Finite Group Databasebeta

← #34047958

Subgroups — #34047958

Order 19072.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×Dic149C211yesmaximal, characteristic, complemented
2C2×#18011867C211yesmaximal, characteristic
3#23749565C211yesmaximal, characteristic, complemented
4#23749561C211yesmaximal, characteristic
5#23749559C212yesmaximal, complemented
6#23749548C212yesmaximal
7#23749544C211yesmaximal, characteristic, complemented
8#23749476C212yesmaximal, complemented
9#23749476C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic149C212yesmaximal
11#393×C149C211yesmaximal, characteristic
12C23×Dic149C2211yescharacteristic
13Q8×Dic149—24no—
14#18011867—24no—
15C2×C4⋊Dic149C2212yes—
16C2×C4⋊Dic149C2211yescharacteristic
17C2×#14558715C2212yes—
18C2×#14558715C2212yes—
19C2×#14558715C2212yes—
20C2×#14558715C2211yescharacteristic
21C2×#14558715C2211yescharacteristic
22C2×C22⋊Dic149C2212yes—
23C2×C22⋊Dic149C2211yescharacteristic
24C2×C4×Dic149C2212yes—
25C2×C4×Dic149C2212yes—
26C2×C4×Dic149C2211yescharacteristic
27C2×C4×Dic149C2211yescharacteristic
28#18011818C2212yes—
29#18011818C2212yes—
30#18011818C2211yescharacteristic
31#18011818C2212yes—
32C22×Q8×C149C2211yescharacteristic
33C2×C22⋊C4×C149C2212yes—
34C2×C22⋊C4×C149C2211yescharacteristic
35#95×C149C2212yes—
36#95×C149C2211yescharacteristic
37C22×Dic149—22no—
38C22×Dic149—22no—
39C22×Dic149Q814yescomplemented
40C22×Dic149C2311yescharacteristic
41C22×Dic149C2312yes—
42C22×Dic149C2311yescharacteristic
43C22×Dic149C2311yescharacteristic
44C22×Dic149C2312yes—
45C22×Dic149C2311yescharacteristic
46C4⋊Dic149—24no—
47C4⋊Dic149—22no—
48#14558715—22no—
49#14558715—24no—
50#14558715—24no—
51#14558715—22no—
52#14558715—24no—
53C22⋊Dic149—22no—
54C22⋊Dic149—24no—
55C4×Dic149—22no—
56C4×Dic149—24no—
57C4×Dic149—24no—
58C4×Dic149D414yes—
59C24×C149C2311yescharacteristic
60C2×Q8×C149—22no—
61C2×Q8×C149—24no—
62C22×C4×C149C2312yes—
63C22×C4×C149C2311yescharacteristic
64C22×C4×C149C2311yescharacteristic
65C22×C4×C149C2312yes—
66C22⋊C4×C149—22no—
67C22⋊C4×C149—24no—
68C2×Dic149—21no—
69C2×Dic149—21no—
70C2×Dic149—22no—
71C2×Dic149—22no—
72C2×Dic149—22no—
73C2×Dic149—28no—
74C2×Dic149—21no—
75C2×Dic149—22no—
76C2×Dic149—22no—
77C2×Dic149—22no—
78C2×Dic149—22no—
79C2×Dic149C2×D412yes—
80C2×Dic149C2×D412yes—
81C2×Dic149C2×Q812yes—
82C2×Dic149C2×Q812yes—
83C2×Dic149#4114yes—
84C2×Dic149#4112yes—
85C23×C149—21no—
86C23×C149—21no—
87C23×C149—22no—
88C23×C149—22no—
89C23×C149C2×Q812yes—
90C23×C149C2411yescharacteristic
91Q8×C149—28no—
92C2×C4×C149—22no—
93C2×C4×C149—22no—
94C2×C4×C149—22no—
95C2×C4×C149—21no—
96C2×C4×C149—22no—
97C2×C4×C149—22no—
98C2×C4×C149—22no—
99C2×C4×C149—22no—
100C2×C4×C149C2×D412yes—
101C2×C4×C149#4114yes—
102C22×C149—22no—
103C22×C149—24no—
104C22×C149—41no—
105C22×C149—42no—
106C22×C149—41no—
107C22×C149C22×D411yescharacteristic
108C22×C149C22×Q811yescharacteristic
109C22×C149C2×#4111yescharacteristic
110C22×C149C2×#4112yes—
111C22×C149C2×#4112yes—
112C4×C149—22no—
113C4×C149—24no—
114C4×C149—42no—
115C4×C149—41no—
116Dic149—22no—
117Dic149—24no—
118Dic149—22no—
119Dic149—41no—
120Dic149—42no—
121Dic149C22⋊Q814yes—
122C2×C149—42no—
123C2×C149C2×C22⋊Q811yescharacteristic
124C2×C149#52511yescharacteristic
125C2×C149#52912yes—
126C2×C149#53211yescharacteristic
127C2×C149#53312yes—
128C149#239611yescharacteristic, complemented, Sylow
129#2396—1491nomaximal, Sylow
130C2×C4×Q8—1491no—
131C2×#117—1491no—
132#400—1491no—
133#399—1491no—
134#398—1492no—
135#396—1492no—
136#395—1491no—
137#393D14911yescharacteristic
138C4×C4⋊C4—1491no—
139C4×C4⋊C4—1492no—
140C4×C22⋊C4—1492no—
141C22×Q8C2×D14911yescharacteristic
142C23×C4—1491no—
143C4×Q8—2984no—
144#117—2984no—
145C2×C4⋊C4—1491no—
146C2×C4⋊C4—1491no—
147C2×C4⋊C4—1492no—
148C2×C4⋊C4—1491no—
149C2×C4⋊C4—1492no—
150C2×C4⋊C4—1492no—
151C2×C4⋊C4—1492no—
152C2×C22⋊C4—1491no—
153C2×C22⋊C4—1492no—
154C2×C22⋊C4C2×D14911yescharacteristic
155C2×C22⋊C4C2×D14912yes—
156C2×C42—1491no—
157C2×C42—1492no—
158C2×C42—1492no—
159C2×C42—1491no—
160#95—1492no—
161#95—1492no—
162#95—1491no—
163#95—1492no—
164#95C2×D14912yes—
165#95C2×D14911yescharacteristic
166C24C22×D14911yescharacteristic
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—2982no—
170C22×C4—2982no—
171C22×C4—1491no—
172C22×C4—1492no—
173C22×C4—1494no—
174C22×C4—1491no—
175C22×C4—1492no—
176C22×C4—1491no—
177C22×C4—1491no—
178C22×C4C22×D14912yes—
179C22×C4C22×D14911yescharacteristic
180C22×C4C22×D14911yescharacteristic
181C22×C4C22×D14912yes—
182C4⋊C4—2982no—
183C4⋊C4—2984no—
184C4⋊C4—2984no—
185C4⋊C4—2984no—
186C4⋊C4—2982no—
187C4⋊C4—2982no—
188C4⋊C4—2984no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—2982no—
192C22⋊C4—2984no—
193C42—2982no—
194C42—2984no—
195C42—2984no—
196C42—1494no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D14912yes—
202C23C23×D14911yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—2981no—
213C2×C4—2982no—
214C2×C4—2982no—
215C2×C4—2981no—
216C2×C4—2982no—
217C2×C4—2988no—
218C2×C4—2981no—
219C2×C4—2982no—
220C2×C4—2982no—
221C2×C4—2982no—
222C2×C4—2982no—
223C2×C4—1492no—
224C2×C4—1492no—
225C2×C4—1494no—
226C2×C4—1492no—
227C2×C4—1492no—
228C2×C4—1492no—
229C2×C4D4×D14912yes—
230C2×C4#1455873514yes—
231C22—24no—
232C22—22no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22C2×D4×D14911yescharacteristic
237C22C2×Q8×D14911yescharacteristic
238C22#41×D14911yescharacteristic
239C22#41×D14912yes—
240C22C2×#1455873512yes—
241C4—22no—
242C4—24no—
243C4—42no—
244C4—41no—
245C4—2984no—
246C4—2982no—
247C4—2982no—
248C4—5962no—
249C4—5961no—
250C4—1494no—
251C2—42no—
252C2C22⋊Q8×D14911yescharacteristic
253C2#2375013111yescharacteristic
254C2#2375015412yes—
255C2#2375017411yescharacteristic
256C2#2375017712yes—
subgroupG/H
center, FrattiniC23C23×D149
commutator, socle, cosocleC23×C149C24
Fitting#393×C149C2
prime radicalC24×C149C23
hypercenter#393D149
hypocenterC149#2396
LCSG ▷ C23×C149 ▷ C149
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic149C2semidirect
3. #23749565C2semidirect
5. #23749559C2semidirect
7. #23749544C2semidirect
8. #23749476C2semidirect
9. #23749476C2semidirect
39. C22×Dic149Q8semidirect
128. C149#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1492
C2×#180118672
#237495652
#237495612
#237495592
#237495482
#237495442
#237494762
C22⋊C4×Dic1492
#393×C1492
#2396149