Finite Group Databasebeta

← #34052754

Subgroups — #34052754

Order 19072.

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

normal subgroups only

This group has 430 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
1C2×#18011910C211yesmaximal, characteristic, complemented
2C2×#18011904C211yesmaximal, characteristic
3#23749565C211yesmaximal, characteristic, complemented
4#23749563C211yesmaximal, characteristic, complemented
5#23749561C211yesmaximal, characteristic
6#23749558C212yesmaximal, complemented
7#23749553C212yesmaximal
8#23749545C212yesmaximal, complemented
9#23749537C212yesmaximal
10#23749492C211yesmaximal, characteristic, complemented
11#399×C149C211yesmaximal, characteristic
12C22×Dic298C2212yes—
13#18011910—24no—
14#18011904—24no—
15C2×C4⋊Dic149C2211yescharacteristic
16C2×C4⋊Dic149C2211yescharacteristic
17C2×#14558715C2211yescharacteristic
18C2×#14558715C2211yescharacteristic
19C2×#14558715C2212yes—
20C2×#14558715C2212yes—
21C2×C22⋊Dic149C2211yescharacteristic
22C2×C22⋊Dic149C2211yescharacteristic
23C2×C22⋊Dic149C2212yes—
24C2×C4×Dic149C2211yescharacteristic
25#18011818C2212yes—
26#18011818C2212yes—
27#18011818C2211yescharacteristic
28#18011818C2212yes—
29#18011818C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32C2×C4⋊C4×C149C2211yescharacteristic
33C2×C4⋊C4×C149C2212yes—
34C2×C22⋊C4×C149C2211yescharacteristic
35C2×C22⋊C4×C149C2212yes—
36#95×C149C2211yescharacteristic
37C2×Dic298—24no—
38C2×Dic298—42no—
39C2×Dic298—42no—
40C22×Dic149C2312yes—
41C22×Dic149C2312yes—
42C22×Dic149C2311yescharacteristic
43C22×Dic149C2311yescharacteristic
44C22×Dic149C2311yescharacteristic
45C22×Dic149C2311yescharacteristic
46C4⋊Dic149—22no—
47C4⋊Dic149—22no—
48#14558715—22no—
49#14558715—24no—
50#14558715—22no—
51#14558715—42no—
52C22⋊Dic149—22no—
53C22⋊Dic149—22no—
54C22⋊Dic149—42no—
55C4×Dic149D414yes—
56C24×C149C2311yescharacteristic
57C22×C4×C149C2312yes—
58C22×C4×C149C2312yes—
59C22×C4×C149C2311yescharacteristic
60C22×C4×C149C2311yescharacteristic
61C4⋊C4×C149—22no—
62C4⋊C4×C149—42no—
63C22⋊C4×C149—22no—
64C22⋊C4×C149—42no—
65Dic298—48no—
66C2×Dic149—22no—
67C2×Dic149—21no—
68C2×Dic149—21no—
69C2×Dic149—22no—
70C2×Dic149—21no—
71C2×Dic149—21no—
72C2×Dic149—22no—
73C2×Dic149—22no—
74C2×Dic149—22no—
75C2×Dic149—22no—
76C2×Dic149—21no—
77C2×Dic149—22no—
78C2×Dic149—21no—
79C2×Dic149—22no—
80C2×Dic149C2×D412yes—
81C2×Dic149C2×D412yes—
82C23×C149—21no—
83C23×C149—22no—
84C23×C149—22no—
85C23×C149—21no—
86C23×C149—21no—
87C23×C149C2411yescharacteristic
88C2×C4×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149—21no—
91C2×C4×C149—22no—
92C2×C4×C149—22no—
93C2×C4×C149—22no—
94C2×C4×C149—22no—
95C2×C4×C149—22no—
96C2×C4×C149—22no—
97C2×C4×C149C2×D412yes—
98C22×C149—41no—
99C22×C149—41no—
100C22×C149—41no—
101C22×C149—42no—
102C22×C149—42no—
103C22×C149C22×D411yescharacteristic
104C22×C149H(5,𝔽2)12yes—
105C22×C1492−1+412yes—
106C22×C1492−1+411yescharacteristic
107C22×C1492−1+411yescharacteristic
108C4×C149—22no—
109C4×C149—41no—
110C4×C149—42no—
111C4×C149—42no—
112Dic149—22no—
113Dic149—22no—
114Dic149—42no—
115Dic149—41no—
116Dic149—41no—
117Dic149—42no—
118C2×C149—81no—
119C2×C149#53511yescharacteristic
120C2×C149#53611yescharacteristic
121C2×C149#53611yescharacteristic
122C2×C149#56312yes—
123C2×C149#56312yes—
124C149#282111yescharacteristic, complemented, Sylow
125#2821—1491nomaximal, Sylow
126C2×C4⋊Q8—1491no—
127C2×#126—1491no—
128#400—1491no—
129#400—1491no—
130#399—1491no—
131#399D14911yescharacteristic
132#398—1492no—
133#397—1492no—
134#395—1492no—
135#393—1492no—
136#383—1491no—
137C22×Q8—1492no—
138C4⋊Q8—2984no—
139#126—2984no—
140C2×C4⋊C4—1491no—
141C2×C4⋊C4—1491no—
142C2×C4⋊C4—1491no—
143C2×C4⋊C4—1492no—
144C2×C4⋊C4—1491no—
145C2×C4⋊C4—1492no—
146C2×C4⋊C4C2×D14911yescharacteristic
147C2×C4⋊C4C2×D14912yes—
148C2×C22⋊C4—1491no—
149C2×C22⋊C4—1491no—
150C2×C22⋊C4—1492no—
151C2×C22⋊C4C2×D14911yescharacteristic
152C2×C22⋊C4C2×D14912yes—
153C2×C42—1491no—
154#95—1492no—
155#95—1492no—
156#95—1491no—
157#95—1492no—
158#95—1492no—
159#95—1492no—
160#95—1492no—
161#95C2×D14911yescharacteristic
162C24C22×D14911yescharacteristic
163C2×Q8—2984no—
164C2×Q8—5962no—
165C2×Q8—5962no—
166C22×C4—1492no—
167C22×C4—1491no—
168C22×C4—1491no—
169C22×C4—1492no—
170C22×C4—1491no—
171C22×C4—1491no—
172C22×C4C22×D14912yes—
173C22×C4C22×D14912yes—
174C22×C4C22×D14911yescharacteristic
175C22×C4C22×D14911yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—2982no—
179C4⋊C4—2982no—
180C4⋊C4—2984no—
181C4⋊C4—2982no—
182C4⋊C4—2982no—
183C4⋊C4—5962no—
184C22⋊C4—22no—
185C22⋊C4—42no—
186C22⋊C4—2982no—
187C22⋊C4—2982no—
188C22⋊C4—5962no—
189C42—1494no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C23×D14911yescharacteristic
196Q8—5968no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—2982no—
207C2×C4—2981no—
208C2×C4—2981no—
209C2×C4—2981no—
210C2×C4—2981no—
211C2×C4—2982no—
212C2×C4—2982no—
213C2×C4—2982no—
214C2×C4—2982no—
215C2×C4—2982no—
216C2×C4—2981no—
217C2×C4—2982no—
218C2×C4—2981no—
219C2×C4—2982no—
220C2×C4—1492no—
221C2×C4—1492no—
222C2×C4D4×D14912yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C2×D4×D14911yescharacteristic
229C22#1801195612yes—
230C22#1801195812yes—
231C22#1801195811yescharacteristic
232C22#1801195911yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—2982no—
238C4—2982no—
239C4—5962no—
240C4—5961no—
241C4—5961no—
242C4—5962no—
243C2—81no—
244C2#2375018811yescharacteristic
245C2#2375019711yescharacteristic
246C2#2375019811yescharacteristic
247C2#2375037612yes—
248C2#2375037812yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#399×C149C2
prime radicalC24×C149C23
hypercenter#399D149
hypocenterC149#2821
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#18011910C2semidirect
3. #23749565C2semidirect
4. #23749563C2semidirect
6. #23749558C2semidirect
8. #23749545C2semidirect
10. #23749492C2semidirect
124. C149#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#180119102
C2×#180119042
#237495652
#237495632
#237495612
#237495582
#237495532
#237495452
#237495372
#237494922
#399×C1492
#2821149