Finite Group Databasebeta

← #34060954

Subgroups — #34060954

Order 19072.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#23750612C211yesmaximal, characteristic
2#23750605C211yesmaximal, characteristic
3#23750528C212yesmaximal, complemented
4#23750527C212yesmaximal, complemented
5#23750197C211yesmaximal, characteristic
6C2×#14558745—21no—
7#18011990—21no—
8#18011972—21no—
9#18011971C2212yes—
10#18011969—21no—
11#18011969C2212yes—
12C2×#14558735C2211yescharacteristic
13C22×Dic298C411yescharacteristic, complemented
14#18011910—21no—
15#18011897—21no—
16#18011895C2211yescharacteristic
17#18011894C2211yescharacteristic
18#18011892C411yescharacteristic, complemented
19#18011889—21no—
20#18011888—21no—
21#18011867C411yescharacteristic, complemented
22#123×C149C411yescharacteristic, complemented
23#14558745—22no—
24#14558745—41no—
25C2×#11924753—41no—
26C4⋊#10597010—41no—
27C22⋊#10597010—22no—
28C4×#10597010—41no—
29#14558735—41no—
30#14558735—41no—
31C2×Dic298—21no—
32C2×Dic298—21no—
33C2×Dic298—41no—
34C2×Dic298—41no—
35C2×Dic298D411yescharacteristic
36C2×C22⋊D149D411yescharacteristic
37C2×C22⋊D149C2311yescharacteristic
38C22×Dic149C2×C411yescharacteristic
39C22×Dic149C2×C411yescharacteristic
40C22×Dic149D411yescharacteristic
41C2×C4×D149D411yescharacteristic
42C4⋊Dic149—21no—
43#14558715—21no—
44#14558715—21no—
45#14558715—21no—
46C22⋊Dic149—21no—
47C22⋊Dic149C2×C411yescharacteristic
48#14558713—21no—
49#14558713—21no—
50C4×Dic149—21no—
51C2×D4×C149C2×C411yescharacteristic
52C22×C4×C149C2×C411yescharacteristic
53C4⋊C4×C149—21no—
54C22⋊C4×C149—21no—
55C22⋊C4×C149C2×C411yescharacteristic
56C2×#10597010—42no—
57#11924753—42no—
58C22×D149C2×D411yescharacteristic
59Dic298—41no—
60Dic298—42no—
61Dic298—41no—
62Dic298—41no—
63C22⋊D149—41no—
64C22⋊D149—41no—
65C2×Dic149—21no—
66C2×Dic149—21no—
67C2×Dic149—21no—
68C2×Dic149—21no—
69C2×Dic149—21no—
70C2×Dic149—21no—
71C2×Dic149—41no—
72C2×Dic149C22⋊C411yescharacteristic
73C2×Dic149C22⋊C411yescharacteristic
74C2×Dic149C2×D411yescharacteristic
75C4×D149—21no—
76C4×D149—21no—
77C23×C149C22⋊C411yescharacteristic
78C23×C149C22×C411yescharacteristic
79D4×C149—41no—
80C2×C4×C149—21no—
81C2×C4×C149—21no—
82C2×C4×C149—41no—
83C2×C4×C149C22⋊C411yescharacteristic
84C22×C149—21no—
85C22×C149—21no—
86C22×C149—41no—
87C22×C149—41no—
88C22×C149C2×C22⋊C411yescharacteristic
89C2×D149—21no—
90C2×D149—41no—
91#10597010—82no—
92C4×C149—21no—
93C4×C149—41no—
94C4×C149—41no—
95Dic149—21no—
96Dic149—21no—
97Dic149—41no—
98Dic149—41no—
99Dic149#9911yescharacteristic
100Dic149#9911yescharacteristic
101C2×C149—21no—
102C2×C149—41no—
103C2×C149—41no—
104C2×C149C2×#9911yescharacteristic
105D149—41no—
106C149#212311yescharacteristic, complemented, Sylow
107#2123—1491nomaximal, Sylow
108#536—1491no—
109#413—1491no—
110#410—1491no—
111#352—1492no—
112#352—1492no—
113C2×#41—1491no—
114C22×Q8—1491no—
115C2×M16—2981no—
116C4⋊Q8—2981no—
117#124—2981no—
118#123—1491no—
119#123—1491no—
120#123—1491no—
121#123#1059701011yescharacteristic, complemented
122C22⋊Q8—2981no—
123C22⋊Q8—2981no—
124#117—2981no—
125#117—1491no—
126#101—2981no—
127#100—1492no—
128#99—2981no—
129#99—1492no—
130#41—5961no—
131#41—5961no—
132C2×Q8—2981no—
133C2×Q8—2981no—
134C2×Q8—5961no—
135C2×Q8—5961no—
136C2×Q8—1491no—
137C2×D4—1491no—
138C2×D4—1491no—
139C2×D4C2×#1059701011yescharacteristic
140C22×C4—1491no—
141C22×C4—1491no—
142C22×C4—1491no—
143C22×C4—1491no—
144C22×C4C2×#1059701011yescharacteristic
145M16—2982no—
146M16—5961no—
147C2×C8—5961no—
148C4⋊C4—21no—
149C4⋊C4—2981no—
150C4⋊C4—2981no—
151C4⋊C4—2981no—
152C4⋊C4—2981no—
153C4⋊C4—5961no—
154C22⋊C4—21no—
155C22⋊C4—2981no—
156C22⋊C4—2981no—
157C22⋊C4—2981no—
158C22⋊C4—2982no—
159C22⋊C4—1491no—
160C22⋊C4C2×#1059701011yescharacteristic
161C42—2981no—
162C42—5961no—
163C23—1491no—
164C23C22⋊#1059701011yescharacteristic
165C23C22×#1059701011yescharacteristic
166Q8—5961no—
167Q8—5962no—
168Q8—5961no—
169Q8—5961no—
170D4—41no—
171D4—5961no—
172D4—5961no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—2981no—
177C2×C4—2981no—
178C2×C4—2981no—
179C2×C4—2981no—
180C2×C4—2981no—
181C2×C4—2981no—
182C2×C4—2981no—
183C2×C4—2981no—
184C2×C4—5961no—
185C2×C4—5962no—
186C2×C4—1491no—
187C2×C4—1491no—
188C2×C4—1491no—
189C2×C4C22⋊#1059701011yescharacteristic
190C8—5962no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—2981no—
196C22—5961no—
197C22C2×C22⋊#1059701011yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—2981no—
202C4—2981no—
203C4—5961no—
204C4—5961no—
205C4—1491no—
206C4—1491no—
207C4—11922no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—5961no—
212C2D149⋊#9911yescharacteristic
subgroupG/H
centerC2D149⋊#99
commutatorC23×C149C22×C4
FrattiniC23C22×#10597010
Fitting#123×C149C4
socleC2×C149C2×#99
prime radicalC2×#14558735C22
cosocleC2×C22⋊D149C23
hypercenter#123#10597010
hypocenterC149#2123
LCSG ▷ C23×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #23750528C2semidirect
4. #23750527C2semidirect
13. C22×Dic298C4semidirect
18. #18011892C4semidirect
21. #18011867C4semidirect
22. #123×C149C4semidirect
106. C149#2123semidirect
121. #123#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506122
#237506052
#237505282
#237505272
#237501972
#2123149