Subgroups — #30613646

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462614	C₂	1	2	yes	maximal
2	#21461457	C₂	1	2	yes	maximal, complemented
3	#21461432	C₂	1	2	yes	maximal, complemented
4	#21461431	C₂	1	2	yes	maximal, complemented
5	#21461406	C₂	1	2	yes	maximal, complemented
6	C₂×#16840626	C₂	1	1	yes	maximal, characteristic
7	#21461127	C₂	1	2	yes	maximal
8	C₈×#11758217	C₂	1	1	yes	maximal, characteristic
9	#21456400	C₂	1	1	yes	maximal, characteristic
10	#16840814	—	2	2	no	—
11	#16840813	C₂²	1	2	yes	—
12	#16840812	—	2	2	no	—
13	#16840812	C₂²	1	2	yes	—
14	#16840811	C₂²	1	2	yes	—
15	Q₈×#92420	C₂²	1	2	yes	—
16	#16840779	C₂²	1	1	yes	characteristic
17	#16840768	—	2	2	no	—
18	C₂×C₄⋊#92420	C₂²	1	2	yes	—
19	#16840726	—	2	2	no	—
20	C₄×#11758217	C₂²	1	1	yes	characteristic
21	#16840626	C₂²	1	1	yes	characteristic
22	#16840626	C₂²	1	2	yes	—
23	#16840626	C₂²	1	1	yes	characteristic
24	#16840612	C₂²	1	2	yes	—
25	#16840612	C₂²	1	2	yes	—
26	Q₈⋊#92420	C₂²	1	2	yes	—
27	Q₈⋊#92420	C₂²	1	2	yes	—
28	#92420⋊C₈	C₂²	1	2	yes	—
29	C₈×#92420	C₂²	1	2	yes	—
30	#16840314	C₂²	1	2	yes	—
31	#16840166	C₄	1	1	yes	characteristic
32	#16840140	C₄	1	1	yes	characteristic
33	#16840111	C₄	1	2	yes	—
34	#16840003	C₄	1	1	yes	characteristic
35	#16840001	C₄	1	2	yes	—
36	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
37	#16839856	C₂²	1	1	yes	characteristic
38	#16839854	C₂²	1	2	yes	—
39	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
40	C₆₁×#467	C₄	1	1	yes	characteristic
41	C₂×#11758217	D₄	1	1	yes	characteristic
42	C₂×#11758217	D₄	1	1	yes	characteristic
43	C₂²×#92420	—	2	2	no	—
44	#14519844	—	2	2	no	—
45	#14519844	—	2	2	no	—
46	#14519844	—	2	1	no	—
47	C₄⋊#92420	—	2	2	no	—
48	C₄⋊#92420	—	2	2	no	—
49	C₄⋊#92420	—	2	2	no	—
50	C₄⋊#92420	C₂³	1	2	yes	—
51	C₄⋊#92420	C₂³	1	2	yes	—
52	C₂²⋊#92420	—	2	1	no	—
53	#14519832	—	2	2	no	—
54	#14519832	—	2	2	no	—
55	#14519832	D₄	1	2	yes	—
56	C₄×#92420	—	2	2	no	—
57	C₄×#92420	—	2	2	no	—
58	C₄×#92420	C₂³	1	2	yes	—
59	C₄×#92420	C₂³	1	2	yes	—
60	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
61	C₂×#11758202	C₂×C₄	1	2	yes	—
62	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
63	#14519776	C₂×C₄	1	2	yes	—
64	#14519775	C₂×C₄	1	2	yes	—
65	#14519767	—	2	2	no	—
66	#14519756	C₂×C₄	1	2	yes	—
67	#14519754	—	2	2	no	—
68	Q₈×Dic₆₁	C₂³	1	2	yes	—
69	#14519742	—	2	2	no	—
70	#14519742	C₂³	1	2	yes	—
71	#14519733	—	2	2	no	—
72	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
73	#14519711	C₂×C₄	1	1	yes	characteristic
74	#14519710	C₂×C₄	1	2	yes	—
75	#14519702	C₂×C₄	1	2	yes	—
76	#14519701	C₂×C₄	1	2	yes	—
77	#14519700	C₂×C₄	1	2	yes	—
78	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
85	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
86	#103×C₆₁	C₂×C₄	1	2	yes	—
87	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₆₁	#41	1	1	yes	characteristic
91	C₂×Dic₁₂₂	—	2	2	no	—
92	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
93	#11758217	—	2	2	no	—
94	#11758217	Q₁₆	1	4	yes	complemented
95	C₂×#92420	—	2	2	no	—
96	C₂×#92420	—	2	2	no	—
97	C₂×#92420	—	2	2	no	—
98	C₂×#92420	—	4	4	no	—
99	C₂×#92420	C₂×D₄	1	2	yes	—
100	C₂×#92420	C₂×D₄	1	2	yes	—
101	Q₁₆×C₆₁	—	4	1	no	—
102	C₂×C₈×C₆₁	—	4	1	no	—
103	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₆₁	—	2	2	no	—
106	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₆₁	—	2	2	no	—
108	C₂²⋊Dic₆₁	—	2	2	no	—
109	#11758202	—	2	4	no	—
110	C₄⋊Dic₆₁	—	2	2	no	—
111	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
112	#11758196	—	2	2	no	—
113	#11758196	—	2	2	no	—
114	#11758196	—	2	2	no	—
115	#11758196	—	2	2	no	—
116	#11758196	—	2	2	no	—
117	C₄×Dic₆₁	—	2	2	no	—
118	C₄×Dic₆₁	—	2	2	no	—
119	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₆₁	#41	1	1	yes	characteristic
122	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
123	C₂×#92418	—	2	2	no	—
124	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
125	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
126	Dic₂₄₄	—	4	1	no	—
127	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₆₁	—	2	2	no	—
129	Q₈×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₆₁	—	2	2	no	—
137	C₂×Dic₆₁	—	2	2	no	—
138	C₂×Dic₆₁	—	4	1	no	—
139	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
140	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₆₁	C₂×Q₁₆	1	2	yes	—
142	C₂×Dic₆₁	#135	1	2	yes	—
143	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
145	Dic₁₂₂	—	2	2	no	—
146	Dic₁₂₂	—	2	2	no	—
147	Dic₁₂₂	—	8	2	no	—
148	#92420	—	2	2	no	—
149	#92420	—	4	2	no	—
150	#92420	—	4	2	no	—
151	#92420	—	4	2	no	—
152	#92420	C₂×Q₁₆	1	4	yes	—
153	C₈×C₆₁	—	2	1	no	—
154	C₈×C₆₁	—	2	1	no	—
155	#92418	—	2	1	no	—
156	#92418	#131	1	2	yes	—
157	#18068	—	61	1	no	maximal, Sylow
158	C₂²×C₆₁	—	2	1	no	—
159	C₂²×C₆₁	—	2	1	no	—
160	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
161	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₆₁	—	2	1	no	—
163	C₄×C₆₁	—	4	2	no	—
164	C₄×C₆₁	—	4	2	no	—
165	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
166	C₄×C₆₁	#568	1	1	yes	characteristic
167	Dic₆₁	—	2	1	no	—
168	Dic₆₁	—	2	1	no	—
169	Dic₆₁	—	4	2	no	—
170	Dic₆₁	—	4	2	no	—
171	Dic₆₁	C₄×Q₁₆	1	2	yes	—
172	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
173	Dic₆₁	C₂×#135	1	1	yes	characteristic
174	#3170	—	61	1	no	—
175	#2962	—	61	2	no	—
176	#1591	—	61	2	no	—
177	#1584	—	61	2	no	—
178	#1584	—	61	2	no	—
179	C₈×Q₁₆	—	61	2	no	—
180	C₂×#335	—	61	1	no	—
181	#1481	—	61	2	no	—
182	C₈×M₁₆	—	61	1	no	—
183	C₂×C₆₁	—	2	2	no	—
184	C₂×C₆₁	#2935	1	1	yes	characteristic
185	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
186	C₂×C₆₁	#2957	1	1	yes	characteristic
187	#533	—	61	2	no	—
188	#500	—	61	1	no	—
189	#494	—	61	1	no	—
190	#487	—	61	2	no	—
191	#467	—	61	1	no	—
192	#467	—	61	2	no	—
193	#467	#6923	1	1	yes	characteristic
194	#446	—	61	2	no	—
195	#446	—	61	2	no	—
196	#446	—	122	2	no	—
197	#446	—	122	2	no	—
198	C₈×Q₈	—	61	2	no	—
199	C₈×Q₈	—	61	2	no	—
200	C₄×Q₁₆	—	61	1	no	—
201	C₄×Q₁₆	—	61	1	no	—
202	C₄×Q₁₆	—	61	2	no	—
203	#431	—	61	1	no	—
204	#424	—	122	2	no	—
205	C₂×C₄⋊C₈	—	61	2	no	—
206	#405	—	122	2	no	—
207	C₄×M₁₆	—	61	1	no	—
208	C₂×C₄×C₈	—	61	1	no	—
209	#335	—	61	1	no	—
210	#335	—	61	2	no	—
211	#335	—	61	1	no	—
212	Q₈⋊C₈	—	61	2	no	—
213	Q₈⋊C₈	—	61	2	no	—
214	Q₈⋊C₈	—	61	2	no	—
215	Q₈⋊C₈	—	61	2	no	—
216	#322	—	61	2	no	—
217	C₈²	—	61	2	no	—
218	C₆₁	#18068	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₁₆	—	61	2	no	—
220	C₂×Q₁₆	—	61	1	no	—
221	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
222	C₂×M₁₆	—	61	1	no	—
223	C₂×M₁₆	—	61	1	no	—
224	C₂²×C₈	—	61	1	no	—
225	C₂²×C₈	—	122	2	no	—
226	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
227	C₄⋊Q₈	—	61	2	no	—
228	C₄⋊Q₈	—	61	2	no	—
229	#125	—	122	2	no	—
230	C₂²⋊Q₈	—	61	2	no	—
231	C₂²⋊Q₈	—	122	2	no	—
232	C₂²⋊Q₈	C₂×#6923	1	2	yes	—
233	C₄×Q₈	—	61	2	no	—
234	C₄×Q₈	—	61	2	no	—
235	C₄×Q₈	—	122	2	no	—
236	#117	—	122	2	no	—
237	C₂×C₄²	—	61	1	no	—
238	#107	—	61	1	no	—
239	#107	—	61	2	no	—
240	#107	C₂×#6923	1	1	yes	characteristic
241	C₄⋊C₈	—	61	2	no	—
242	C₄⋊C₈	—	61	2	no	—
243	C₄⋊C₈	—	122	2	no	—
244	C₄⋊C₈	—	122	1	no	—
245	C₄⋊C₈	—	122	2	no	—
246	C₄⋊C₈	—	122	2	no	—
247	C₄⋊C₈	—	122	2	no	—
248	C₄⋊C₈	—	122	2	no	—
249	#103	—	61	2	no	—
250	#103	—	61	2	no	—
251	#103	—	61	2	no	—
252	#103	C₂×#6923	1	2	yes	—
253	C₂²⋊C₈	—	122	1	no	—
254	#97	—	61	2	no	—
255	#97	—	122	2	no	—
256	#97	—	122	2	no	—
257	C₄×C₈	—	61	2	no	—
258	C₄×C₈	—	61	1	no	—
259	C₄×C₈	—	61	1	no	—
260	C₄×C₈	—	61	1	no	—
261	C₄×C₈	—	61	1	no	—
262	C₄×C₈	—	61	2	no	—
263	C₄×C₈	—	122	2	no	—
264	C₄×C₈	—	122	2	no	—
265	C₂×Q₈	—	61	2	no	—
266	C₂×Q₈	—	122	2	no	—
267	C₂×Q₈	C₂²×#6923	1	2	yes	—
268	C₂²×C₄	—	61	1	no	—
269	C₂²×C₄	—	61	1	no	—
270	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
271	Q₁₆	—	4	1	no	—
272	Q₁₆	—	122	4	no	—
273	Q₁₆	—	244	1	no	—
274	M₁₆	—	61	4	no	—
275	M₁₆	—	122	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	61	2	no	—
278	C₂×C₈	—	61	1	no	—
279	C₂×C₈	—	61	2	no	—
280	C₂×C₈	—	61	1	no	—
281	C₂×C₈	—	122	2	no	—
282	C₂×C₈	—	122	2	no	—
283	C₂×C₈	—	122	2	no	—
284	C₂×C₈	—	122	2	no	—
285	C₂×C₈	—	244	4	no	—
286	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
287	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	61	2	no	—
290	C₄⋊C₄	—	122	2	no	—
291	C₄⋊C₄	—	122	2	no	—
292	C₄⋊C₄	—	122	2	no	—
293	C₄⋊C₄	—	122	2	no	—
294	C₄⋊C₄	—	122	2	no	—
295	C₄⋊C₄	—	122	2	no	—
296	C₄⋊C₄	C₂²×#6923	1	2	yes	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	122	2	no	—
299	C₄²	—	61	1	no	—
300	C₄²	—	61	1	no	—
301	C₄²	—	61	1	no	—
302	C₄²	—	61	1	no	—
303	C₄²	—	122	2	no	—
304	C₄²	—	122	2	no	—
305	C₂³	D₄×#6923	1	1	yes	characteristic
306	Q₈	—	2	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	122	2	no	—
309	Q₈	—	122	2	no	—
310	Q₈	—	488	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	61	2	no	—
316	C₂×C₄	—	61	2	no	—
317	C₂×C₄	—	61	1	no	—
318	C₂×C₄	—	61	1	no	—
319	C₂×C₄	—	61	1	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	122	2	no	—
322	C₂×C₄	—	122	2	no	—
323	C₂×C₄	—	244	1	no	—
324	C₂×C₄	D₄×#6923	1	1	yes	characteristic
325	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	61	4	no	—
329	C₈	—	61	2	no	—
330	C₈	—	122	2	no	—
331	C₈	—	122	1	no	—
332	C₈	—	244	2	no	—
333	C₈	—	244	2	no	—
334	C₈	—	244	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	Q₁₆×#6923	1	2	yes	—
338	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	61	1	no	—
343	C₄	—	61	1	no	—
344	C₄	—	61	2	no	—
345	C₄	—	122	1	no	—
346	C₄	—	122	1	no	—
347	C₄	—	244	2	no	—
348	C₄	—	244	2	no	—
349	C₄	D₆₁⋊#131	1	1	yes	characteristic
350	C₄	#16840840	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	#21462563	1	1	yes	characteristic
353	C₂	C₂×Q₁₆×#6923	1	1	yes	characteristic
354	C₂	#21462598	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#467	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	#467	#6923
hypocenter	C₆₁	#18068
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
2. #21461457	C₂	semidirect
3. #21461432	C₂	semidirect
4. #21461431	C₂	semidirect
5. #21461406	C₂	semidirect
94. #11758217	Q₁₆	semidirect
218. C₆₁	#18068	semidirect

Maximal subgroups

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

subgroup	index
#21462614	2
#21461457	2
#21461432	2
#21461431	2
#21461406	2
C₂×#16840626	2
#21461127	2
C₈×#11758217	2
#21456400	2
#18068	61