Finite Group Databasebeta

← #34054917

Subgroups — #34054917

Order 19072.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750244C211yesmaximal, characteristic
2#23750101C211yesmaximal, characteristic, complemented
3#23749964C211yesmaximal, characteristic, complemented
4#23749948C211yesmaximal, characteristic
5#23749920C211yesmaximal, characteristic, complemented
6#23749911C211yesmaximal, characteristic, complemented
7#23749844C211yesmaximal, characteristic, complemented
8#23749779C211yesmaximal, characteristic, complemented
9#23749762C211yesmaximal, characteristic
10#23749732C211yesmaximal, characteristic, complemented
11#23749722C211yesmaximal, characteristic, complemented
12#23749717C211yesmaximal, characteristic, complemented
13SD16×Dic149C211yesmaximal, characteristic, complemented
14#23749685C211yesmaximal, characteristic, complemented
15#464×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17C2×#14558726C2211yescharacteristic
18C2×#14558724C2211yescharacteristic
19#18011911C2211yescharacteristic
20#18011904—21no—
21#18011897—21no—
22#18011895—21no—
23#18011888—21no—
24#18011887C2211yescharacteristic
25#18011882C2211yescharacteristic
26Q8×Dic149C2211yescharacteristic
27Q8×Dic149C2211yescharacteristic
28C4×Dic298—21no—
29C4×Dic298C2211yescharacteristic
30#18011876—21no—
31#18011876C2211yescharacteristic
32D4×Dic149C2211yescharacteristic
33#18011869—21no—
34#18011869C2211yescharacteristic
35#18011868—21no—
36#18011868C2211yescharacteristic
37#18011844C2211yescharacteristic
38#18011844C2211yescharacteristic
39#18011843C2211yescharacteristic
40#18011843C2211yescharacteristic
41C4⋊#11924746C2211yescharacteristic
42#18011836C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011835C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011832C2211yescharacteristic
48#18011823C2211yescharacteristic
49#18011823C2211yescharacteristic
50#18011821C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C2×Q16×C149C2211yescharacteristic
53C2×SD16×C149C2211yescharacteristic
54#124×C149C2211yescharacteristic
55C4×Q8×C149C2211yescharacteristic
56C4⋊C8×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×Dic298—21no—
61C2×Dic298C2311yescharacteristic
62Q8×D149—22no—
63Q8×D149—22no—
64C2×C22⋊D149—21no—
65C22×Dic149—21no—
66C2×C4×D149—21no—
67C2×C4×D149C2311yescharacteristic
68#14558726—22no—
69#14558724—22no—
70C2×#11924746C2311yescharacteristic
71C2×#11924746C2311yescharacteristic
72C4⋊Dic149—21no—
73C4⋊Dic149—21no—
74C4⋊Dic149—21no—
75C4⋊Dic149C2311yescharacteristic
76C4⋊Dic149C2311yescharacteristic
77C4⋊Dic149C2311yescharacteristic
78#14558715—21no—
79#14558715—21no—
80#14558715—21no—
81#14558715D411yescharacteristic
82#14558715D411yescharacteristic
83C22⋊Dic149—21no—
84C22⋊Dic149—21no—
85#14558713—21no—
86#14558713—21no—
87#14558713D411yescharacteristic
88#14558713D411yescharacteristic
89C4×Dic149—21no—
90C4×Dic149—21no—
91C4×Dic149—21no—
92C4×Dic149—21no—
93C4×Dic149—21no—
94C4×Dic149C2311yescharacteristic
95C2×Q8×C149C2311yescharacteristic
96C2×Q8×C149C2311yescharacteristic
97C2×D4×C149C2311yescharacteristic
98Q16×C149—22no—
99SD16×C149—22no—
100C2×C8×C149C2311yescharacteristic
101C2×C8×C149C2311yescharacteristic
102C4⋊C4×C149—21no—
103C4⋊C4×C149C2311yescharacteristic
104C22⋊C4×C149—21no—
105C22⋊C4×C149—21no—
106C42×C149—21no—
107C42×C149C2311yescharacteristic
108C22×D149C2×D411yescharacteristic
109Dic298—24no—
110Dic298—21no—
111Dic298—81no—
112Dic298#4112yes—
113C22⋊D149—81no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149—21no—
118C2×Dic149—81no—
119C2×Dic149C2×D411yescharacteristic
120C2×Dic149C2×D411yescharacteristic
121C2×Dic149C2×D411yescharacteristic
122C4×D149—24no—
123C4×D149—21no—
124C4×D149—21no—
125#11924746—21no—
126#11924746—21no—
127C23×C149—21no—
128Q8×C149—21no—
129Q8×C149—21no—
130Q8×C149—21no—
131Q8×C149#4112yes—
132D4×C149—21no—
133D4×C149—21no—
134C2×C4×C149—21no—
135C2×C4×C149—21no—
136C2×C4×C149—21no—
137C2×C4×C149C2×D411yescharacteristic
138C2×C4×C149C2×D411yescharacteristic
139C2×C4×C149C2411yescharacteristic
140C8×C149—21no—
141C8×C149—21no—
142C22×C149—41no—
143C22×C149—41no—
144C22×C149—41no—
145C22×C149C22×D411yescharacteristic
146C2×D149—21no—
147C2×D149—21no—
148C2×D149—21no—
149C4×C149—22no—
150C4×C149—21no—
151C4×C149—21no—
152C4×C149—41no—
153C4×C149—41no—
154C4×C149C2×#4111yescharacteristic
155C4×C149C2×#4111yescharacteristic
156Dic149—22no—
157Dic149—21no—
158Dic149—21no—
159Dic149—21no—
160Dic149—41no—
161Dic149—41no—
162Dic149—41no—
163C2×C149—81no—
164C2×C149#52511yescharacteristic
165C2×C149#57711yescharacteristic
166C2×C149#57811yescharacteristic
167D149—41no—
168C149#312111yescharacteristic, complemented, Sylow
169#3121—1491nomaximal, Sylow
170#542—1491no—
171#519—1491no—
172#485—1491no—
173#482—1491no—
174#477—1491no—
175#475—1491no—
176#464—1491no—
177#464D14911yescharacteristic, complemented
178#451—1491no—
179#448—1491no—
180#441—1491no—
181#440—1491no—
182C4×Q16—1491no—
183C4×SD16—1491no—
184#433—1491no—
185C22×Q8—1491no—
186C2×Q16—1491no—
187C2×Q16C2×D14911yescharacteristic
188C2×SD16—1491no—
189C2×SD16C2×D14911yescharacteristic
190C4⋊Q8—1491no—
191#126—2981no—
192#124—2981no—
193#124C2×D14911yescharacteristic
194#123—2981no—
195C22⋊Q8—2981no—
196C22⋊Q8—1491no—
197#121—1491no—
198C4×Q8—2981no—
199C4×Q8—2981no—
200C4×Q8—1491no—
201C4×Q8—1491no—
202C4×Q8—1491no—
203C4×Q8—1491no—
204C4×Q8C2×D14911yescharacteristic
205C4×D4—1491no—
206#117—2981no—
207#117—2981no—
208#117—1491no—
209#117—1491no—
210#107—1491no—
211#107—1491no—
212#106—1491no—
213#106—1491no—
214C4⋊C8—1491no—
215C4⋊C8C2×D14911yescharacteristic
216#103—1491no—
217#103—1491no—
218#103—1491no—
219#103—1491no—
220#103—1491no—
221#103C2×D14911yescharacteristic
222#102—1491no—
223#102C2×D14911yescharacteristic
224C22⋊C8—1491no—
225C22⋊C8—1491no—
226#97—1491no—
227C4×C8—1491no—
228C2×Q8—2981no—
229C2×Q8—2982no—
230C2×Q8—2982no—
231C2×Q8—2981no—
232C2×Q8—1491no—
233C2×Q8C22×D14911yescharacteristic
234C2×Q8C22×D14911yescharacteristic
235C2×D4—2981no—
236C2×D4C22×D14911yescharacteristic
237C22×C4—2981no—
238C22×C4—2981no—
239C22×C4—1491no—
240Q16—22no—
241Q16—2982no—
242SD16—22no—
243SD16—2982no—
244C2×C8—1491no—
245C2×C8—1491no—
246C2×C8C22×D14911yescharacteristic
247C2×C8C22×D14911yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—2981no—
250C4⋊C4—2981no—
251C4⋊C4—2981no—
252C4⋊C4—2981no—
253C4⋊C4—2981no—
254C4⋊C4—2981no—
255C4⋊C4—1491no—
256C4⋊C4—1491no—
257C4⋊C4—1491no—
258C4⋊C4—1491no—
259C4⋊C4—1491no—
260C4⋊C4C22×D14911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2981no—
264C22⋊C4—2981no—
265C22⋊C4—2981no—
266C22⋊C4—2981no—
267C22⋊C4—1491no—
268C22⋊C4—1491no—
269C42—21no—
270C42—2981no—
271C42—2981no—
272C42—2981no—
273C42—2981no—
274C42—2981no—
275C42—1491no—
276C42C22×D14911yescharacteristic
277C23—21no—
278C23—1491no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—2984no—
283Q8—2981no—
284Q8—1492no—
285Q8—11921no—
286Q8#1455873512yes—
287D4—21no—
288D4—21no—
289D4—11921no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2984no—
294C2×C4—2981no—
295C2×C4—2981no—
296C2×C4—2981no—
297C2×C4—2981no—
298C2×C4—2981no—
299C2×C4—2981no—
300C2×C4—1491no—
301C2×C4—1491no—
302C2×C4—1491no—
303C2×C4—11921no—
304C2×C4D4×D14911yescharacteristic
305C2×C4D4×D14911yescharacteristic
306C2×C4C23×D14911yescharacteristic
307C8—21no—
308C8—21no—
309C8—2981no—
310C8—2981no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—2981no—
315C22—2981no—
316C22—2981no—
317C22C2×D4×D14911yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—2981no—
324C4—2981no—
325C4—2982no—
326C4—2981no—
327C4—5961no—
328C4—5961no—
329C4—5961no—
330C4#41×D14911yescharacteristic
331C4C2×#1455873511yescharacteristic
332C2—81no—
333C2—5961no—
334C2#2375013511yescharacteristic
335C2#2375043211yescharacteristic
336C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#464×C149C2
socleC22×C149C22×D4
prime radical#18011882C22
hypercenter#464D149
hypocenterC149#3121
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #23750101C2semidirect
3. #23749964C2semidirect
5. #23749920C2semidirect
6. #23749911C2semidirect
7. #23749844C2semidirect
8. #23749779C2semidirect
10. #23749732C2semidirect
11. #23749722C2semidirect
12. #23749717C2semidirect
13. SD16×Dic149C2semidirect
14. #23749685C2semidirect
15. #464×C149C2semidirect
168. C149#3121semidirect
177. #464D149semidirect

Maximal subgroups

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

subgroupindex
#237502442
#237501012
#237499642
#237499482
#237499202
#237499112
#237498442
#237497792
#237497622
#237497322
#237497222
#237497172
SD16×Dic1492
#237496852
#464×C1492
#3121149