Subgroups — #30616507

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461988	C₂	1	2	yes	maximal
2	#21461916	C₂	1	2	yes	maximal
3	#21461872	C₂	1	4	yes	maximal
4	C₄×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#92420	C₂	1	2	yes	maximal
6	#21461533	C₂	1	2	yes	maximal
7	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#92420	C₂²	1	4	yes	—
10	C₂×C₄⋊#92420	C₂²	1	2	yes	—
11	C₂×C₄×#92420	C₂²	1	2	yes	—
12	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
13	C₂×C₄×#92420	C₂²	1	2	yes	—
14	C₂×C₄×#92420	C₂²	1	2	yes	—
15	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
16	#16840628	C₂²	1	4	yes	—
17	#16840628	C₂²	1	4	yes	—
18	#16840628	C₂²	1	2	yes	—
19	#16840628	C₂²	1	4	yes	—
20	C₂×Q₈×Dic₆₁	C₄	1	1	yes	characteristic
21	C₂×Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×Dic₆₁	C₄	1	2	yes	—
23	C₄⋊C₄×Dic₆₁	C₄	1	1	yes	characteristic
24	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₆₁	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊Dic₆₁	C₄	1	2	yes	—
27	C₄×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
28	C₄²×Dic₆₁	C₂²	1	2	yes	—
29	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
31	C₂²×#92420	C₂³	1	4	yes	—
32	C₂²×#92420	C₂³	1	2	yes	—
33	C₂²×#92420	C₂³	1	2	yes	—
34	C₄⋊#92420	—	2	8	no	—
35	C₄⋊#92420	—	2	4	no	—
36	C₄×#92420	—	2	4	no	—
37	C₄×#92420	—	2	4	no	—
38	C₄×#92420	—	2	2	no	—
39	C₄×#92420	—	2	4	no	—
40	C₄×#92420	Q₈	1	4	yes	complemented
41	Q₈×Dic₆₁	—	2	4	no	—
42	Q₈×Dic₆₁	—	2	4	no	—
43	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
55	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
56	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
58	#14519684	C₂×C₄	1	2	yes	—
59	#14519684	C₂×C₄	1	2	yes	—
60	#14519684	C₂×C₄	1	2	yes	—
61	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₆₁	C₈	1	8	yes	complemented
63	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂×Q₈×C₆₁	—	2	2	no	—
68	C₂×Q₈×C₆₁	—	2	2	no	—
69	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
81	C₂×#92420	—	2	2	no	—
82	C₂×#92420	—	2	2	no	—
83	C₂×#92420	—	2	2	no	—
84	C₂×#92420	—	2	2	no	—
85	C₂×#92420	—	2	4	no	—
86	C₂×#92420	—	2	4	no	—
87	C₂×#92420	—	2	4	no	—
88	C₂×#92420	C₂×Q₈	1	4	yes	—
89	C₂×#92420	#41	1	4	yes	—
90	C₄⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
91	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
92	C₄²×C₆₁	C₂×C₈	1	8	yes	—
93	C₄²×C₆₁	C₂×C₈	1	4	yes	—
94	C₄⋊Dic₆₁	—	2	2	no	—
95	C₄⋊Dic₆₁	—	2	2	no	—
96	C₄⋊Dic₆₁	—	2	4	no	—
97	C₄⋊Dic₆₁	—	2	4	no	—
98	C₄×Dic₆₁	—	2	2	no	—
99	C₄×Dic₆₁	—	2	4	no	—
100	C₄×Dic₆₁	—	2	4	no	—
101	C₄×Dic₆₁	—	2	2	no	—
102	C₄×Dic₆₁	—	2	4	no	—
103	C₄×Dic₆₁	—	2	2	no	—
104	C₄×Dic₆₁	—	2	4	no	—
105	C₄×Dic₆₁	—	2	4	no	—
106	C₄×Dic₆₁	M₁₆	1	4	yes	—
107	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
108	C₄×Dic₆₁	#41	1	4	yes	—
109	C₄×Dic₆₁	#41	1	2	yes	—
110	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₆₁	—	2	8	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	#117	1	2	yes	—
119	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
121	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₆₁	#131	1	2	yes	—
127	C₂×Dic₆₁	—	2	1	no	—
128	C₂×Dic₆₁	—	2	1	no	—
129	C₂×Dic₆₁	—	2	1	no	—
130	C₂×Dic₆₁	—	2	1	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	2	2	no	—
133	C₂×Dic₆₁	—	2	2	no	—
134	C₂×Dic₆₁	—	2	2	no	—
135	C₂×Dic₆₁	—	2	2	no	—
136	C₂×Dic₆₁	—	2	2	no	—
137	C₂×Dic₆₁	#117	1	2	yes	—
138	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
139	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
140	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
141	C₂×Dic₆₁	#131	1	2	yes	—
142	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
143	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₆₁	C₂×#41	1	2	yes	—
145	C₂×Dic₆₁	C₂×#41	1	2	yes	—
146	#92420	—	2	4	no	—
147	#92420	—	2	4	no	—
148	#92420	—	4	4	no	—
149	#21339	—	61	1	no	maximal, Sylow
150	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
151	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂²×C₆₁	C₄×#41	1	1	yes	characteristic
153	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
154	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
155	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
156	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
157	C₄×C₆₁	—	2	4	no	—
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	—	2	4	no	—
160	C₄×C₆₁	#431	1	4	yes	—
161	C₄×C₆₁	C₈×Q₈	1	4	yes	—
162	Dic₆₁	—	2	2	no	—
163	Dic₆₁	—	2	2	no	—
164	Dic₆₁	—	4	2	no	—
165	Dic₆₁	—	4	2	no	—
166	Dic₆₁	#446	1	4	yes	—
167	Dic₆₁	#529	1	2	yes	—
168	Dic₆₁	#533	1	2	yes	—
169	C₄²×Q₈	—	61	1	no	—
170	#1984	—	61	2	no	—
171	#1939	—	61	2	no	—
172	#1916	—	61	4	no	—
173	C₄×C₄⋊C₈	—	61	1	no	—
174	C₄×C₄⋊C₈	—	61	2	no	—
175	#1744	—	61	2	no	—
176	C₄²×C₈	—	61	1	no	—
177	C₂×C₆₁	#2295	1	1	yes	characteristic
178	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
179	C₂×C₆₁	#2922	1	1	yes	characteristic
180	C₂×C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂×C₆₁	C₂×#446	1	1	yes	characteristic
182	C₂×C₆₁	C₈×#41	1	1	yes	characteristic
183	C₂×C₆₁	#2965	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	61	1	no	—
185	C₂×C₄×Q₈	—	61	1	no	—
186	C₂×C₄×Q₈	#6923	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	61	4	no	—
188	C₂×C₄⋊C₈	—	61	2	no	—
189	C₂×C₄×C₈	—	61	1	no	—
190	C₂×C₄×C₈	—	61	2	no	—
191	C₂×C₄×C₈	—	61	2	no	—
192	C₂×C₄×C₈	—	61	1	no	—
193	C₂×C₄×C₈	—	61	2	no	—
194	C₄×C₄⋊C₄	—	61	2	no	—
195	C₄×C₄⋊C₄	—	61	1	no	—
196	C₄×C₄⋊C₄	—	61	1	no	—
197	C₄×C₄⋊C₄	—	61	2	no	—
198	C₄×C₄⋊C₄	—	61	2	no	—
199	C₄×C₄⋊C₄	—	61	1	no	—
200	C₄³	—	61	1	no	—
201	C₄³	—	61	2	no	—
202	#336	—	61	2	no	—
203	#336	—	61	4	no	—
204	#336	—	61	4	no	—
205	#336	—	61	4	no	—
206	C₆₁	#21339	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	C₂×#6923	1	1	yes	characteristic
208	C₂²×C₈	—	61	2	no	—
209	C₂²×C₈	—	61	2	no	—
210	C₂²×C₈	—	61	4	no	—
211	C₄×Q₈	—	122	4	no	—
212	C₄×Q₈	—	122	4	no	—
213	C₄×Q₈	#92420	1	8	yes	complemented
214	C₂×C₄⋊C₄	—	61	2	no	—
215	C₂×C₄⋊C₄	—	61	2	no	—
216	C₂×C₄⋊C₄	—	61	1	no	—
217	C₂×C₄⋊C₄	—	61	1	no	—
218	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
220	C₂×C₄²	—	61	1	no	—
221	C₂×C₄²	—	61	1	no	—
222	C₂×C₄²	—	61	1	no	—
223	C₂×C₄²	—	61	2	no	—
224	C₂×C₄²	—	61	2	no	—
225	C₂×C₄²	—	61	1	no	—
226	C₂×C₄²	—	61	2	no	—
227	C₂×C₄²	—	61	1	no	—
228	C₂×C₄²	—	61	2	no	—
229	C₂×C₄²	—	61	1	no	—
230	C₂×C₄²	—	61	2	no	—
231	C₂×C₄²	C₂×#6923	1	2	yes	—
232	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
233	C₄⋊C₈	—	122	4	no	—
234	C₄⋊C₈	—	122	8	no	—
235	C₄×C₈	—	61	4	no	—
236	C₄×C₈	—	122	4	no	—
237	C₄×C₈	—	122	2	no	—
238	C₄×C₈	—	122	4	no	—
239	C₄×C₈	—	122	4	no	—
240	#95	—	61	2	no	—
241	#95	—	61	2	no	—
242	#95	—	61	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	C₂×#92420	1	4	yes	—
246	C₂²×C₄	—	61	1	no	—
247	C₂²×C₄	—	61	1	no	—
248	C₂²×C₄	—	61	1	no	—
249	C₂²×C₄	—	61	1	no	—
250	C₂²×C₄	—	61	2	no	—
251	C₂²×C₄	—	61	2	no	—
252	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
253	C₂²×C₄	C₂²×#6923	1	2	yes	—
254	C₂²×C₄	C₂²×#6923	1	2	yes	—
255	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
256	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
257	C₂×C₈	—	61	4	no	—
258	C₂×C₈	—	61	4	no	—
259	C₂×C₈	—	122	2	no	—
260	C₂×C₈	—	122	2	no	—
261	C₂×C₈	—	122	2	no	—
262	C₂×C₈	—	122	2	no	—
263	C₂×C₈	—	122	4	no	—
264	C₂×C₈	—	122	4	no	—
265	C₂×C₈	—	122	4	no	—
266	C₄⋊C₄	—	122	4	no	—
267	C₄⋊C₄	—	122	4	no	—
268	C₄⋊C₄	—	122	2	no	—
269	C₄⋊C₄	—	122	2	no	—
270	C₄⋊C₄	C₂×#92420	1	8	yes	—
271	C₄⋊C₄	C₂×#92420	1	4	yes	—
272	C₄²	—	61	4	no	—
273	C₄²	—	61	2	no	—
274	C₄²	—	61	2	no	—
275	C₄²	—	61	4	no	—
276	C₄²	—	122	2	no	—
277	C₄²	—	122	4	no	—
278	C₄²	—	122	2	no	—
279	C₄²	—	122	4	no	—
280	C₄²	—	122	4	no	—
281	C₄²	—	122	4	no	—
282	C₄²	—	122	2	no	—
283	C₄²	—	122	4	no	—
284	C₄²	C₂×#92420	1	8	yes	—
285	C₄²	C₂×#92420	1	4	yes	—
286	C₂³	C₂³×#6923	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	61	2	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	1	no	—
297	C₂×C₄	—	61	1	no	—
298	C₂×C₄	—	61	2	no	—
299	C₂×C₄	—	61	2	no	—
300	C₂×C₄	—	61	2	no	—
301	C₂×C₄	—	61	2	no	—
302	C₂×C₄	—	61	2	no	—
303	C₂×C₄	—	122	1	no	—
304	C₂×C₄	—	122	1	no	—
305	C₂×C₄	—	122	1	no	—
306	C₂×C₄	—	122	1	no	—
307	C₂×C₄	—	122	2	no	—
308	C₂×C₄	—	122	2	no	—
309	C₂×C₄	—	122	2	no	—
310	C₂×C₄	—	122	2	no	—
311	C₂×C₄	—	122	2	no	—
312	C₂×C₄	—	122	2	no	—
313	C₂×C₄	#14519856	1	2	yes	—
314	C₂×C₄	Q₈×#6923	1	2	yes	—
315	C₂×C₄	C₂²×#92420	1	4	yes	—
316	C₂×C₄	C₂²×#92420	1	2	yes	—
317	C₂×C₄	C₂²×#92420	1	4	yes	—
318	C₂×C₄	C₂²×#92420	1	2	yes	—
319	C₂×C₄	C₂²×#92420	1	2	yes	—
320	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
321	C₂×C₄	#14519865	1	2	yes	—
322	C₈	—	122	4	no	—
323	C₈	—	122	4	no	—
324	C₈	—	244	4	no	—
325	C₂²	C₂×#14519856	1	1	yes	characteristic
326	C₂²	C₂×Q₈×#6923	1	1	yes	characteristic
327	C₂²	#41×#6923	1	1	yes	characteristic
328	C₂²	C₂³×#92420	1	1	yes	characteristic
329	C₂²	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
330	C₂²	D₆₁⋊#131	1	1	yes	characteristic
331	C₂²	C₂×#14519865	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	61	2	no	—
336	C₄	—	61	2	no	—
337	C₄	—	61	4	no	—
338	C₄	—	122	2	no	—
339	C₄	—	122	2	no	—
340	C₄	—	244	2	no	—
341	C₄	—	244	2	no	—
342	C₄	#16840778	1	4	yes	—
343	C₄	Q₈×#92420	1	4	yes	—
344	C₂	#21462278	1	1	yes	characteristic
345	C₂	C₂×#16840778	1	1	yes	characteristic
346	C₂	#21462504	1	1	yes	characteristic
347	C₂	C₂×Q₈×#92420	1	1	yes	characteristic
348	C₂	D₆₁⋊#446	1	1	yes	characteristic
349	C₂	#41×#92420	1	1	yes	characteristic
350	C₂	#21462627	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂³×C₈
Frattini	C₂³	C₂³×#6923
Fitting	C₂×C₄×Q₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂³×C₄
cosocle	C₂²×Dic₆₁	C₂⁴
hypercenter	C₂×C₄×Q₈	#6923
hypocenter	C₆₁	#21339
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
40. C₄×#92420	Q₈	semidirect
62. C₄×Q₈×C₆₁	C₈	semidirect
206. C₆₁	#21339	semidirect
213. C₄×Q₈	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461988	2
#21461916	2
#21461872	2
C₄×C₄⋊#92420	2
#21461533	2
C₄²×#92420	2
C₄×Q₈×Dic₆₁	2
#21339	61