Finite Group Databasebeta

← #34057603

Subgroups — #34057603

Order 19072.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic298C211yesmaximal, characteristic, complemented
2#23750335C211yesmaximal, characteristic
3#23750034C212yesmaximal, complemented
4#23750030C211yesmaximal, characteristic, complemented
5#23749933C212yesmaximal, complemented
6#23749921C211yesmaximal, characteristic, complemented
7#23749916C211yesmaximal, characteristic
8#23749906C212yesmaximal
9#23749727C212yesmaximal
10C4×#14558725C211yesmaximal, characteristic
11#446×C149C211yesmaximal, characteristic, complemented
12C2×#14558725C2211yescharacteristic
13#18011911C2212yes—
14#18011911C2211yescharacteristic
15#18011911C2211yescharacteristic
16#18011910—21no—
17#18011910—22no—
18#18011903C2212yes—
19#18011889—22no—
20#18011887—21no—
21Q8×Dic149—21no—
22C4×Dic298C2211yescharacteristic
23C4×Dic298C2212yes—
24C4×D596C2212yes—
25C4×D596C2211yescharacteristic
26C4⋊C4×D149—21no—
27#18011845C2212yes—
28#18011845C2212yes—
29#18011845C2212yes—
30#18011843C2212yes—
31#18011843C2211yescharacteristic
32#18011836C2212yes—
33#18011836C2211yescharacteristic
34#18011833C2211yescharacteristic
35#18011833C2212yes—
36C4×Q8×C149C2211yescharacteristic
37C4⋊C8×C149C2212yes—
38C4⋊C8×C149C2211yescharacteristic
39#97×C149C2212yes—
40C4×C8×C149C2211yescharacteristic
41C2×Dic298—22no—
42C2×Dic298—21no—
43C2×Dic298—21no—
44C2×Dic298C2311yescharacteristic
45C2×D596C2311yescharacteristic
46C2×C4×D149—21no—
47C2×C4×D149—22no—
48#14558725Q814yescomplemented
49C4⋊Dic149—22no—
50C4⋊Dic149—21no—
51C4⋊Dic149—21no—
52C4⋊Dic149C2311yescharacteristic
53C4⋊Dic149C2312yes—
54C4⋊Dic149C2312yes—
55C4⋊Dic149C2311yescharacteristic
56#14558715—22no—
57#14558715—21no—
58#14558715—22no—
59#14558715—22no—
60#14558713—22no—
61#14558713—21no—
62C4×Dic149—22no—
63C4×Dic149—21no—
64C2×Q8×C149D411yescharacteristic
65C2×C8×C149C2311yescharacteristic
66C2×C8×C149C2312yes—
67C2×C8×C149C2311yescharacteristic
68C4⋊C4×C149D411yescharacteristic
69C4⋊C4×C149D412yes—
70C42×C149C2311yescharacteristic
71C42×C149C2312yes—
72C22×D149—21no—
73Dic298—21no—
74Dic298—81no—
75Dic298—82no—
76Dic298—81no—
77Dic298C2×Q812yes—
78D596—21no—
79D596C2×Q812yes—
80C2×Dic149—21no—
81C2×Dic149—22no—
82C2×Dic149—21no—
83C2×Dic149—21no—
84C2×Dic149—22no—
85C4×D149—42no—
86C4×D149—82no—
87Q8×C149—22no—
88C2×C4×C149C2×D412yes—
89C2×C4×C149C2×D411yescharacteristic
90C2×C4×C149C2×D412yes—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2411yescharacteristic
93C8×C149—21no—
94C8×C149—22no—
95C8×C149C2×Q812yes—
96C22×C149C22×D411yescharacteristic
97C2×D149—22no—
98C2×D149—41no—
99C2×D149—41no—
100C4×C149—22no—
101C4×C149—22no—
102C4×C149—21no—
103C4×C149#13712yes—
104C4×C149C22×Q811yescharacteristic
105C4×C1492−1+411yescharacteristic
106Dic149—22no—
107Dic149—41no—
108Dic149—42no—
109Dic149—41no—
110Dic149—42no—
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#500—1491no—
121#479—1492no—
122#477—1491no—
123#476—1491no—
124#474—1492no—
125#446D14911yescharacteristic, complemented
126#440—1492no—
127C4×SD16—1491no—
128C2×SD16—1491no—
129C4⋊Q8—2981no—
130C4⋊Q8—2982no—
131C4⋊Q8—1491no—
132C4⋊Q8—1492no—
133C4⋊Q8—1491no—
134#125—1492no—
135C22⋊Q8—2981no—
136C22⋊Q8—2982no—
137C4×Q8—2981no—
138C4×Q8—1491no—
139C4×Q8—1492no—
140C4×Q8C2×D14911yescharacteristic
141C4×D4—1491no—
142C4×D4—1492no—
143C2×C4⋊C4—2981no—
144#107—1492no—
145#107—1492no—
146#107—1492no—
147#106—1491no—
148#106—1492no—
149C4⋊C8C2×D14912yes—
150C4⋊C8C2×D14911yescharacteristic
151#103—1491no—
152#103—1492no—
153#102—1491no—
154#102—1492no—
155#97C2×D14912yes—
156C4×C8C2×D14911yescharacteristic
157C2×Q8—2981no—
158C2×Q8—2981no—
159C2×Q8—2982no—
160C2×Q8—1491no—
161C2×Q8D59611yescharacteristic
162C2×D4—1491no—
163C22×C4—2981no—
164C22×C4—2982no—
165SD16—1494no—
166C2×C8C22×D14911yescharacteristic
167C2×C8C22×D14912yes—
168C2×C8C22×D14911yescharacteristic
169C4⋊C4—2982no—
170C4⋊C4—2981no—
171C4⋊C4—2982no—
172C4⋊C4—2981no—
173C4⋊C4—2981no—
174C4⋊C4—2982no—
175C4⋊C4—2982no—
176C4⋊C4—1491no—
177C4⋊C4—1492no—
178C4⋊C4—1491no—
179C4⋊C4—1492no—
180C4⋊C4D59611yescharacteristic
181C4⋊C4D59612yes—
182C22⋊C4—2981no—
183C22⋊C4—2982no—
184C42—2982no—
185C42—2981no—
186C42C22×D14911yescharacteristic
187C42C22×D14912yes—
188C23—2981no—
189Q8—22no—
190Q8—2981no—
191Q8—1492no—
192Q8—11922no—
193Q8—11921no—
194Q8—11921no—
195D4—2981no—
196D4—1492no—
197C2×C4—2981no—
198C2×C4—2982no—
199C2×C4—2981no—
200C2×C4—2981no—
201C2×C4—2982no—
202C2×C4—5962no—
203C2×C4—11922no—
204C2×C4C2×D59612yes—
205C2×C4C2×D59611yescharacteristic
206C2×C4C2×D59612yes—
207C2×C4C2×D59611yescharacteristic
208C2×C4C23×D14911yescharacteristic
209C8—21no—
210C8—22no—
211C8Q8×D14912yes—
212C22—2982no—
213C22—5961no—
214C22—5961no—
215C22C22×D59611yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—2982no—
220C4—5961no—
221C4—5962no—
222C4—5961no—
223C4—5962no—
224C4#1801194412yes—
225C4C2×Q8×D14911yescharacteristic
226C4#1801195811yescharacteristic
227C2—5962no—
228C2Q8×D59611yescharacteristic
229C2C2×#1801194411yescharacteristic
230C2#2375043011yescharacteristic
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#446×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#446D149
hypocenterC149#3385
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. Q8×Dic298C2semidirect
3. #23750034C2semidirect
4. #23750030C2semidirect
5. #23749933C2semidirect
6. #23749921C2semidirect
11. #446×C149C2semidirect
48. #14558725Q8semidirect
115. C149#3385semidirect
125. #446D149semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237503352
#237500342
#237500302
#237499332
#237499212
#237499162
#237499062
#237497272
C4×#145587252
#446×C1492
#3385149