Subgroups — #29649151

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454936	C₂	1	1	yes	maximal, characteristic
2	#21454930	C₂	1	1	yes	maximal, characteristic
3	#21444608	C₂	1	2	yes	maximal, complemented
4	#21444604	C₂	1	2	yes	maximal, complemented
5	#21444599	C₂	1	2	yes	maximal, complemented
6	#21444594	C₂	1	2	yes	maximal, complemented
7	#21444571	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444571	C₂	1	1	yes	maximal, characteristic, complemented
9	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2922	C₂	1	1	yes	maximal, characteristic
12	#16839893	—	2	1	no	—
13	Q₈×#92418	—	2	1	no	—
14	#16839840	—	2	1	no	—
15	#16839834	—	2	2	no	—
16	D₄×#92418	—	2	2	no	—
17	D₄×#92418	—	2	1	no	—
18	#16839819	C₂²	1	2	yes	—
19	#16839815	C₂²	1	2	yes	—
20	#16839815	C₂²	1	1	yes	characteristic
21	#16839815	C₂²	1	1	yes	characteristic
22	#16839786	C₂²	1	2	yes	—
23	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#92418	C₂²	1	2	yes	—
25	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
26	#16839515	C₂²	1	2	yes	—
27	#16839515	C₂²	1	2	yes	—
28	#16839515	C₂²	1	2	yes	—
29	#16839515	C₂²	1	1	yes	characteristic
30	#16839515	C₂²	1	1	yes	characteristic
31	#16839515	C₂²	1	2	yes	—
32	#16839513	C₂²	1	1	yes	characteristic
33	#16839513	C₂²	1	2	yes	—
34	#16839513	C₂²	1	1	yes	characteristic
35	#16839513	C₂²	1	2	yes	—
36	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#433	C₄	1	4	yes	—
38	C₆₁×#431	C₂²	1	2	yes	—
39	C₆₁×#431	C₂²	1	2	yes	—
40	C₆₁×#424	C₄	1	2	yes	—
41	C₆₁×#408	C₄	1	2	yes	—
42	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
44	C₂×#11758194	—	2	2	no	—
45	C₂²×#92418	—	2	1	no	—
46	C₂²×#92418	—	2	2	no	—
47	C₂²×#92418	—	2	1	no	—
48	C₄⋊#92418	—	2	1	no	—
49	C₄⋊#92418	—	2	2	no	—
50	C₄⋊#92418	—	2	1	no	—
51	C₄⋊#92418	—	2	2	no	—
52	C₄⋊#92418	C₂³	1	1	yes	characteristic
53	C₄⋊#92418	C₂³	1	1	yes	characteristic
54	C₄⋊#92418	C₂³	1	1	yes	characteristic
55	C₄⋊#92418	C₂³	1	2	yes	—
56	C₄⋊#92418	C₂³	1	2	yes	—
57	C₄⋊#92418	C₂³	1	1	yes	characteristic
58	C₂²⋊#92418	—	2	1	no	—
59	C₂²⋊#92418	—	2	2	no	—
60	C₂²⋊#92418	—	2	1	no	—
61	C₂²⋊#92418	—	2	2	no	—
62	#14519688	—	2	2	no	—
63	C₄×#92418	—	2	1	no	—
64	C₄×#92418	—	2	2	no	—
65	C₄×#92418	—	2	1	no	—
66	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₆₁	Q₈	1	1	yes	characteristic
71	C₄×D₄×C₆₁	D₄	1	2	yes	—
72	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
73	C₄×D₄×C₆₁	Q₈	1	2	yes	—
74	C₄×D₄×C₆₁	Q₈	1	1	yes	characteristic
75	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
76	#117×C₆₁	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
83	#97×C₆₁	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₁	C₂³	1	2	yes	—
87	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
88	#41×C₆₁	—	2	4	no	—
89	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
98	M₁₆×C₆₁	—	2	4	no	—
99	C₂×C₈×C₆₁	—	2	4	no	—
100	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
103	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
104	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
109	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	2	yes	—
110	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₁	C₂×D₄	1	2	yes	—
114	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
115	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
116	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
117	#11758194	—	8	2	no	—
118	C₂×#92418	—	2	1	no	—
119	C₂×#92418	—	2	1	no	—
120	C₂×#92418	—	2	1	no	—
121	C₂×#92418	—	2	2	no	—
122	C₂×#92418	—	2	1	no	—
123	C₂×#92418	—	2	2	no	—
124	C₂×#92418	—	8	1	no	—
125	C₂×#92418	—	8	2	no	—
126	C₂×#92418	—	8	1	no	—
127	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₆₁	—	2	2	no	—
130	D₄×C₆₁	—	2	4	no	—
131	D₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	2	4	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	4	1	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	#131	1	4	yes	—
146	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₆₁	—	2	2	no	—
150	C₈×C₆₁	—	2	2	no	—
151	C₈×C₆₁	—	2	4	no	—
152	#92418	—	4	2	no	—
153	#92418	—	4	1	no	—
154	#92418	—	4	2	no	—
155	#92418	—	4	1	no	—
156	#92418	—	4	1	no	—
157	#92418	—	4	1	no	—
158	#17812	—	61	1	no	maximal, Sylow
159	C₂²×C₆₁	—	2	1	no	—
160	C₂²×C₆₁	—	2	1	no	—
161	C₂²×C₆₁	—	2	2	no	—
162	C₂²×C₆₁	—	2	2	no	—
163	C₂²×C₆₁	—	2	2	no	—
164	C₂²×C₆₁	#424	1	2	yes	—
165	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	—	2	1	no	—
168	C₄×C₆₁	—	2	2	no	—
169	C₄×C₆₁	—	2	1	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	—	2	2	no	—
172	C₄×C₆₁	#424	1	2	yes	—
173	C₄×C₆₁	C₂×#131	1	2	yes	—
174	C₄×C₆₁	#576	1	1	yes	characteristic
175	C₄×C₆₁	#578	1	1	yes	characteristic
176	#2989	—	61	1	no	—
177	#2988	—	61	1	no	—
178	#2922	D₆₁	1	1	yes	characteristic
179	#1574	—	61	2	no	—
180	#1573	—	61	2	no	—
181	#1572	—	61	2	no	—
182	#1571	—	61	2	no	—
183	#1565	—	61	1	no	—
184	#1565	—	61	1	no	—
185	#1564	—	61	1	no	—
186	C₂×#335	—	61	1	no	—
187	C₂×C₆₁	—	2	2	no	—
188	C₂×C₆₁	—	4	2	no	—
189	C₂×C₆₁	C₂×#424	1	1	yes	characteristic
190	C₂×C₆₁	#2913	1	1	yes	characteristic
191	C₂×C₆₁	#2916	1	1	yes	characteristic
192	C₄×#41	C₂×D₆₁	1	1	yes	characteristic
193	#446	—	122	1	no	—
194	C₈×Q₈	—	122	1	no	—
195	#436	—	122	1	no	—
196	#435	—	122	2	no	—
197	C₈×D₄	—	122	2	no	—
198	C₈×D₄	—	122	1	no	—
199	#433	Dic₆₁	1	4	yes	—
200	#432	—	61	2	no	—
201	#431	—	61	1	no	—
202	#431	—	61	1	no	—
203	#431	—	61	2	no	—
204	#431	C₂×D₆₁	1	2	yes	—
205	#431	C₂×D₆₁	1	2	yes	—
206	#424	Dic₆₁	1	2	yes	—
207	#423	—	61	2	no	—
208	C₂×C₄⋊C₈	—	61	1	no	—
209	C₂×C₄⋊C₈	—	61	1	no	—
210	C₂×C₄⋊C₈	—	61	2	no	—
211	#408	Dic₆₁	1	2	yes	—
212	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
214	#335	—	61	2	no	—
215	#335	—	61	2	no	—
216	#335	—	61	1	no	—
217	#335	—	61	1	no	—
218	#335	—	61	2	no	—
219	#335	—	61	2	no	—
220	#334	—	61	1	no	—
221	#334	—	61	1	no	—
222	#334	—	61	2	no	—
223	#334	—	61	2	no	—
224	C₆₁	#17812	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	C₂×Dic₆₁	1	1	yes	characteristic
226	C₂×M₁₆	—	122	2	no	—
227	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
228	C₂²×C₈	—	122	1	no	—
229	C₂²×C₈	—	122	2	no	—
230	C₂²×C₈	—	122	1	no	—
231	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
232	C₄×Q₈	Dic₁₂₂	1	1	yes	characteristic
233	C₄×Q₈	D₂₄₄	1	1	yes	characteristic
234	C₄×D₄	Dic₁₂₂	1	2	yes	—
235	C₄×D₄	Dic₁₂₂	1	1	yes	characteristic
236	C₄×D₄	D₂₄₄	1	2	yes	—
237	C₄×D₄	D₂₄₄	1	1	yes	characteristic
238	#117	C₂×Dic₆₁	1	1	yes	characteristic
239	#117	C₂×Dic₆₁	1	2	yes	—
240	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
242	C₄⋊C₈	—	61	1	no	—
243	C₄⋊C₈	—	61	1	no	—
244	C₄⋊C₈	—	61	1	no	—
245	C₄⋊C₈	—	61	2	no	—
246	C₄⋊C₈	—	61	2	no	—
247	C₄⋊C₈	—	61	1	no	—
248	C₄⋊C₈	—	122	1	no	—
249	C₄⋊C₈	—	122	2	no	—
250	C₄⋊C₈	—	122	1	no	—
251	C₄⋊C₈	—	122	2	no	—
252	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
253	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
254	C₂²⋊C₈	—	122	1	no	—
255	C₂²⋊C₈	—	122	2	no	—
256	C₂²⋊C₈	—	122	1	no	—
257	C₂²⋊C₈	—	122	2	no	—
258	C₂²⋊C₈	C₂×Dic₆₁	1	4	yes	—
259	C₂²⋊C₈	C₂×Dic₆₁	1	4	yes	—
260	#97	—	122	2	no	—
261	#97	C₂×Dic₆₁	1	2	yes	—
262	C₄×C₈	—	122	1	no	—
263	C₄×C₈	—	122	2	no	—
264	C₄×C₈	—	122	1	no	—
265	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
266	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₆₁	1	2	yes	—
268	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
269	#41	—	2	4	no	—
270	C₂×Q₈	C₄⋊Dic₆₁	1	1	yes	characteristic
271	C₂×D₄	C₄⋊Dic₆₁	1	2	yes	—
272	C₂×D₄	C₄⋊Dic₆₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₁₂₂	1	2	yes	—
274	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
277	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
278	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	488	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	122	1	no	—
283	C₂×C₈	—	122	1	no	—
284	C₂×C₈	—	122	1	no	—
285	C₂×C₈	—	122	2	no	—
286	C₂×C₈	—	122	1	no	—
287	C₂×C₈	—	122	2	no	—
288	C₂×C₈	—	488	1	no	—
289	C₂×C₈	—	488	2	no	—
290	C₂×C₈	—	488	1	no	—
291	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
292	C₂×C₈	C₂²×Dic₆₁	1	2	yes	—
293	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
294	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
295	C₄⋊C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
296	C₄⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
297	C₄⋊C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
298	C₂²⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
299	C₂²⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
300	C₂²⋊C₄	C₄⋊Dic₆₁	1	2	yes	—
301	C₄²	C₂×Dic₁₂₂	1	2	yes	—
302	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
303	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
304	C₄²	C₂×D₂₄₄	1	2	yes	—
305	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
306	C₄²	C₂²×Dic₆₁	1	2	yes	—
307	C₄²	C₂³×D₆₁	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊Dic₆₁	1	2	yes	—
309	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
321	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
323	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
326	C₂×C₄	#14519784	1	4	yes	—
327	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
329	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	244	2	no	—
334	C₈	—	244	1	no	—
335	C₈	—	244	2	no	—
336	C₈	—	244	1	no	—
337	C₈	—	244	1	no	—
338	C₈	—	244	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	#16839789	1	2	yes	—
345	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	#16839789	1	2	yes	—
353	C₄	C₂×#14519784	1	2	yes	—
354	C₄	#16840567	1	1	yes	characteristic
355	C₄	#16840579	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×#16839789	1	1	yes	characteristic
359	C₂	#21454419	1	1	yes	characteristic
360	C₂	#21454430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#2922	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#2922	D₆₁
hypocenter	C₆₁	#17812
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #21444608	C₂	semidirect
4. #21444604	C₂	semidirect
5. #21444599	C₂	semidirect
6. #21444594	C₂	semidirect
7. #21444571	C₂	semidirect
8. #21444571	C₂	semidirect
9. #21444567	C₂	semidirect
10. C₂×#16839515	C₂	semidirect
224. C₆₁	#17812	semidirect

Maximal subgroups

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

subgroup	index
#21454936	2
#21454930	2
#21444608	2
#21444604	2
#21444599	2
#21444594	2
#21444571	2
#21444567	2
C₂×#16839515	2
C₆₁×#2922	2
#17812	61