Finite Group Databasebeta

← #34056035

Subgroups — #34056035

Order 19072.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#23750180C211yesmaximal, characteristic
2#23750179C211yesmaximal, characteristic
3#23749919C212yesmaximal, complemented
4#23749917C211yesmaximal, characteristic, complemented
5#23749867C212yesmaximal
6#23749865C212yesmaximal
7#23749836C211yesmaximal, characteristic, complemented
8#23749835C211yesmaximal, characteristic, complemented
9#23749834C212yesmaximal, complemented
10C4⋊C8×D149C211yesmaximal, characteristic
11#477×C149C211yesmaximal, characteristic, complemented
12C2×Dic596C2212yes—
13C2×#14558726C2212yes—
14C2×C8×D149C2212yes—
15#18011911C2211yescharacteristic
16#18011910C2211yescharacteristic
17#18011909C2211yescharacteristic
18#18011902—21no—
19#18011901—21no—
20#18011891C2211yescharacteristic
21#18011891C2212yes—
22#18011889—21no—
23#18011887—21no—
24#18011887C2211yescharacteristic
25Q8×Dic149—21no—
26C4×Dic298—21no—
27C4×Dic298C2211yescharacteristic
28#18011876C2212yes—
29#18011869—21no—
30#18011868—21no—
31C42×D149C2211yescharacteristic
32#18011844C2212yes—
33C4⋊#11924746C2211yescharacteristic
34#18011840C2212yes—
35#18011836C2212yes—
36#18011834C2212yes—
37#18011834C2212yes—
38C4⋊Q8×C149C2211yescharacteristic
39C4×Q8×C149C2211yescharacteristic
40#107×C149C2212yes—
41C4⋊C8×C149C2211yescharacteristic
42#103×C149C2212yes—
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45C2×Dic298C2312yes—
46C2×C4×D149D411yescharacteristic
47C2×C4×D149D411yescharacteristic
48C2×C4×D149C2311yescharacteristic
49Dic596—42no—
50#14558726—42no—
51C2×#11924746C2312yes—
52C8×D149—42no—
53C4⋊Dic149—21no—
54C4⋊Dic149—22no—
55C4⋊Dic149—21no—
56C4⋊Dic149—21no—
57C4⋊Dic149C2311yescharacteristic
58#14558715—21no—
59#14558715—21no—
60#14558715—21no—
61#14558715—22no—
62#14558715—21no—
63#14558715—21no—
64#14558713—21no—
65#14558713—21no—
66#14558713—22no—
67C4×Dic149—22no—
68C4×Dic149—21no—
69C4×Dic149D411yescharacteristic
70C4×Dic149D411yescharacteristic
71C4×Dic149C2311yescharacteristic
72C2×Q8×C149—21no—
73C2×Q8×C149C2311yescharacteristic
74C2×C8×C149C2312yes—
75C4⋊C4×C149—21no—
76C4⋊C4×C149—21no—
77C4⋊C4×C149C2311yescharacteristic
78C4⋊C4×C149C2312yes—
79C42×C149—21no—
80C42×C149C2311yescharacteristic
81C22×D149C2×D411yescharacteristic
82Dic298—22no—
83Dic298—21no—
84Dic298—21no—
85Dic298—22no—
86Dic298—42no—
87C2×Dic149—21no—
88C2×Dic149—21no—
89C2×Dic149—22no—
90C2×Dic149C2×D411yescharacteristic
91C2×Dic149C2×D411yescharacteristic
92C2×Dic149C2×D411yescharacteristic
93C4×D149—21no—
94C4×D149—21no—
95C4×D149—41no—
96C4×D149Q1614yescomplemented
97#11924746—22no—
98Q8×C149—21no—
99Q8×C149—21no—
100Q8×C149—42no—
101C2×C4×C149—21no—
102C2×C4×C149—21no—
103C2×C4×C149—22no—
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2411yescharacteristic
107C8×C149—22no—
108C22×C149C22×D411yescharacteristic
109C2×D149—21no—
110C2×D149—21no—
111C2×D149C2×Q1612yes—
112C4×C149—21no—
113C4×C149—41no—
114C4×C149—42no—
115C4×C149—41no—
116C4×C149C2×Q1612yes—
117C4×C149H(5,𝔽2)11yescharacteristic
118C4×C1492−1+411yescharacteristic
119Dic149—21no—
120Dic149—21no—
121Dic149—41no—
122Dic149—42no—
123Dic149—41no—
124Dic149C2×Q1612yes—
125C2×C149#53711yescharacteristic
126C2×C149C22×Q1611yescharacteristic
127C2×C149#57511yescharacteristic
128D149—22no—
129C149#324911yescharacteristic, complemented, Sylow
130#3249—1491nomaximal, Sylow
131#533—1491no—
132#533—1491no—
133#477—1491no—
134#477—1492no—
135#477D14911yescharacteristic, complemented
136#467—1492no—
137#467—1492no—
138#462—1491no—
139#462—1491no—
140#462—1492no—
141C2×C4⋊C8—1491no—
142C2×Q16—1492no—
143C2×Q16—1492no—
144C22×C8—1492no—
145C4⋊Q8—1491no—
146C4⋊Q8—1491no—
147C4⋊Q8—1491no—
148C4⋊Q8C2×D14911yescharacteristic
149#125—2981no—
150#125—2981no—
151C22⋊Q8—2981no—
152C22⋊Q8—2981no—
153C22⋊Q8—1491no—
154C22⋊Q8—1491no—
155C22⋊Q8—1492no—
156C4×Q8—2981no—
157C4×Q8—2981no—
158C4×Q8—1492no—
159C4×Q8—1491no—
160C4×Q8C2×D14911yescharacteristic
161#117—2981no—
162#117—2981no—
163C2×C42—1491no—
164#107—1492no—
165#107C2×D14912yes—
166C4⋊C8—1492no—
167C4⋊C8—1491no—
168C4⋊C8C2×D14911yescharacteristic
169#103—1492no—
170#103—1492no—
171#103—1492no—
172#103C2×D14912yes—
173C2×Q8—21no—
174C2×Q8—2981no—
175C2×Q8—1491no—
176C2×Q8—1492no—
177C2×Q8C22×D14911yescharacteristic
178C22×C4—1491no—
179C22×C4—1491no—
180C22×C4—1491no—
181Q16—5962no—
182Q16—5962no—
183C2×C8—5962no—
184C2×C8—1492no—
185C2×C8C22×D14912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2981no—
189C4⋊C4—2981no—
190C4⋊C4—2981no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—2982no—
194C4⋊C4—2982no—
195C4⋊C4—2981no—
196C4⋊C4—2981no—
197C4⋊C4—2981no—
198C4⋊C4—1491no—
199C4⋊C4C22×D14911yescharacteristic
200C4⋊C4C22×D14912yes—
201C22⋊C4—2981no—
202C22⋊C4—2981no—
203C22⋊C4—2982no—
204C42—21no—
205C42—2982no—
206C42—2981no—
207C42—1491no—
208C42—1491no—
209C42—1491no—
210C42C22×D14911yescharacteristic
211C23—1491no—
212Q8—21no—
213Q8—21no—
214Q8—42no—
215Q8—2982no—
216Q8—2981no—
217Q8—2981no—
218Q8—2982no—
219Q8—5962no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—2981no—
224C2×C4—2981no—
225C2×C4—2981no—
226C2×C4—2982no—
227C2×C4—2981no—
228C2×C4—5961no—
229C2×C4—1491no—
230C2×C4—1491no—
231C2×C4—1491no—
232C2×C4—1494no—
233C2×C4D4×D14911yescharacteristic
234C2×C4D4×D14911yescharacteristic
235C2×C4C23×D14911yescharacteristic
236C8—22no—
237C8—2982no—
238C22—2981no—
239C22—2981no—
240C22—1492no—
241C22C2×D4×D14911yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—2981no—
247C4—2981no—
248C4—5961no—
249C4—5961no—
250C4—5962no—
251C4—1492no—
252C4Q16×D14912yes—
253C4#1801195711yescharacteristic
254C4#1801195911yescharacteristic
255C2—2982no—
256C2#2375020511yescharacteristic
257C2C2×Q16×D14911yescharacteristic
258C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#477×C149C2
socleC22×C149C22×D4
prime radicalC22×D149C2×D4
hypercenter#477D149
hypocenterC149#3249
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #23749919C2semidirect
4. #23749917C2semidirect
7. #23749836C2semidirect
8. #23749835C2semidirect
9. #23749834C2semidirect
11. #477×C149C2semidirect
96. C4×D149Q16semidirect
129. C149#3249semidirect
135. #477D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237501792
#237499192
#237499172
#237498672
#237498652
#237498362
#237498352
#237498342
C4⋊C8×D1492
#477×C1492
#3249149