Finite Group Databasebeta

← #34057571

Subgroups — #34057571

Order 19072.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750350C211yesmaximal, characteristic, complemented
2#23750336C211yesmaximal, characteristic
3#23750025C212yesmaximal, complemented
4#23749935C212yesmaximal, complemented
5#23749924C212yesmaximal
6#23749917C211yesmaximal, characteristic, complemented
7#23749902C211yesmaximal, characteristic
8C8×Dic298C211yesmaximal, characteristic, complemented
9C4×#14558725C211yesmaximal, characteristic
10#23749712C212yesmaximal
11#498×C149C211yesmaximal, characteristic, complemented
12C2×#14558725C2211yescharacteristic
13#18011909—21no—
14#18011902C2212yes—
15#18011902C2212yes—
16#18011901—22no—
17#18011891—21no—
18#18011887—22no—
19Q8×Dic149—21no—
20C4×Dic298C2211yescharacteristic
21C4×Dic298C2211yescharacteristic
22#18011876C2212yes—
23C4×D596C2211yescharacteristic
24#18011873C2212yes—
25C4⋊C4×D149—21no—
26#18011844C2212yes—
27#18011844C2212yes—
28#18011843C2211yescharacteristic
29C4⋊#11924746C2211yescharacteristic
30#18011840C2212yes—
31#18011836C2211yescharacteristic
32#18011834C2212yes—
33#18011833C2211yescharacteristic
34#18011831C2212yes—
35C8×Dic149C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4⋊Q8×C149C2211yescharacteristic
38#106×C149C2212yes—
39#106×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41C2×Dic298D411yescharacteristic
42C2×Dic298C2311yescharacteristic
43C2×D596C2311yescharacteristic
44C2×C4×D149—22no—
45C2×C4×D149—21no—
46#14558725Q814yescomplemented
47C2×#11924746C2312yes—
48C4⋊Dic149—22no—
49C4⋊Dic149—22no—
50C4⋊Dic149D411yescharacteristic
51C4⋊Dic149C2311yescharacteristic
52C4⋊Dic149C2311yescharacteristic
53#14558715—22no—
54#14558715—22no—
55#14558715—22no—
56#14558715—21no—
57#14558715D412yes—
58#14558713—22no—
59#14558713—21no—
60C4×Dic149—22no—
61C4×Dic149—21no—
62C4×Dic149C2312yes—
63C2×Q8×C149—21no—
64C2×Q8×C149—21no—
65C2×C8×C149C2311yescharacteristic
66C2×C8×C149C2311yescharacteristic
67C4⋊C4×C149—21no—
68C4⋊C4×C149—21no—
69C4⋊C4×C149C2312yes—
70C4⋊C4×C149C2312yes—
71C42×C149C2311yescharacteristic
72C22×D149—21no—
73Dic298—22no—
74Dic298—21no—
75Dic298C2×Q812yes—
76D596—21no—
77D596C2×Q812yes—
78C2×Dic149—21no—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C2×Dic149C2×D412yes—
82C2×Dic149C2×D412yes—
83C4×D149—42no—
84C4×D149—82no—
85#11924746—22no—
86Q8×C149—42no—
87Q8×C149—81no—
88C2×C4×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149C2×D411yescharacteristic
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2411yescharacteristic
93C8×C149—21no—
94C8×C149C2×Q812yes—
95C22×C149C22×D411yescharacteristic
96C2×D149—22no—
97C2×D149—41no—
98C2×D149—41no—
99C4×C149—21no—
100C4×C149—42no—
101C4×C149—42no—
102C4×C149#13512yes—
103C4×C149C22×Q811yescharacteristic
104C4×C1492−1+411yescharacteristic
105Dic149—22no—
106Dic149—22no—
107Dic149—22no—
108Dic149—41no—
109Dic149—41no—
110C2×C149D4×Q811yescharacteristic
111C2×C149C2×#13511yescharacteristic
112C2×C149#57711yescharacteristic
113D149—42no—
114C149#338111yescharacteristic, complemented, Sylow
115#3381—1491nomaximal, Sylow
116#557—1491no—
117#554—1491no—
118#499—1492no—
119#498D14911yescharacteristic, complemented
120#479—1492no—
121#478—1492no—
122#477—1491no—
123#474—1491no—
124C8×Q8—1491no—
125C4×SD16—1492no—
126C4×SD16—1491no—
127C2×SD16—1491no—
128C4⋊Q8—2981no—
129C4⋊Q8C2×D14911yescharacteristic
130C4⋊Q8C2×D14911yescharacteristic
131#125—2982no—
132#125—1492no—
133#125—1492no—
134C22⋊Q8—2981no—
135C22⋊Q8—2982no—
136C4×Q8—2981no—
137C4×Q8—1492no—
138C4×Q8—1491no—
139C4×Q8—1491no—
140C4×D4—1492no—
141C4×D4—1491no—
142C2×C4⋊C4—2981no—
143#107—1492no—
144#107—1492no—
145#106—1491no—
146#106C2×D14912yes—
147#106C2×D14912yes—
148C4⋊C8—1492no—
149C4⋊C8—1491no—
150#103—1491no—
151#103—1492no—
152#102—1491no—
153#102—1492no—
154C4×C8—1492no—
155C4×C8C2×D14911yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—1491no—
159C2×Q8—1491no—
160C2×D4—1491no—
161C22×C4—2982no—
162C22×C4—2981no—
163SD16—1494no—
164C2×C8—1492no—
165C2×C8C22×D14911yescharacteristic
166C2×C8C22×D14911yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—2982no—
170C4⋊C4—2982no—
171C4⋊C4—2982no—
172C4⋊C4—2982no—
173C4⋊C4—2982no—
174C4⋊C4—2981no—
175C4⋊C4—1491no—
176C4⋊C4—1491no—
177C4⋊C4—1491no—
178C4⋊C4—1492no—
179C4⋊C4C22×D14912yes—
180C4⋊C4C22×D14912yes—
181C22⋊C4—2982no—
182C22⋊C4—2981no—
183C42—2982no—
184C42—2981no—
185C42—1492no—
186C42C22×D14911yescharacteristic
187C23—2981no—
188Q8—42no—
189Q8—81no—
190Q8—2982no—
191Q8—2981no—
192Q8—1492no—
193D4—2981no—
194D4—1492no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—2981no—
198C2×C4—2981no—
199C2×C4—2981no—
200C2×C4—5962no—
201C2×C4—1492no—
202C2×C4—1492no—
203C2×C4—11922no—
204C2×C4D4×D14911yescharacteristic
205C2×C4D4×D14911yescharacteristic
206C2×C4C23×D14911yescharacteristic
207C8—21no—
208C8—2982no—
209C8Q8×D14912yes—
210C22—2982no—
211C22—5961no—
212C22—5961no—
213C22C2×D4×D14911yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—2982no—
218C4—2982no—
219C4—2982no—
220C4—5961no—
221C4—5961no—
222C4#1801193212yes—
223C4C2×Q8×D14911yescharacteristic
224C4#1801195911yescharacteristic
225C2—5962no—
226C2#2375029011yescharacteristic
227C2C2×#1801193211yescharacteristic
228C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#498×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#498D149
hypocenterC149#3381
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #23750350C2semidirect
3. #23750025C2semidirect
4. #23749935C2semidirect
6. #23749917C2semidirect
8. C8×Dic298C2semidirect
11. #498×C149C2semidirect
46. #14558725Q8semidirect
114. C149#3381semidirect
119. #498D149semidirect

Maximal subgroups

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

subgroupindex
#237503502
#237503362
#237500252
#237499352
#237499242
#237499172
#237499022
C8×Dic2982
C4×#145587252
#237497122
#498×C1492
#3381149