Finite Group Databasebeta

← #29600263

Subgroups — #29600263

Order 15616.

498 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 71 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#21455167C211yesmaximal, characteristic
2#21444411C211yesmaximal, characteristic, complemented
3#21443969C214yesmaximal, complemented
4C61×#2126C211yesmaximal, characteristic, complemented
5#16840598C411yescharacteristic, complemented
6#16840560—21no—
7#16839944C412yescomplemented
8#16839939—21no—
9#16839934C2212yes—
10#16839929—21no—
11C2×#14519705—21no—
12#16839746C411yescharacteristic, complemented
13C4×#11758194—21no—
14#16839502—21no—
15#16839502C2212yescomplemented
16C61×#532C2211yescharacteristic
17C61×#410—21no—
18C61×#354C2212yescomplemented
19C61×#353—21no—
20#14519825C2×C412yes—
21#14519824—41no—
22#14519823—41no—
23#14519822—21no—
24#14519822—41no—
25C2×#11758222—21no—
26C2×#11758222—41no—
27#41×D61—42no—
28#14519808—22no—
29#14519807—41no—
30#14519803—22no—
31#14519802—41no—
32#14519797—42no—
33C2×#11758194—21no—
34C2×#11758194—41no—
35#14519705—22no—
36#14519705—22no—
37#14519696C2×C412yes—
38#14519694—21no—
39#14519688—22no—
40C4×#92418—41no—
41C2×#41×C61C2×C411yescharacteristic
42C2×#41×C61D411yescharacteristic
43C4⋊Q8×C61C2×C411yescharacteristic
44#127×C61D411yescharacteristic
45#124×C61D411yescharacteristic
46#124×C61C2311yescharacteristic
47#121×C61—41no—
48C4×D4×C61—41no—
49#117×C61—41no—
50C2×C42×C61D411yescharacteristic
51#99×C61—22no—
52#99×C61—41no—
53#41×C61—41no—
54#41×C61—41no—
55#41×C61—41no—
56#41×C61—41no—
57C2×Q8×C61C22⋊C411yescharacteristic
58C2×Q8×C61C22×C411yescharacteristic
59C2×D4×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—41no—
62C2×D4×C61C22⋊C411yescharacteristic
63C2×D4×C61C2×D411yescharacteristic
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C2×C22⋊D61—42no—
69C2×C22⋊D61—41no—
70#11758226—42no—
71Q8×D61—42no—
72#11758224—42no—
73#11758224—44no—
74D4×D61—42no—
75D4×D61—44no—
76#11758222—22no—
77#11758222—22no—
78#11758222—41no—
79#11758222—42no—
80#11758222—41no—
81#11758222—42no—
82#11758222—42no—
83C2×D244—21no—
84C2×D244—41no—
85C2×C4×D61—41no—
86C2×C4×D61—42no—
87C2×Dic122—21no—
88C2×Dic122—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C42×C61—21no—
95C42×C61—81no—
96C42×C61C22⋊C411yescharacteristic
97C42×C61C2×D411yescharacteristic
98#11758202—42no—
99#11758201—42no—
100#11758200—42no—
101#11758199—42no—
102#11758194—22no—
103#11758194—41no—
104#11758194—44no—
105C2×#92418—42no—
106C2×#92418—41no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61#9911yescharacteristic
110C22×D61—41no—
111C22×D61—42no—
112Q8×C61—41no—
113Q8×C61—41no—
114D4×C61—41no—
115D4×C61—41no—
116D4×C61—42no—
117D4×C61—82no—
118D4×C61—81no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—82no—
131C2×C4×C61#9911yescharacteristic
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C22⋊D61—42no—
134C22⋊D61—44no—
135C22⋊D61—42no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138D244—22no—
139D244—42no—
140D244—41no—
141C4×D61—42no—
142C4×D61—44no—
143C4×D61—42no—
144Dic122—22no—
145Dic122—41no—
146Dic122—42no—
147#92418—42no—
148#92418—44no—
149#13568—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—81no—
155C22×C61—81no—
156C22×C61C2×#9911yescharacteristic
157C2×D61—42no—
158C2×D61—161no—
159C2×D61—162no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—162no—
166C4×C61AΣL(1,𝔽16)11yescharacteristic
167C4×C61AΣL(1,𝔽16)11yescharacteristic
168Dic61—42no—
169#3018—611no—
170#2126D6111yescharacteristic, complemented
171#1525—611no—
172#1404—614no—
173C2×C61—21no—
174C2×C61—41no—
175C2×C61—81no—
176C2×C61—81no—
177C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
178D61—82no—
179#585—611no—
180#575—1221no—
181#532C2×D6111yescharacteristic
182#456—612no—
183#455—1221no—
184#454—612no—
185#453—1221no—
186C2×#104—1221no—
187#413—611no—
188#410—21no—
189C4×M16—1221no—
190#354C2×D6112yescomplemented
191#353—21no—
192#329—612no—
193#329—1221no—
194C61#1356811yescharacteristic, complemented, Sylow
1952−1+4—612no—
1962−1+4—2441no—
197H(5,𝔽2)—1221no—
198H(5,𝔽2)—2441no—
199H(5,𝔽2)—2441no—
200C2×#41—1221no—
201C2×#41—2441no—
202C2×#41—2442no—
203C2×#41C4×D6111yescharacteristic
204C2×#41C22⋊D6111yescharacteristic
205#137—1222no—
206#137—2441no—
207AGL(1,ℤ/8)—1222no—
208AGL(1,ℤ/8)—2441no—
209#135—2442no—
210C2×M16—1221no—
211C2×M16—2441no—
212C4⋊Q8C4×D6111yescharacteristic
213#127C22⋊D6111yescharacteristic
214#124D24411yescharacteristic
215#124C22×D6111yescharacteristic
216#121—41no—
217C4×D4—41no—
218#117—41no—
219C2×C42D24411yescharacteristic
220#104—1222no—
221#104—1222no—
222#101—612no—
223#100—1221no—
224#99—22no—
225#99—41no—
226#97—1222no—
227C4×C8—2441no—
228#41—41no—
229#41—41no—
230#41—41no—
231#41—41no—
232#41—1222no—
233#41—1222no—
234#41—2441no—
235#41—2442no—
236#41—2442no—
237#41—2442no—
238#41—2442no—
239#41—2444no—
240#41—2442no—
241#41—2441no—
242C2×Q8—1221no—
243C2×Q8—2441no—
244C2×Q8—2442no—
245C2×Q8#1175819811yescharacteristic
246C2×Q8C2×C4×D6111yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—1221no—
251C2×D4—2441no—
252C2×D4—2442no—
253C2×D4—2444no—
254C2×D4—2442no—
255C2×D4—2441no—
256C2×D4#1175819811yescharacteristic
257C2×D4C2×C22⋊D6111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—2441no—
262C22×C4—2442no—
263C22×C4#1175819811yescharacteristic
264Q16—2442no—
265SD16—2442no—
266SD16—2442no—
267D8—2442no—
268M16—1222no—
269M16—2441no—
270M16—2444no—
271C2×C8—2442no—
272C2×C8—2441no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C42—21no—
279C42—81no—
280C42#1175819811yescharacteristic
281C42C2×D24411yescharacteristic
282C23—21no—
283C23—21no—
284C23—2441no—
285C23—2442no—
286C23#1451969111yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—1222no—
290Q8—2442no—
291Q8—2441no—
292D4—41no—
293D4—41no—
294D4—42no—
295D4—82no—
296D4—81no—
297D4—1222no—
298D4—2442no—
299D4—2442no—
300D4—2444no—
301D4—2442no—
302D4—2441no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—82no—
315C2×C4—2442no—
316C2×C4—2444no—
317C2×C4—2441no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4#1451969111yescharacteristic
321C2×C4C2×#1175819811yescharacteristic
322C8—2442no—
323C8—2444no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22—9761no—
331C22—9762no—
332C22—2442no—
333C22C2×#1451969111yescharacteristic
334C4—22no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—162no—
340C4—2442no—
341C4#1683954811yescharacteristic
342C4#1683954811yescharacteristic
343C2—21no—
344C2—41no—
345C2—81no—
346C2—81no—
347C2—4882no—
348C2C2×#1683954811yescharacteristic
subgroupG/H
centerC4#16839548
commutatorC2×Q8×C61C22×C4
Frattini#124C22×D61
FittingC61×#2126C2
socleC2×C61C2×AΣL(1,𝔽16)
prime radical#21455167C2
cosocle#124×C61C23
hypercenter#2126D61
hypocenterC61#13568
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
2. #21444411C2semidirect
3. #21443969C2semidirect
4. C61×#2126C2semidirect
5. #16840598C4semidirect
7. #16839944C4semidirect
12. #16839746C4semidirect
15. #16839502C22semidirect
18. C61×#354C22semidirect
170. #2126D61semidirect
190. #354C2×D61semidirect
194. C61#13568semidirect

Maximal subgroups

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

subgroupindex
#214551672
#214444112
#214439692
C61×#21262
#1356861