Finite Group Databasebeta

← #34055511

Subgroups — #34055511

Order 19072.

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

normal subgroups only

This group has 410 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
1C22⋊Q8×D149C212yesmaximal
2#23749957C212yesmaximal, complemented
3#23749954C212yesmaximal, complemented
4#23749883C211yesmaximal, characteristic, complemented
5#23749881C212yesmaximal, complemented
6#23749782C212yesmaximal, complemented
7#23749779C212yesmaximal, complemented
8#23749588C211yesmaximal, characteristic
9#470×C149C211yesmaximal, characteristic, complemented
10C2×Q8×D149C2212yes—
11C22×C4×D149C2211yescharacteristic
12C2×Dic596C2211yescharacteristic, complemented
13C2×#14558726C2212yescomplemented
14C2×#14558720C2211yescharacteristic
15#18011891C2212yes—
16#18011890C2212yes—
17#18011889—22no—
18#18011888—22no—
19#18011887C2212yes—
20C4⋊C4×D149—22no—
21C4⋊C4×D149C2212yes—
22C22⋊C4×D149—22no—
23#18011843C2211yescharacteristic, complemented
24#18011842C2212yescomplemented
25#18011836C2212yescomplemented
26#18011835C2212yescomplemented
27#18011834C2212yescomplemented
28#18011823C2212yes—
29#18011823C2212yes—
30C2×Q16×C149C2211yescharacteristic, complemented
31C2×M16×C149C2211yescharacteristic
32C22⋊Q8×C149C2212yes—
33#106×C149C2211yescharacteristic, complemented
34#103×C149C2212yescomplemented
35C23×D149D411yescharacteristic
36C2×Dic298—22no—
37C2×Dic298C2312yes—
38Q8×D149—42no—
39Q8×D149—42no—
40C22×Dic149D411yescharacteristic
41C2×C4×D149—22no—
42C2×C4×D149—22no—
43C2×C4×D149—41no—
44C2×C4×D149—41no—
45C2×C4×D149D412yes—
46C2×C4×D149C2312yes—
47Dic596—41no—
48#14558726—42no—
49#14558720—41no—
50C2×#11924746C2312yes—
51C4⋊Dic149—22no—
52C4⋊Dic149C2312yes—
53#14558715—22no—
54#14558715—22no—
55#14558715—22no—
56C22⋊Dic149—22no—
57#14558713—22no—
58#14558713—22no—
59C2×Q8×C149C2312yes—
60C22×C4×C149C2311yescharacteristic
61Q16×C149—41no—
62M16×C149—41no—
63C2×C8×C149C2311yescharacteristic
64C2×C8×C149C2311yescharacteristic
65C4⋊C4×C149—22no—
66C4⋊C4×C149C2312yes—
67C22⋊C4×C149—22no—
68C22×D149—24no—
69C22×D149—41no—
70C22×D149C2×D412yes—
71Dic298—22no—
72Dic298—22no—
73Dic298—44no—
74C2×Dic149—22no—
75C2×Dic149—22no—
76C2×Dic149—41no—
77C2×Dic149C2×D412yes—
78C4×D149—22no—
79C4×D149—22no—
80C4×D149—42no—
81C4×D149—44no—
82C4×D149—44no—
83#11924746—22no—
84C23×C149C2×D411yescharacteristic
85Q8×C149—22no—
86Q8×C149—22no—
87C2×C4×C149—21no—
88C2×C4×C149—21no—
89C2×C4×C149—22no—
90C2×C4×C149—22no—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2411yescharacteristic
93C8×C149—21no—
94C8×C149—21no—
95C22×C149—21no—
96C22×C149—22no—
97C22×C149C22×D411yescharacteristic
98C2×D149—22no—
99C2×D149—22no—
100C2×D149—44no—
101C2×D149#13714yes—
102C4×C149—21no—
103C4×C149—42no—
104C4×C149—42no—
105C4×C149H(5,𝔽2)11yescharacteristic
106C4×C149H(5,𝔽2)11yescharacteristic
107Dic149—22no—
108Dic149—42no—
109Dic149—42no—
110C2×C149—41no—
111C2×C149#53411yescharacteristic
112C2×C149C2×#13712yes—
113D149—24no—
114C149#319511yescharacteristic, complemented, Sylow
115#3195—1491nomaximal, Sylow
116C2×C22⋊Q8—1492no—
117#483—1492no—
118#483—1492no—
119#470—1491no—
120#470—1492no—
121#470D14911yescharacteristic, complemented
122#451—1492no—
123#451—1492no—
124#407—1491no—
125C22×Q8—1492no—
126C23×C4—1491no—
127C2×Q16—1492no—
128C2×Q16—1491no—
129C2×Q16C2×D14911yescharacteristic, complemented
130C2×M16—1491no—
131C2×M16C2×D14911yescharacteristic
132C22⋊Q8—2982no—
133C22⋊Q8—2982no—
134C22⋊Q8—1492no—
135C22⋊Q8—1492no—
136C22⋊Q8—1492no—
137C22⋊Q8C2×D14912yes—
138C2×C4⋊C4—2982no—
139C2×C4⋊C4—1492no—
140C2×C22⋊C4—2982no—
141#106—1491no—
142#106—1492no—
143#106C2×D14911yescharacteristic, complemented
144#103—1492no—
145#103—1492no—
146#103—1492no—
147#103C2×D14912yescomplemented
148C22⋊C8—1492no—
149C22⋊C8—1492no—
150C24—1491no—
151C2×Q8—2982no—
152C2×Q8—5962no—
153C2×Q8—5962no—
154C2×Q8—1492no—
155C2×Q8C22×D14912yes—
156C22×C4—2982no—
157C22×C4—2982no—
158C22×C4—5961no—
159C22×C4—5961no—
160C22×C4—1492no—
161C22×C4—1491no—
162C22×C4—1492no—
163C22×C4C22×D14911yescharacteristic
164Q16—41no—
165Q16—5961no—
166Q16—5962no—
167M16—41no—
168M16—5961no—
169C2×C8—1492no—
170C2×C8C22×D14911yescharacteristic
171C2×C8C22×D14911yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—2982no—
174C4⋊C4—2982no—
175C4⋊C4—2982no—
176C4⋊C4—2982no—
177C4⋊C4—1492no—
178C4⋊C4C22×D14912yes—
179C22⋊C4—22no—
180C22⋊C4—2982no—
181C22⋊C4—2982no—
182C22⋊C4—2982no—
183C23—2984no—
184C23—5961no—
185C23—1492no—
186C23D4×D14911yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—2982no—
190Q8—2982no—
191Q8—5964no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—2982no—
197C2×C4—2982no—
198C2×C4—2982no—
199C2×C4—2982no—
200C2×C4—5962no—
201C2×C4—5964no—
202C2×C4—5964no—
203C2×C4—5961no—
204C2×C4—1492no—
205C2×C4D4×D14911yescharacteristic
206C2×C4C23×D14911yescharacteristic
207C8—21no—
208C8—21no—
209C8—2982no—
210C22—22no—
211C22—21no—
212C22—2982no—
213C22—2982no—
214C22—5964no—
215C22—1494no—
216C22C2×D4×D14911yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—2982no—
221C4—5962no—
222C4—5962no—
223C4#1801195611yescharacteristic
224C4#1801195711yescharacteristic
225C2—41no—
226C2—2984no—
227C2#2375018311yescharacteristic
228C2#137×D14912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#470×C149C2
socleC22×C149C22×D4
prime radicalC23×D149D4
hypercenter#470D149
hypocenterC149#3195
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #23749957C2semidirect
3. #23749954C2semidirect
4. #23749883C2semidirect
5. #23749881C2semidirect
6. #23749782C2semidirect
7. #23749779C2semidirect
9. #470×C149C2semidirect
12. C2×Dic596C22semidirect
13. C2×#14558726C22semidirect
23. #18011843C22semidirect
24. #18011842C22semidirect
25. #18011836C22semidirect
26. #18011835C22semidirect
27. #18011834C22semidirect
30. C2×Q16×C149C22semidirect
33. #106×C149C22semidirect
34. #103×C149C22semidirect
114. C149#3195semidirect
121. #470D149semidirect
129. C2×Q16C2×D149semidirect
143. #106C2×D149semidirect
147. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1492
#237499572
#237499542
#237498832
#237498812
#237497822
#237497792
#237495882
#470×C1492
#3195149