Finite Group Databasebeta

← #34057606

Subgroups — #34057606

Order 19072.

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

normal subgroups only

This group has 340 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#23750355C211yesmaximal, characteristic, complemented
2#23750336C211yesmaximal, characteristic
3#23750033C212yesmaximal, complemented
4#23749931C212yesmaximal, complemented
5#23749917C211yesmaximal, characteristic, complemented
6#23749912C211yesmaximal, characteristic
7#23749904C212yesmaximal
8#23749755C211yesmaximal, characteristic, complemented
9#23749724C212yesmaximal
10C4×#14558725C211yesmaximal, characteristic
11#500×C149C211yesmaximal, characteristic, complemented
12C2×#14558725C2211yescharacteristic
13#18011910—21no—
14#18011910C2212yes—
15#18011909—22no—
16#18011902C2212yes—
17#18011891—21no—
18#18011887—22no—
19C4×Dic298C2211yescharacteristic
20C4×Dic298C2211yescharacteristic
21#18011876—21no—
22#18011876C2212yes—
23C4×D596C2211yescharacteristic
24#18011873C2212yes—
25C4⋊C4×D149—21no—
26#18011844C2212yes—
27#18011843C2211yescharacteristic
28#18011842C2212yes—
29C4⋊#11924746C2211yescharacteristic
30#18011840C2212yes—
31#18011836C2211yescharacteristic
32#18011834C2212yes—
33#18011833C2211yescharacteristic
34#18011831C2212yes—
35#18011821C2212yes—
36C4⋊Q8×C149C2211yescharacteristic
37C4⋊Q8×C149C2211yescharacteristic
38#107×C149C2212yes—
39#107×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41C2×Dic298—22no—
42C2×Dic298D411yescharacteristic
43C2×Dic298C2311yescharacteristic
44C2×D596C2311yescharacteristic
45C2×C4×D149—22no—
46C2×C4×D149—21no—
47#14558725Q814yescomplemented
48C2×#11924746C2312yes—
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—
75Dic298—82no—
76Dic298C2×Q812yes—
77D596—21no—
78D596C2×Q812yes—
79C2×Dic149—21no—
80C2×Dic149—21no—
81C2×Dic149—21no—
82C2×Dic149C2×D412yes—
83C2×Dic149C2×D412yes—
84C4×D149—42no—
85C4×D149—82no—
86#11924746—22no—
87Q8×C149—81no—
88Q8×C149—81no—
89C2×C4×C149—22no—
90C2×C4×C149—22no—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2×D411yescharacteristic
93C2×C4×C149C2411yescharacteristic
94C8×C149—21no—
95C8×C149C2×Q812yes—
96C22×C149C22×D411yescharacteristic
97C2×D149—22no—
98C2×D149—41no—
99C2×D149—41no—
100C4×C149—21no—
101C4×C149—42no—
102C4×C149—42no—
103C4×C149#13712yes—
104C4×C149C22×Q811yescharacteristic
105C4×C1492−1+411yescharacteristic
106Dic149—22no—
107Dic149—22no—
108Dic149—22no—
109Dic149—41no—
110Dic149—41no—
111C2×C149D4×Q811yescharacteristic
112C2×C149C2×#13711yescharacteristic
113C2×C149#57611yescharacteristic
114D149—42no—
115C149#338511yescharacteristic, complemented, Sylow
116#3385—1491nomaximal, Sylow
117Q82—1491no—
118#554—1491no—
119#501—1492no—
120#500D14911yescharacteristic, complemented
121#479—1492no—
122#477—1491no—
123#476—1491no—
124#474—1492no—
125#446—1491no—
126#440—1492no—
127C4×SD16—1491no—
128C2×SD16—1491no—
129C4⋊Q8—2981no—
130C4⋊Q8—2982no—
131C4⋊Q8—1492no—
132C4⋊Q8C2×D14911yescharacteristic
133C4⋊Q8C2×D14911yescharacteristic
134#125—1492no—
135C22⋊Q8—2981no—
136C22⋊Q8—2982no—
137C4×Q8—2981no—
138C4×Q8—1492no—
139C4×Q8—1491no—
140C4×Q8—1491no—
141C4×D4—1492no—
142C4×D4—1491no—
143C2×C4⋊C4—2981no—
144#107—1492no—
145#107C2×D14912yes—
146#107C2×D14912yes—
147#106—1491no—
148#106—1492no—
149C4⋊C8—1492no—
150C4⋊C8—1491no—
151#103—1491no—
152#103—1492no—
153#102—1491no—
154#102—1492no—
155#97—1492no—
156C4×C8C2×D14911yescharacteristic
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—2982no—
160C2×Q8—1491no—
161C2×Q8—1491no—
162C2×D4—1491no—
163C22×C4—2982no—
164C22×C4—2981no—
165SD16—1494no—
166C2×C8—1492no—
167C2×C8C22×D14911yescharacteristic
168C2×C8C22×D14911yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—2982no—
172C4⋊C4—2982no—
173C4⋊C4—2982no—
174C4⋊C4—2982no—
175C4⋊C4—2981no—
176C4⋊C4—1491no—
177C4⋊C4—1491no—
178C4⋊C4—1491no—
179C4⋊C4—1492no—
180C4⋊C4C22×D14912yes—
181C4⋊C4C22×D14912yes—
182C22⋊C4—2982no—
183C22⋊C4—2981no—
184C42—2982no—
185C42—2981no—
186C42—1492no—
187C42C22×D14911yescharacteristic
188C23—2981no—
189Q8—81no—
190Q8—81no—
191Q8—2982no—
192Q8—2981no—
193Q8—1492no—
194Q8—11922no—
195D4—2981no—
196D4—1492no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—2981no—
200C2×C4—2981no—
201C2×C4—2981no—
202C2×C4—5962no—
203C2×C4—1492no—
204C2×C4—1492no—
205C2×C4—11922no—
206C2×C4D4×D14911yescharacteristic
207C2×C4D4×D14911yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—21no—
210C8—2982no—
211C8Q8×D14912yes—
212C22—2982no—
213C22—5961no—
214C22—5961no—
215C22C2×D4×D14911yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—2982no—
220C4—2982no—
221C4—2982no—
222C4—5961no—
223C4—5961no—
224C4#1801194312yes—
225C4C2×Q8×D14911yescharacteristic
226C4#1801195911yescharacteristic
227C2—5962no—
228C2#2375029011yescharacteristic
229C2C2×#1801194311yescharacteristic
230C2#2375042611yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#500×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#500D149
hypocenterC149#3385
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #23750355C2semidirect
3. #23750033C2semidirect
4. #23749931C2semidirect
5. #23749917C2semidirect
8. #23749755C2semidirect
11. #500×C149C2semidirect
47. #14558725Q8semidirect
115. C149#3385semidirect
120. #500D149semidirect

Maximal subgroups

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

subgroupindex
#237503552
#237503362
#237500332
#237499312
#237499172
#237499122
#237499042
#237497552
#237497242
C4×#145587252
#500×C1492
#3385149