Finite Group Databasebeta

← #29600233

Subgroups — #29600233

Order 15616.

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

normal subgroups only

This group has 486 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#21447484C211yesmaximal, characteristic, complemented
3#21443965C214yesmaximal, complemented
4C61×#1543C211yesmaximal, characteristic, complemented
5#16840598C411yescharacteristic, complemented
6#16840560—21no—
7#16839944—21no—
8#16839939C412yescomplemented
9#16839934—21no—
10#16839929C2212yes—
11C2×#14519705—21no—
12#16839746C411yescharacteristic, complemented
13#16839736—21no—
14#16839557—21no—
15#16839555C2212yescomplemented
16C61×#532C2211yescharacteristic
17C61×#431—21no—
18C61×#333—21no—
19C61×#331C2212yescomplemented
20#14519825—21no—
21#14519824—41no—
22#14519823—41no—
23#14519822—41no—
24#14519822C2×C412yes—
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—
34#14519736—41no—
35#14519705—22no—
36#14519705—22no—
37#14519696—21no—
38#14519694C2×C412yes—
39#14519692—22no—
40#14519692—41no—
41C2×#41×C61C2×C411yescharacteristic
42C2×#41×C61D411yescharacteristic
43C4⋊Q8×C61D411yescharacteristic
44#127×C61C2311yescharacteristic
45#124×C61C2×C411yescharacteristic
46#124×C61D411yescharacteristic
47#121×C61—41no—
48C4×D4×C61—41no—
49C2×C42×C61D411yescharacteristic
50C4⋊C8×C61—22no—
51C22⋊C8×C61—41no—
52C4×C8×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—41no—
69C2×C22⋊D61—42no—
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—42no—
81#11758222—42no—
82#11758222—41no—
83C2×D244—21no—
84C2×D244—41no—
85C2×C4×D61—41no—
86C2×C4×D61—42no—
87C2×Dic122—21no—
88C2×Dic122—41no—
89C2×C8×C61—42no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C42×C61—21no—
94C42×C61C22⋊C411yescharacteristic
95C42×C61C2×D411yescharacteristic
96C22⋊Dic61—42no—
97#11758202—42no—
98#11758201—42no—
99#11758200—42no—
100#11758199—42no—
101C4⋊Dic61—81no—
102C4×Dic61—81no—
103#11758194—22no—
104#11758194—41no—
105C2×#92418—41no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61#9911yescharacteristic
109C22×D61—41no—
110C22×D61—42no—
111Q8×C61—41no—
112Q8×C61—41no—
113D4×C61—41no—
114D4×C61—41no—
115D4×C61—42no—
116D4×C61—82no—
117D4×C61—81no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61#9911yescharacteristic
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C22⋊D61—42no—
132C22⋊D61—44no—
133C22⋊D61—42no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—82no—
137D244—22no—
138D244—42no—
139D244—41no—
140C4×D61—42no—
141C4×D61—44no—
142C4×D61—42no—
143Dic122—22no—
144Dic122—42no—
145Dic122—41no—
146C8×C61—82no—
147#92418—42no—
148#13560—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61C2×#9911yescharacteristic
156C2×D61—42no—
157C2×D61—161no—
158C2×D61—162no—
159C4×C61—22no—
160C4×C61—21no—
161C4×C61—41no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61AΣL(1,𝔽16)11yescharacteristic
165C4×C61AΣL(1,𝔽16)11yescharacteristic
166Dic61—42no—
167Dic61—162no—
168#3018—611no—
169#2126—611no—
170#1543D6111yescharacteristic, complemented
171#1402—614no—
172C2×C61—21no—
173C2×C61—41no—
174C2×C61—81no—
175C2×C61—81no—
176C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
177D61—82no—
178#585—611no—
179#575—1221no—
180#532C2×D6111yescharacteristic
181#456—1221no—
182#455—612no—
183#454—1221no—
184#453—612no—
185#431—21no—
186C2×#104—1221no—
187#413—611no—
188#410—1221no—
189#354—1221no—
190#353—612no—
191#333—21no—
192#331C2×D6112yescomplemented
193C61#1356011yescharacteristic, complemented, Sylow
1942−1+4—1221no—
1952−1+4—2441no—
196H(5,𝔽2)—612no—
197H(5,𝔽2)—2441no—
198H(5,𝔽2)—2441no—
199C2×#41—1221no—
200C2×#41—2441no—
201C2×#41—2442no—
202C2×#41C4×D6111yescharacteristic
203C2×#41D24411yescharacteristic
204#137—1222no—
205#137—2441no—
206AGL(1,ℤ/8)—1222no—
207AGL(1,ℤ/8)—2441no—
208#135—2442no—
209C2×M16—1221no—
210C4⋊Q8C22⋊D6111yescharacteristic
211#127C22×D6111yescharacteristic
212#124C4×D6111yescharacteristic
213#124D24411yescharacteristic
214#121—41no—
215C4×D4—41no—
216#117—2441no—
217C2×C42C22⋊D6111yescharacteristic
218C4⋊C8—22no—
219#104—1222no—
220#104—1222no—
221#101—1221no—
222#100—612no—
223#99—1222no—
224#99—2441no—
225C22⋊C8—41no—
226C4×C8—41no—
227#41—41no—
228#41—41no—
229#41—41no—
230#41—41no—
231#41—1222no—
232#41—1222no—
233#41—2441no—
234#41—2442no—
235#41—2442no—
236#41—2442no—
237#41—2444no—
238#41—2442no—
239#41—2441no—
240#41—2442no—
241C2×Q8—1221no—
242C2×Q8—2441no—
243C2×Q8—2442no—
244C2×Q8#1175819811yescharacteristic
245C2×Q8#1175819811yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—1221no—
250C2×D4—2441no—
251C2×D4—2442no—
252C2×D4—2444no—
253C2×D4—2441no—
254C2×D4—2442no—
255C2×D4C2×C4×D6111yescharacteristic
256C2×D4C2×D24411yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2441no—
261C22×C4—2442no—
262C22×C4#1175819811yescharacteristic
263Q16—2442no—
264SD16—2442no—
265SD16—2442no—
266D8—2442no—
267M16—1222no—
268M16—2441no—
269C2×C8—42no—
270C2×C8—2441no—
271C4⋊C4—41no—
272C4⋊C4—4881no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—2442no—
276C42—21no—
277C42—4881no—
278C42#1175819811yescharacteristic
279C42C2×C22⋊D6111yescharacteristic
280C23—21no—
281C23—21no—
282C23—2441no—
283C23—2442no—
284C23#1451969111yescharacteristic
285Q8—41no—
286Q8—41no—
287Q8—1222no—
288Q8—2442no—
289Q8—2441no—
290D4—41no—
291D4—41no—
292D4—42no—
293D4—82no—
294D4—81no—
295D4—1222no—
296D4—2442no—
297D4—2442no—
298D4—2444no—
299D4—2442no—
300D4—2441no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—4882no—
313C2×C4—2442no—
314C2×C4—2444no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4#1451969111yescharacteristic
319C2×C4C2×#1175819811yescharacteristic
320C8—82no—
321C8—2442no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22—9761no—
329C22—9762no—
330C22—2442no—
331C22C2×#1451969111yescharacteristic
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—9762no—
338C4—2442no—
339C4#1683955011yescharacteristic
340C4#1683955011yescharacteristic
341C2—21no—
342C2—41no—
343C2—81no—
344C2—81no—
345C2—4882no—
346C2C2×#1683955011yescharacteristic
subgroupG/H
centerC4#16839550
commutatorC2×D4×C61C22×C4
Frattini#127C22×D61
FittingC61×#1543C2
socleC2×C61C2×AΣL(1,𝔽16)
prime radical#21455167C2
cosocle#127×C61C23
hypercenter#1543D61
hypocenterC61#13560
LCSG ▷ C2×D4×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1543

Complements

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

NH ≅ G/Ntype
2. #21447484C2semidirect
3. #21443965C2semidirect
4. C61×#1543C2semidirect
5. #16840598C4semidirect
8. #16839939C4semidirect
12. #16839746C4semidirect
15. #16839555C22semidirect
19. C61×#331C22semidirect
170. #1543D61semidirect
192. #331C2×D61semidirect
193. C61#13560semidirect

Maximal subgroups

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

subgroupindex
#214551672
#214474842
#214439652
C61×#15432
#1356061