Finite Group Databasebeta

← #34055714

Subgroups — #34055714

Order 19072.

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

normal subgroups only

This group has 386 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
1C4⋊Q8×D149C211yesmaximal, characteristic, complemented
2#23750154C211yesmaximal, characteristic
3#23750020C211yesmaximal, characteristic, complemented
4#23749994C211yesmaximal, characteristic, complemented
5#23749993C211yesmaximal, characteristic, complemented
6#23749833C212yesmaximal, complemented
7#23749828C212yesmaximal, complemented
8#23749779C212yesmaximal, complemented
9#23749762C212yesmaximal
10#23749680C211yesmaximal, characteristic, complemented
11#497×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149C2212yes—
13C2×#14558726C2212yes—
14C2×#14558724C2212yes—
15#18011911C2211yescharacteristic
16#18011910—21no—
17#18011910—21no—
18#18011909C2212yes—
19#18011904—21no—
20#18011903C2211yescharacteristic
21#18011893—21no—
22#18011887C2211yescharacteristic
23#18011882C2211yescharacteristic
24Q8×Dic149C2211yescharacteristic
25D4×Dic149C2211yescharacteristic
26#18011871—21no—
27#18011867—21no—
28C4⋊C4×D149—21no—
29C4⋊C4×D149—21no—
30C42×D149C2211yescharacteristic
31#18011840C2212yes—
32#18011836C2212yes—
33#18011836C2212yes—
34#18011835C2212yes—
35#18011835C2212yes—
36#18011823C2212yes—
37#18011822C2211yescharacteristic
38C2×Q16×C149C2212yes—
39C2×SD16×C149C2212yes—
40C4⋊Q8×C149C2211yescharacteristic
41#124×C149C2211yescharacteristic
42#97×C149C2211yescharacteristic
43C2×Dic298—22no—
44C2×Dic298C2312yes—
45Q8×D149—42no—
46Q8×D149—42no—
47C2×C22⋊D149—21no—
48C22×Dic149—21no—
49C2×C4×D149—22no—
50C2×C4×D149D411yescharacteristic
51C2×C4×D149D411yescharacteristic
52C2×C4×D149C2311yescharacteristic
53#14558726—42no—
54#14558724—42no—
55C2×#11924746C2312yes—
56C4⋊Dic149—21no—
57C4⋊Dic149—21no—
58C4⋊Dic149—21no—
59C4⋊Dic149C2311yescharacteristic
60C4⋊Dic149C2311yescharacteristic
61#14558715—21no—
62#14558715—22no—
63#14558715—22no—
64#14558715—21no—
65#14558715—21no—
66C22⋊Dic149—21no—
67C22⋊Dic149—21no—
68#14558713—21no—
69C4×Dic149—21no—
70C4×Dic149D411yescharacteristic
71C4×Dic149D411yescharacteristic
72C4×Dic149C2311yescharacteristic
73C2×Q8×C149C2312yes—
74C2×Q8×C149C2311yescharacteristic
75C2×D4×C149C2311yescharacteristic
76Q16×C149—42no—
77SD16×C149—42no—
78C2×C8×C149C2312yes—
79C4⋊C4×C149—21no—
80C4⋊C4×C149—21no—
81C22⋊C4×C149—21no—
82C22⋊C4×C149—21no—
83C42×C149C2311yescharacteristic
84C22×D149C2×D411yescharacteristic
85Dic298—22no—
86Dic298—22no—
87Dic298—44no—
88C22⋊D149—42no—
89C2×Dic149—22no—
90C2×Dic149—21no—
91C2×Dic149—21no—
92C2×Dic149—42no—
93C2×Dic149C2×D411yescharacteristic
94C2×Dic149C2×D411yescharacteristic
95C2×Dic149C2×D411yescharacteristic
96C4×D149—22no—
97C4×D149—22no—
98C4×D149—22no—
99C4×D149—44no—
100#11924746—22no—
101C23×C149—21no—
102Q8×C149—22no—
103Q8×C149—22no—
104Q8×C149—22no—
105D4×C149—22no—
106C2×C4×C149—22no—
107C2×C4×C149—21no—
108C2×C4×C149C2×D411yescharacteristic
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149C2411yescharacteristic
111C8×C149—22no—
112C22×C149—41no—
113C22×C149—41no—
114C22×C149—41no—
115C22×C149C22×D411yescharacteristic
116C2×D149—21no—
117C2×D149—21no—
118C2×D149#13712yes—
119C4×C149—21no—
120C4×C149—21no—
121C4×C149—41no—
122C4×C149—42no—
123C4×C149H(5,𝔽2)12yes—
124Dic149—21no—
125Dic149—21no—
126Dic149—42no—
127Dic149—41no—
128Dic149—41no—
129Dic149#13712yes—
130C2×C149—81no—
131C2×C149#53511yescharacteristic
132C2×C149C2×#13711yescharacteristic
133C2×C149C2×#13711yescharacteristic
134D149—22no—
135C149#321811yescharacteristic, complemented, Sylow
136#3218—1491nomaximal, Sylow
137C2×C4⋊Q8—1491no—
138#529—1491no—
139#497—1491no—
140#497D14911yescharacteristic, complemented
141#491—1491no—
142#491—1491no—
143#462—1492no—
144#461—1492no—
145#451—1492no—
146#448—1492no—
147#432—1491no—
148C22×Q8—1492no—
149C2×Q16—1492no—
150C2×Q16C2×D14912yes—
151C2×SD16—1492no—
152C2×SD16C2×D14912yes—
153C4⋊Q8—2981no—
154C4⋊Q8—2981no—
155C4⋊Q8—1492no—
156C4⋊Q8—1491no—
157C4⋊Q8C2×D14911yescharacteristic
158#126—2981no—
159#125—1491no—
160#124C2×D14911yescharacteristic
161#123—2981no—
162C22⋊Q8—1491no—
163#121—1491no—
164C4×Q8—1491no—
165C4×D4—2981no—
166C4×D4—1491no—
167#117—2981no—
168C2×C4⋊C4—2981no—
169C2×C4⋊C4—2981no—
170C2×C42—1491no—
171C4⋊C8—1492no—
172#103—1492no—
173#103—1492no—
174#103—1492no—
175#103—1492no—
176C22⋊C8—1492no—
177#97—1491no—
178#97C2×D14911yescharacteristic
179C2×Q8—2982no—
180C2×Q8—5962no—
181C2×Q8—5962no—
182C2×Q8—1492no—
183C2×Q8C22×D14912yes—
184C2×Q8C22×D14911yescharacteristic
185C2×D4—2981no—
186C2×D4C22×D14911yescharacteristic
187C22×C4—2982no—
188C22×C4—2981no—
189C22×C4—1491no—
190C22×C4—1491no—
191C22×C4—1491no—
192Q16—42no—
193Q16—5962no—
194SD16—42no—
195SD16—5962no—
196C2×C8—1492no—
197C2×C8C22×D14912yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2982no—
201C4⋊C4—2982no—
202C4⋊C4—2981no—
203C4⋊C4—2981no—
204C4⋊C4—2981no—
205C4⋊C4—2981no—
206C4⋊C4—2981no—
207C4⋊C4—2981no—
208C4⋊C4—1491no—
209C4⋊C4—1491no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—2981no—
213C22⋊C4—2981no—
214C22⋊C4—2981no—
215C42—2981no—
216C42—1491no—
217C42—1491no—
218C42—1491no—
219C42C22×D14911yescharacteristic
220C23—21no—
221C23—1491no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—2982no—
226Q8—2982no—
227Q8—5964no—
228D4—22no—
229D4—5962no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—2982no—
235C2×C4—2981no—
236C2×C4—2981no—
237C2×C4—2982no—
238C2×C4—5964no—
239C2×C4—5962no—
240C2×C4—1491no—
241C2×C4—1491no—
242C2×C4—1491no—
243C2×C4D4×D14911yescharacteristic
244C2×C4D4×D14911yescharacteristic
245C2×C4C23×D14911yescharacteristic
246C8—22no—
247C8—2982no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—2981no—
252C22—2981no—
253C22—1492no—
254C22C2×D4×D14911yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—2981no—
260C4—2981no—
261C4—5961no—
262C4—5961no—
263C4—5962no—
264C4—1492no—
265C4#1801195612yes—
266C2—81no—
267C2—2982no—
268C2#2375018811yescharacteristic
269C2#137×D14911yescharacteristic
270C2#137×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#497×C149C2
socleC22×C149C22×D4
prime radical#18011882C22
hypercenter#497D149
hypocenterC149#3218
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D149C2semidirect
3. #23750020C2semidirect
4. #23749994C2semidirect
5. #23749993C2semidirect
6. #23749833C2semidirect
7. #23749828C2semidirect
8. #23749779C2semidirect
10. #23749680C2semidirect
11. #497×C149C2semidirect
135. C149#3218semidirect
140. #497D149semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1492
#237501542
#237500202
#237499942
#237499932
#237498332
#237498282
#237497792
#237497622
#237496802
#497×C1492
#3218149