Finite Group Databasebeta

← #34046713

Subgroups — #34046713

Order 19072.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#23750374C211yesmaximal, characteristic, complemented
2#23749810C211yesmaximal, characteristic, complemented
3#23749807C211yesmaximal, characteristic, complemented
4#23749800C211yesmaximal, characteristic
5#23749423C211yesmaximal, characteristic, complemented
6#23749421C211yesmaximal, characteristic, complemented
7#356×C149C211yesmaximal, characteristic, complemented
8#18011959C2211yescharacteristic
9#18011944—21no—
10#18011942—21no—
11#18011939—21no—
12#18011909C2211yescharacteristic
13#18011906—21no—
14#18011905—21no—
15#18011901—21no—
16#18011900C2211yescharacteristic
17#18011896—21no—
18#18011889—21no—
19#18011887—21no—
20#18011839—21no—
21#18011837—21no—
22#18011837—21no—
23#18011830C2211yescharacteristic
24C4⋊Q8×C149C2211yescharacteristic
25#101×C149C2211yescharacteristic
26#101×C149C2211yescharacteristic
27#14558737—41no—
28#14558736—21no—
29#14558736—21no—
30#14558736—21no—
31C2×Dic298D411yescharacteristic
32Q8×D149—41no—
33C2×D596D411yescharacteristic
34C2×C4×D149—21no—
35Dic596—41no—
36#14558726—41no—
37#14558725—41no—
38#14558725—41no—
39#14558723—41no—
40D1192—41no—
41#14558720—21no—
42C4⋊Dic149—21no—
43#14558715—21no—
44#14558715—21no—
45#14558715—21no—
46#14558715—21no—
47#14558715—21no—
48#14558713—21no—
49#14558713—21no—
50#14558713—21no—
51#14558713—21no—
52#14558713—21no—
53C4×Dic149D411yescharacteristic
54C4×Dic149D411yescharacteristic
55C2×Q8×C149D411yescharacteristic
56C2×Q8×C149C2311yescharacteristic
57M16×C149—21no—
58M16×C149—21no—
59C4⋊C4×C149—21no—
60C4⋊C4×C149—21no—
61C42×C149D411yescharacteristic
62C22×D149—21no—
63Dic298—21no—
64Dic298—21no—
65Dic298—21no—
66Dic298—41no—
67D596—21no—
68D596—21no—
69D596—21no—
70D596—41no—
71C22⋊D149—41no—
72C2×Dic149—21no—
73C2×Dic149—21no—
74C2×Dic149—21no—
75C4×D149—41no—
76C4×D149—41no—
77C4×D149—41no—
78C4×D149—81no—
79#11924746—41no—
80Q8×C149—41no—
81Q8×C149—41no—
82C2×C4×C149—21no—
83C2×C4×C149—21no—
84C2×C4×C149C2×D411yescharacteristic
85C2×C4×C149C2×D411yescharacteristic
86C2×C4×C149C2×D411yescharacteristic
87C8×C149—41no—
88C8×C149—41no—
89C22×C149#12011yescharacteristic
90C2×D149—42no—
91C2×D149—41no—
92C2×D149—41no—
93C4×C149—21no—
94C4×C149—21no—
95C4×C149—21no—
96C4×C149—41no—
97C4×C149—41no—
98Dic149—41no—
99Dic149—41no—
100Dic149—41no—
101Dic149—41no—
102C2×C149—21no—
103C2×C149B(4,𝔽2)11yescharacteristic
104D149—81no—
105D149—81no—
106C149#220311yescharacteristic, complemented, Sylow
107#2203—1491nomaximal, Sylow
108#563—1491no—
109#456—1491no—
110#456—1491no—
111#454—1491no—
112#356—1491no—
113#356D14911yescharacteristic, complemented
114#355—1491no—
1152−1+4—1491no—
116#137—2981no—
117#137—2981no—
118AGL(1,ℤ/8)—2981no—
119C4⋊Q8—1491no—
120C4⋊Q8C2×D14911yescharacteristic
121#126—2981no—
122#126—2981no—
123#125—2981no—
124#124—1491no—
125#123—2981no—
126C22⋊Q8—2981no—
127C22⋊Q8—2981no—
128#104—2981no—
129#104—2981no—
130#104—2981no—
131#101—1491no—
132#101C2×D14911yescharacteristic
133#101C2×D14911yescharacteristic
134#41—2981no—
135#41—2981no—
136#41—2981no—
137#41—5961no—
138C2×Q8—5961no—
139C2×Q8—1491no—
140C2×Q8D59611yescharacteristic
141C2×Q8C22×D14911yescharacteristic
142C2×D4—1491no—
143C22×C4—2981no—
144Q16—5961no—
145Q16—5961no—
146SD16—5961no—
147SD16—5961no—
148SD16—5961no—
149D8—5961no—
150M16—21no—
151M16—21no—
152M16—2981no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—2981no—
156C4⋊C4—2981no—
157C4⋊C4—2981no—
158C4⋊C4—2981no—
159C4⋊C4—2981no—
160C4⋊C4—2981no—
161C22⋊C4—2981no—
162C22⋊C4—2981no—
163C22⋊C4—2981no—
164C22⋊C4—2981no—
165C22⋊C4—2981no—
166C42—1491no—
167C42—1491no—
168C42D59611yescharacteristic
169C23—2981no—
170Q8—41no—
171Q8—41no—
172Q8—2981no—
173Q8—2981no—
174Q8—2981no—
175Q8—5961no—
176D4—2981no—
177D4—2981no—
178D4—2981no—
179D4—5961no—
180D4—5961no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—2981no—
184C2×C4—2981no—
185C2×C4—2981no—
186C2×C4—5961no—
187C2×C4—5961no—
188C2×C4—5961no—
189C2×C4—11921no—
190C2×C4D4×D14911yescharacteristic
191C2×C4D4×D14911yescharacteristic
192C2×C4C2×D59611yescharacteristic
193C8—41no—
194C8—41no—
195C8—5961no—
196C22—5962no—
197C22—5961no—
198C22—5961no—
199C22#1801188111yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—5961no—
206C4—5961no—
207C4—5961no—
208C4—5961no—
209C2—21no—
210C2—11921no—
211C2—11921no—
212C2#2374981311yescharacteristic
subgroupG/H
centerC2#23749813
commutator, cosocleC2×Q8×C149C23
FrattiniC2×Q8C22×D149
Fitting#356×C149C2
socleC2×C149B(4,𝔽2)
prime radical#23749800C2
hypercenter#356D149
hypocenterC149#2203
LCSG ▷ C2×Q8×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #23750374C2semidirect
2. #23749810C2semidirect
3. #23749807C2semidirect
5. #23749423C2semidirect
6. #23749421C2semidirect
7. #356×C149C2semidirect
106. C149#2203semidirect
113. #356D149semidirect

Maximal subgroups

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

subgroupindex
#237503742
#237498102
#237498072
#237498002
#237494232
#237494212
#356×C1492
#2203149