Finite Group Databasebeta

← #34055768

Subgroups — #34055768

Order 19072.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#23750201C211yesmaximal, characteristic
2#23750017C211yesmaximal, characteristic, complemented
3#23750004C211yesmaximal, characteristic, complemented
4#23749995C211yesmaximal, characteristic, complemented
5#23749982C211yesmaximal, characteristic, complemented
6#23749847C212yesmaximal, complemented
7#23749835C212yesmaximal, complemented
8#23749785C212yesmaximal
9#23749780C212yesmaximal, complemented
10#23749686C211yesmaximal, characteristic, complemented
11#541×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×Dic298C2211yescharacteristic
14C2×#14558726C2212yes—
15C2×#14558726C2212yes—
16C2×#14558726C2212yes—
17C2×#14558723C2212yes—
18#18011911C2211yescharacteristic
19#18011911C2211yescharacteristic
20#18011899C2211yescharacteristic
21#18011899C2211yescharacteristic
22#18011895—22no—
23#18011891—22no—
24C4⋊#11924746C2212yes—
25#18011834C2212yes—
26#18011834C2212yes—
27#18011834C2212yes—
28#18011831C2212yes—
29C22⋊#11924746C2212yes—
30#18011822C2211yescharacteristic
31C4×#11924746C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33#126×C149—22no—
34#125×C149—22no—
35#125×C149C2211yescharacteristic
36C22⋊Q8×C149C2212yes—
37C4×Q8×C149C2212yes—
38#117×C149C2211yescharacteristic
39#14558736—42no—
40C2×Dic298—21no—
41C2×Dic298—42no—
42C2×Dic298C2311yescharacteristic
43C2×Dic298C2311yescharacteristic
44C2×Dic298C2311yescharacteristic
45C2×D596C2311yescharacteristic
46C2×C22⋊D149—21no—
47C22×Dic149—21no—
48C2×C4×D149—21no—
49#14558726—42no—
50#14558726—42no—
51#14558726—42no—
52#14558723—42no—
53C2×#11924746C2312yes—
54C2×#11924746C2312yes—
55C4⋊Dic149—22no—
56C4⋊Dic149—22no—
57#14558713—22no—
58#14558713—22no—
59C2×Q8×C149C2312yes—
60C22×C4×C149C2311yescharacteristic
61C4⋊C4×C149—22no—
62C4⋊C4×C149—22no—
63C4⋊C4×C149—22no—
64C4⋊C4×C149—22no—
65C4⋊C4×C149D412yes—
66C4⋊C4×C149C2312yes—
67C22⋊C4×C149—22no—
68C22⋊C4×C149D412yes—
69C42×C149—22no—
70C42×C149C2311yescharacteristic
71C42×C149C2311yescharacteristic
72C22×D149—21no—
73Dic298—22no—
74Dic298—22no—
75Dic298—22no—
76Dic298—42no—
77D596—22no—
78C22⋊D149—42no—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C2×Dic149—21no—
82C2×Dic149—42no—
83C4×D149—42no—
84#11924746—22no—
85#11924746—22no—
86C23×C149C2×D411yescharacteristic
87Q8×C149—22no—
88Q8×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149—22no—
91C2×C4×C149—22no—
92C2×C4×C149C2×D411yescharacteristic
93C2×C4×C149C2×D412yes—
94C2×C4×C149C2×D412yes—
95C2×C4×C149C2411yescharacteristic
96C22×C149—21no—
97C22×C149—22no—
98C22×C149C22×D411yescharacteristic
99C2×D149—41no—
100C2×D149—42no—
101C4×C149—21no—
102C4×C149—22no—
103C4×C149—22no—
104C4×C149—42no—
105C4×C149—42no—
106C4×C149H(5,𝔽2)12yes—
107Dic149—41no—
108Dic149—41no—
109Dic149—41no—
110C2×C149—41no—
111C2×C149#53511yescharacteristic
112C2×C149#57812yes—
113D149—81no—
114C149#322311yescharacteristic, complemented, Sylow
115#3223—1491nomaximal, Sylow
116#541D14911yescharacteristic, complemented
117#536—1491no—
118#497—1491no—
119#494—1491no—
120#491—1491no—
121#488—1491no—
122#464—1492no—
123#462—1492no—
124#452—1492no—
125#451—1492no—
126#433—1491no—
127C2×#41—1491no—
128C22×Q8—1491no—
129C2×Q16—1492no—
130C2×Q16—1492no—
131C2×Q16—1492no—
132C2×SD16—1492no—
133C4⋊Q8—1491no—
134C4⋊Q8—1491no—
135C4⋊Q8C2×D14911yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D14911yescharacteristic
139#124—1491no—
140#124—1491no—
141#123—2982no—
142C22⋊Q8—2982no—
143C22⋊Q8C2×D14912yes—
144C4×Q8C2×D14912yes—
145#117C2×D14911yescharacteristic
146C4⋊C8—1492no—
147#103—1492no—
148#103—1492no—
149#103—1492no—
150#102—1492no—
151C22⋊C8—1492no—
152#97—1491no—
153C4×C8—1491no—
154#41—5962no—
155C2×Q8—2981no—
156C2×Q8—5962no—
157C2×Q8—1491no—
158C2×Q8—1491no—
159C2×Q8—1491no—
160C2×Q8C22×D14912yes—
161C2×D4—2981no—
162C2×D4—1491no—
163C22×C4—2981no—
164C22×C4—2981no—
165C22×C4C22×D14911yescharacteristic
166Q16—5962no—
167Q16—5962no—
168Q16—5962no—
169SD16—5962no—
170C2×C8—1492no—
171C2×C8—1492no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—2982no—
177C4⋊C4—2982no—
178C4⋊C4C22⋊D14912yes—
179C4⋊C4C22×D14912yes—
180C22⋊C4—22no—
181C22⋊C4—2982no—
182C22⋊C4—2982no—
183C22⋊C4C22⋊D14912yes—
184C42—22no—
185C42C22×D14911yescharacteristic
186C42C22×D14911yescharacteristic
187C23—2981no—
188C23C2×C22⋊D14911yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—2982no—
192Q8—2982no—
193Q8—2982no—
194Q8—5962no—
195D4—2982no—
196D4—5962no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2981no—
201C2×C4—2981no—
202C2×C4—2981no—
203C2×C4—5962no—
204C2×C4—5962no—
205C2×C4C2×C22⋊D14912yes—
206C2×C4C2×C22⋊D14912yes—
207C2×C4C2×C22⋊D14911yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—2982no—
210C8—2982no—
211C22—22no—
212C22—21no—
213C22—5962no—
214C22—5961no—
215C22C22×C22⋊D14911yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—5961no—
222C4—5961no—
223C4—5961no—
224C4#1801195712yes—
225C2—41no—
226C2—11921no—
227C2#2375018911yescharacteristic
228C2#2375044012yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#541×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558736C22
hypercenter#541D149
hypocenterC149#3223
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #23750017C2semidirect
3. #23750004C2semidirect
4. #23749995C2semidirect
5. #23749982C2semidirect
6. #23749847C2semidirect
7. #23749835C2semidirect
9. #23749780C2semidirect
10. #23749686C2semidirect
11. #541×C149C2semidirect
114. C149#3223semidirect
116. #541D149semidirect

Maximal subgroups

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

subgroupindex
#237502012
#237500172
#237500042
#237499952
#237499822
#237498472
#237498352
#237497852
#237497802
#237496862
#541×C1492
#3223149