteristic
34#18011823C2212yes—
35C4×#11924746C2211yescharacteristic
36C4⋊Q8×C149C2211yescharacteristic
37C4×Q8×C149C2211yescharacteristic
38#106×C149C2212yes—
39C4⋊C8×C149C2211yescharacteristic
40#103×C149C2212yes—
41C2×Dic298—21no—
42C2×Dic298C2311yescharacteristic
43C2×D596D411yescharacteristic
44C2×C4×D149—22no—
45C2×C4×D149C2312yes—
46#14558726—22no—
47#14558719—24no—
48C2×#11924746C2311yescharacteristic
49C2×#11924746C2311yescharacteristic
50C4⋊Dic149—22no—
51C4⋊Dic149—21no—
52C4⋊Dic149D411yescharacteristic
53C4⋊Dic149C2312yes—
54C4⋊Dic149C2311yescharacteristic
55#14558715—22no—
56#14558715—22no—
57#14558715—21no—
58#14558713—22no—
59#14558713—22no—
60#14558713—22no—
61#14558713D412yes—
62C4×Dic149—22no—
63C4×Dic149—21no—
64C2×Q8×C149—21no—
65C2×Q8×C149C2311yescharacteristic
66C2×C8×C149C2312yes—
67C4⋊C4×C149—21no—
68C4⋊C4×C149—21no—
69C4⋊C4×C149C2312yes—
70C4⋊C4×C149C2311yescharacteristic
71C42×C149—21no—
72C42×C149C2311yescharacteristic
73C22×D149C2×D412yes—
74Dic298—21no—
75Dic298—21no—
76Dic298—81no—
77D596—22no—
78C2×Dic149—22no—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C2×Dic149C2×D412yes—
82C4×D149—22no—
83C4×D149—22no—
84C4×D149—82no—
85#11924746—21no—
86#11924746#4112yes—
87Q8×C149—21no—
88Q8×C149—21no—
89Q8×C149—42no—
90C2×C4×C149—21no—
91C2×C4×C149—21no—
92C2×C4×C149—22no—
93C2×C4×C149C2×D411yescharacteristic
94C2×C4×C149C2×D411yescharacteristic
95C2×C4×C149C2411yescharacteristic
96C8×C149—22no—
97C22×C149C22×D411yescharacteristic
98C2×D149—22no—
99C2×D149—22no—
100C2×D149—22no—
101C4×C149—21no—
102C4×C149—41no—
103C4×C149—42no—
104C4×C149—41no—
105C4×C149#13712yes—
106C4×C149C2×#4111yescharacteristic
107C4×C1492−1+411yescharacteristic
108Dic149—22no—
109Dic149—41no—
110Dic149—42no—
111Dic149—41no—
112C2×C149#54711yescharacteristic
113C2×C149C2×#13711yescharacteristic
114C2×C149#57711yescharacteristic
115D149—42no—
116C149#335611yescharacteristic, complemented, Sylow
117#3356—1491nomaximal, Sylow
118#556—1491no—
119#554—1491no—
120#498—1491no—
121#485—1492no—
122#483—1492no—
123#477—1491no—
124#475D14911yescharacteristic, complemented
125#470—1492no—
126C4×Q16—1491no—
127#436—1491no—
128#428—1492no—
129C2×Q16—1491no—
130C2×M16—1492no—
131C4⋊Q8—1491no—
132C4⋊Q8C2×D14911yescharacteristic
133#126—2982no—
134#125—2981no—
135#124—2981no—
136C22⋊Q8—2982no—
137C22⋊Q8—1492no—
138C22⋊Q8—1492no—
139C4×Q8—2981no—
140C4×Q8—1491no—
141C4×Q8C2×D14911yescharacteristic
142C4×D4—2981no—
143C4×D4—1491no—
144#117—1492no—
145C2×C4⋊C4—1492no—
146#107—1492no—
147#107—1491no—
148#106—1492no—
149#106—1492no—
150#106C2×D14912yes—
151C4⋊C8C2×D14911yescharacteristic
152#103—1491no—
153#103—1492no—
154#103—1491no—
155#103C2×D14912yes—
156C22⋊C8—1492no—
157C4×C8—1491no—
158C2×Q8—21no—
159C2×Q8—2981no—
160C2×Q8—1491no—
161C2×Q8C22×D14911yescharacteristic
162C2×D4—1491no—
163C22×C4—2982no—
164C22×C4—1492no—
165Q16—2982no—
166M16—2984no—
167C2×C8—1491no—
168C2×C8—1491no—
169C2×C8C22×D14912yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—2982no—
173C4⋊C4—2982no—
174C4⋊C4—2982no—
175C4⋊C4—2981no—
176C4⋊C4—2981no—
177C4⋊C4—1492no—
178C4⋊C4—1491no—
179C4⋊C4—1491no—
180C4⋊C4C22×D14911yescharacteristic
181C4⋊C4C22×D14912yes—
182C22⋊C4—2982no—
183C22⋊C4—2982no—
184C22⋊C4—2982no—
185C22⋊C4—1492no—
186C42—21no—
187C42—2982no—
188C42—2981no—
189C42C22×D14911yescharacteristic
190C23—1492no—
191Q8—21no—
192Q8—21no—
193Q8—42no—
194Q8—2981no—
195Q8—2981no—
196Q8—11921no—
197D4—2982no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—2982no—
202C2×C4—2982no—
203C2×C4—2981no—
204C2×C4—2981no—
205C2×C4—2982no—
206C2×C4—1492no—
207C2×C4—11922no—
208C2×C4D4×D14911yescharacteristic
209C2×C4D4×D14911yescharacteristic
210C2×C4C23×D14911yescharacteristic
211C8—22no—
212C8—2981no—
213C8—1492no—
214C22—2982no—
215C22—2982no—
216C22—2982no—
217C22C2×D4×D14911yescharacteristic
218C4—21no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—2982no—
223C4—5962no—
224C4—5961no—
225C4—5961no—
226C4#1801194012yes—
227C4#41×D14911yescharacteristic
228C4#1801195911yescharacteristic
229C2—5962no—
230C2#2375027711yescharacteristic
231C2C2×#1801194011yescharacteristic
232C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#475×C149C2
socleC22×C149C22×D4
prime radicalC2×D596D4
hypercenter#475D149
hypocenterC149#3356
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #23750021C2semidirect
4. #23749965C2semidirect
5. #23749957C2semidirect
6. #23749921C2semidirect
7. #23749882C2semidirect
8. C4×#14558726C2semidirect
10. #23749664C2semidirect
11. #475×C149C2semidirect
116. C149#3356semidirect
124. #475D149semidirect

Maximal subgroups

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

subgroupindex
#237503492
#237503352
#237500212
#237499652
#237499572
#237499212
#237498822
C4×#145587262
#237497022
#237496642
#475×C1492
#3356149