Finite Group Databasebeta

← #34052750

Subgroups — #34052750

Order 19072.

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

normal subgroups only

This group has 430 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
1C2×#18011905C211yesmaximal, characteristic
2#23749563C212yesmaximal, complemented
3#23749560C212yesmaximal
4#23749556C212yesmaximal, complemented
5#23749551C212yesmaximal
6#23749544C212yesmaximal, complemented
7#23749536C212yesmaximal
8#23749493C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C149C211yesmaximal, characteristic, complemented
10#18011905—24no—
11C2×#14558715C2211yescharacteristic
12C2×#14558715C2211yescharacteristic
13C2×#14558715C2211yescharacteristic
14C2×#14558715C2214yes—
15C2×#14558713C2211yescharacteristic
16C2×#14558713C2211yescharacteristic
17C2×#14558713C2211yescharacteristic
18C2×#14558713C2214yes—
19#18011818C2212yes—
20#18011818C2212yes—
21#18011818C2212yes—
22#18011818C2214yes—
23#18011818C2214yes—
24C22×Q8×C149C2212yes—
25C4⋊Q8×C149—24no—
26C2×C4⋊C4×C149C2212yes—
27C2×C4⋊C4×C149C2212yes—
28C2×C42×C149C2211yescharacteristic
29C23×D149C2311yescharacteristic
30C22×Dic149C2314yes—
31C22×Dic149C2311yescharacteristic
32C22×Dic149C2311yescharacteristic
33C22×Dic149C2311yescharacteristic
34#14558715—22no—
35#14558715—22no—
36#14558715—22no—
37#14558715—44no—
38#14558713—22no—
39#14558713—22no—
40#14558713—22no—
41#14558713—44no—
42C2×Q8×C149—24no—
43C2×Q8×C149—44no—
44C22×C4×C149C2314yes—
45C22×C4×C149C2311yescharacteristic
46C22×C4×C149C2311yescharacteristic
47C22×C4×C149C2311yescharacteristic
48C4⋊C4×C149—24no—
49C4⋊C4×C149—24no—
50C42×C149D414yes—
51C22×D149—21no—
52C22×D149—22no—
53C22×D149—22no—
54C22×D149—22no—
55C2×Dic149—21no—
56C2×Dic149—21no—
57C2×Dic149—21no—
58C2×Dic149—24no—
59C2×Dic149—22no—
60C2×Dic149—22no—
61C2×Dic149—22no—
62C2×Dic149—24no—
63C2×Dic149—24no—
64C23×C149C2411yescharacteristic
65Q8×C149—48no—
66C2×C4×C149—22no—
67C2×C4×C149—22no—
68C2×C4×C149—22no—
69C2×C4×C149—24no—
70C2×C4×C149—24no—
71C2×C4×C149—24no—
72C2×C4×C149C2×D412yes—
73C2×C4×C149C2×D412yes—
74C2×C4×C149C2×D412yes—
75C22×C149C22×D411yescharacteristic
76C22×C149H(5,𝔽2)12yes—
77C22×C1492−1+412yes—
78C22×C1492−1+412yes—
79C2×D149—41no—
80C2×D149—41no—
81C2×D149—41no—
82C2×D149—44no—
83C4×C149—22no—
84C4×C149—22no—
85C4×C149—22no—
86C4×C149—44no—
87Dic149—41no—
88Dic149—41no—
89Dic149—41no—
90Dic149—44no—
91C2×C149#53511yescharacteristic
92C2×C149#53611yescharacteristic
93C2×C149#53611yescharacteristic
94C2×C149#56314yes—
95D149—81no—
96C149#282111yescharacteristic, complemented, Sylow
97#2821—1491nomaximal, Sylow
98C2×C4⋊Q8D14911yescharacteristic, complemented
99C2×#126—1491no—
100#400—1492no—
101#399—1492no—
102#398—1492no—
103#397—1492no—
104#395—1492no—
105#393—1492no—
106#383—1491no—
107C22×Q8C2×D14912yes—
108C4⋊Q8—24no—
109#126—2984no—
110C2×C4⋊C4—1491no—
111C2×C4⋊C4—1491no—
112C2×C4⋊C4—1491no—
113C2×C4⋊C4—1494no—
114C2×C4⋊C4C2×D14912yes—
115C2×C4⋊C4C2×D14912yes—
116C2×C22⋊C4—1491no—
117C2×C22⋊C4—1491no—
118C2×C22⋊C4—1491no—
119C2×C22⋊C4—1494no—
120C2×C42C2×D14911yescharacteristic
121#95—1492no—
122#95—1492no—
123#95—1492no—
124#95—1494no—
125#95—1494no—
126C24—1491no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—1494no—
130C22×C4—1491no—
131C22×C4—1491no—
132C22×C4—1491no—
133C22×C4C22×D14914yes—
134C22×C4C22×D14911yescharacteristic
135C22×C4C22×D14911yescharacteristic
136C22×C4C22×D14911yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—2982no—
140C4⋊C4—2982no—
141C4⋊C4—2982no—
142C4⋊C4—5964no—
143C22⋊C4—2982no—
144C22⋊C4—2982no—
145C22⋊C4—2982no—
146C22⋊C4—5964no—
147C42C22⋊D14914yes—
148C23—2981no—
149C23—2982no—
150C23—2982no—
151C23—2982no—
152C23C23×D14911yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—2981no—
161C2×C4—2981no—
162C2×C4—2981no—
163C2×C4—2984no—
164C2×C4—2982no—
165C2×C4—2982no—
166C2×C4—2982no—
167C2×C4—2984no—
168C2×C4—2984no—
169C2×C4C2×C22⋊D14912yes—
170C2×C4C2×C22⋊D14912yes—
171C2×C4C2×C22⋊D14912yes—
172C22—5961no—
173C22—5961no—
174C22—5961no—
175C22—5964no—
176C22C22×C22⋊D14911yescharacteristic
177C22#1801195612yes—
178C22#1801195912yes—
179C22#1801195912yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—5961no—
185C4—5961no—
186C4—5961no—
187C4—5964no—
188C2—11921no—
189C2#2375019211yescharacteristic
190C2#2375019911yescharacteristic
191C2#2375019911yescharacteristic
192C2#2375037414yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
FittingC2×C4⋊Q8×C149C2
prime radicalC23×D149C23
hypercenterC2×C4⋊Q8D149
hypocenterC149#2821
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23749563C2semidirect
4. #23749556C2semidirect
6. #23749544C2semidirect
8. #23749493C2semidirect
9. C2×C4⋊Q8×C149C2semidirect
96. C149#2821semidirect
98. C2×C4⋊Q8D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119052
#237495632
#237495602
#237495562
#237495512
#237495442
#237495362
#237494932
C2×C4⋊Q8×C1492
#2821149