Subgroups — #29699716

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519799	C₂	1	1	yes	maximal, characteristic
2	#21454484	C₂	1	2	yes	maximal, complemented
3	#21446712	C₂	1	2	yes	maximal
4	#21446468	C₂	1	4	yes	maximal, complemented
5	#21445991	C₂	1	1	yes	maximal, characteristic, complemented
6	#21445840	C₂	1	2	yes	maximal, complemented
7	#21445703	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₂²	1	1	yes	characteristic
10	C₄×#11758222	C₂²	1	2	yes	—
11	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
13	C₄×#11758190	C₄	1	2	yes	complemented
14	C₄×#11758190	C₄	1	2	yes	complemented
15	C₄×D₄₈₈	C₄	1	1	yes	characteristic, complemented
16	C₄×D₄₈₈	C₄	1	1	yes	characteristic, complemented
17	#16839822	—	2	4	no	—
18	#16839818	—	2	2	no	—
19	#16839814	—	2	2	no	—
20	C₂×#14519713	C₂²	1	2	yes	—
21	C₂×#14519713	C₂²	1	2	yes	—
22	#16839800	C₂²	1	1	yes	characteristic
23	#16839786	C₂²	1	2	yes	—
24	C₂×#14519705	C₂²	1	4	yes	—
25	#16839759	C₂²	1	2	yes	—
26	#16839728	C₂²	1	4	yes	—
27	#16839717	C₂²	1	4	yes	—
28	#16839522	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#336	C₂²	1	2	yes	—
34	C₂×#11758222	C₂³	1	2	yes	—
35	#14519799	—	2	2	no	—
36	#14519799	—	2	2	no	—
37	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
38	C₂×#11758190	C₂×C₄	1	2	yes	—
39	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
40	C₂×#11758194	C₂³	1	4	yes	—
41	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
44	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₆₁	—	2	2	no	—
46	C₄×C₂²⋊D₆₁	—	2	2	no	—
47	#14519736	C₂³	1	2	yes	—
48	#14519734	—	2	2	no	—
49	C₄²×D₆₁	—	2	2	no	—
50	#14519713	—	2	4	no	—
51	#14519713	—	4	2	no	—
52	#14519711	C₂×C₄	1	1	yes	characteristic
53	#14519711	C₂×C₄	1	1	yes	characteristic
54	#14519709	C₂×C₄	1	2	yes	—
55	C₄⋊#92418	—	2	2	no	—
56	C₄⋊#92418	—	2	2	no	—
57	#14519705	—	4	4	no	—
58	#14519702	C₂×C₄	1	2	yes	—
59	#14519702	C₂×C₄	1	2	yes	—
60	#14519699	C₂×C₄	1	2	yes	—
61	#14519699	C₂×C₄	1	2	yes	—
62	#14519689	—	2	4	no	—
63	#14519689	—	2	4	no	—
64	#14519687	—	2	4	no	—
65	C₈×Dic₆₁	—	2	4	no	—
66	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
70	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
73	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
74	#97×C₆₁	—	2	4	no	—
75	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	#95×C₆₁	—	2	1	no	—
80	C₂²×C₄×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₂²⋊D₆₁	—	2	2	no	—
85	#11758222	—	2	2	no	—
86	#11758222	—	2	2	no	—
87	#11758222	—	2	2	no	—
88	#11758222	—	2	2	no	—
89	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₆₁	—	2	2	no	—
91	C₂×C₄×D₆₁	—	2	2	no	—
92	C₂×C₄×D₆₁	—	2	2	no	—
93	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	1	no	—
96	C₂×C₈×C₆₁	—	2	1	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₆₁	#41	1	2	yes	—
104	C₂×C₈×C₆₁	#41	1	2	yes	—
105	C₄⋊C₄×C₆₁	—	4	2	no	—
106	C₄²×C₆₁	—	2	2	no	—
107	C₄²×C₆₁	—	2	2	no	—
108	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₆₁	—	2	2	no	—
113	#11758198	—	2	2	no	—
114	#11758198	—	2	2	no	—
115	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
116	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
117	#11758196	—	2	2	no	—
118	#11758196	—	2	2	no	—
119	C₄×Dic₆₁	—	2	2	no	—
120	C₄×Dic₆₁	—	2	2	no	—
121	C₄×Dic₆₁	—	2	2	no	—
122	#11758194	—	2	4	no	—
123	#11758194	—	2	4	no	—
124	C₂×#92418	—	2	4	no	—
125	Dic₂₄₄	—	2	2	no	—
126	D₄₈₈	—	2	2	no	—
127	#11758190	—	2	4	no	—
128	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
129	C₂²×D₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²⋊D₆₁	—	4	2	no	—
146	C₂²⋊D₆₁	—	4	2	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	—	2	2	no	—
150	D₂₄₄	—	2	2	no	—
151	D₂₄₄	—	2	2	no	—
152	C₄×D₆₁	—	4	2	no	—
153	C₄×D₆₁	—	4	2	no	—
154	C₄×D₆₁	—	4	4	no	—
155	C₄×D₆₁	—	8	2	no	—
156	Dic₁₂₂	—	2	2	no	—
157	Dic₁₂₂	—	2	2	no	—
158	C₈×C₆₁	—	2	1	no	—
159	C₈×C₆₁	—	2	1	no	—
160	C₈×C₆₁	—	2	4	no	—
161	C₈×C₆₁	#117	1	2	yes	—
162	C₈×C₆₁	#117	1	2	yes	—
163	#92418	—	4	4	no	—
164	#22803	—	61	1	no	maximal, Sylow
165	C₂²×C₆₁	—	2	1	no	—
166	C₂²×C₆₁	—	2	1	no	—
167	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₆₁	#555	1	1	yes	characteristic
169	C₂²×C₆₁	#556	1	1	yes	characteristic
170	C₂×D₆₁	—	4	2	no	—
171	C₂×D₆₁	—	4	2	no	—
172	C₂×D₆₁	—	4	2	no	—
173	C₄×C₆₁	—	2	1	no	—
174	C₄×C₆₁	—	2	1	no	—
175	C₄×C₆₁	—	2	1	no	—
176	C₄×C₆₁	—	8	2	no	—
177	C₄×C₆₁	#444	1	2	yes	—
178	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
179	C₄×C₆₁	#520	1	1	yes	characteristic
180	C₄×C₆₁	#547	1	1	yes	characteristic
181	C₄×C₆₁	#550	1	1	yes	characteristic
182	Dic₆₁	—	4	2	no	—
183	Dic₆₁	—	4	2	no	—
184	Dic₆₁	—	4	2	no	—
185	C₄×#135	—	61	1	no	—
186	#2925	—	61	2	no	—
187	#1975	—	61	2	no	—
188	#1929	—	61	4	no	—
189	#1848	—	61	1	no	—
190	#1806	—	61	2	no	—
191	#1773	—	61	2	no	—
192	#1770	D₆₁	1	1	yes	characteristic, complemented
193	C₂×C₆₁	—	2	1	no	—
194	C₂×C₆₁	—	2	1	no	—
195	C₂×C₆₁	#2354	1	1	yes	characteristic
196	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
197	C₂×C₆₁	#2957	1	1	yes	characteristic
198	D₆₁	—	8	2	no	—
199	C₂×#135	—	61	1	no	—
200	C₄×#41	—	61	2	no	—
201	C₄×Q₁₆	—	61	1	no	—
202	C₄×Q₁₆	—	61	1	no	—
203	C₄×SD₁₆	—	61	2	no	—
204	C₄×SD₁₆	—	61	2	no	—
205	C₄×D₈	—	61	1	no	—
206	C₄×D₈	—	61	1	no	—
207	#433	—	122	4	no	—
208	#432	—	122	2	no	—
209	#431	—	122	2	no	—
210	C₂×#108	—	61	2	no	—
211	C₂×#108	—	61	2	no	—
212	#427	—	61	1	no	—
213	#423	—	61	2	no	—
214	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
215	C₂×#104	—	61	4	no	—
216	#416	—	61	2	no	—
217	#408	—	61	4	no	—
218	#405	—	61	4	no	—
219	C₂×#97	C₂×D₆₁	1	2	yes	—
220	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
221	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
222	#339	—	61	4	no	—
223	#336	C₂×D₆₁	1	2	yes	—
224	C₆₁	#22803	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	—	61	2	no	—
226	#135	—	122	2	no	—
227	#135	—	122	2	no	—
228	C₂×Q₁₆	—	61	1	no	—
229	C₂×SD₁₆	—	61	2	no	—
230	C₂×D₈	—	61	1	no	—
231	C₂×M₁₆	—	61	4	no	—
232	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
234	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
235	C₄×Q₈	—	61	2	no	—
236	C₄×Q₈	—	61	2	no	—
237	C₄×D₄	—	61	2	no	—
238	C₄×D₄	—	61	2	no	—
239	C₄×D₄	—	122	2	no	—
240	C₄×D₄	—	122	2	no	—
241	#117	—	61	2	no	—
242	#117	—	122	2	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	122	2	no	—
245	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
246	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
247	#108	—	122	4	no	—
248	#108	—	244	2	no	—
249	#107	—	61	1	no	—
250	#107	—	61	1	no	—
251	#106	—	61	2	no	—
252	C₄⋊C₈	—	122	2	no	—
253	C₄⋊C₈	—	122	2	no	—
254	C₄⋊C₈	D₂₄₄	1	2	yes	—
255	C₄⋊C₈	D₂₄₄	1	2	yes	—
256	#104	—	244	4	no	—
257	#103	—	61	2	no	—
258	#103	—	61	2	no	—
259	#102	—	61	2	no	—
260	#102	—	61	2	no	—
261	C₂²⋊C₈	—	122	4	no	—
262	C₂²⋊C₈	—	122	4	no	—
263	#97	—	2	4	no	—
264	#97	—	122	4	no	—
265	C₄×C₈	—	122	4	no	—
266	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
267	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
268	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
269	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
270	#95	—	2	1	no	—
271	#41	—	122	2	no	—
272	#41	—	122	2	no	—
273	#41	—	122	2	no	—
274	#41	—	122	2	no	—
275	C₂×Q₈	—	61	2	no	—
276	C₂×D₄	—	61	2	no	—
277	C₂×D₄	—	122	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	122	2	no	—
280	C₂²×C₄	—	122	2	no	—
281	C₂²×C₄	—	122	2	no	—
282	C₂²×C₄	#11758222	1	1	yes	characteristic
283	C₂²×C₄	#11758222	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
285	Q₁₆	—	122	2	no	—
286	SD₁₆	—	122	4	no	—
287	D₈	—	122	2	no	—
288	M₁₆	—	122	4	no	—
289	M₁₆	—	122	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	122	4	no	—
295	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
299	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
300	C₂×C₈	#11758224	1	2	yes	—
301	C₂×C₈	#11758226	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	61	2	no	—
304	C₄⋊C₄	—	61	2	no	—
305	C₄⋊C₄	—	122	2	no	—
306	C₄⋊C₄	—	122	2	no	—
307	C₂²⋊C₄	—	122	2	no	—
308	C₂²⋊C₄	—	122	2	no	—
309	C₂²⋊C₄	—	122	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	122	2	no	—
313	C₄²	—	122	2	no	—
314	C₄²	—	122	2	no	—
315	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
317	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
318	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
319	C₂³	—	122	2	no	—
320	C₂³	C₂×#11758222	1	1	yes	characteristic
321	Q₈	—	122	2	no	—
322	Q₈	—	122	2	no	—
323	D₄	—	122	2	no	—
324	D₄	—	122	2	no	—
325	D₄	—	244	2	no	—
326	D₄	—	244	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	122	2	no	—
333	C₂×C₄	—	122	2	no	—
334	C₂×C₄	—	122	2	no	—
335	C₂×C₄	—	488	2	no	—
336	C₂×C₄	—	244	4	no	—
337	C₂×C₄	—	244	2	no	—
338	C₂×C₄	—	244	2	no	—
339	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
342	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
345	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
346	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
347	C₂×C₄	#14519823	1	1	yes	characteristic
348	C₂×C₄	#14519824	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	4	no	—
352	C₈	—	244	4	no	—
353	C₈	#14519735	1	2	yes	—
354	C₈	#14519735	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	244	2	no	—
358	C₂²	—	244	2	no	—
359	C₂²	—	244	2	no	—
360	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
361	C₂²	#16840477	1	1	yes	characteristic
362	C₂²	#16840485	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	244	2	no	—
368	C₄	—	244	2	no	—
369	C₄	—	244	2	no	—
370	C₄	#16839885	1	2	yes	—
371	C₄	C₂×#14519735	1	1	yes	characteristic
372	C₄	#16840242	1	1	yes	characteristic
373	C₄	#16840416	1	1	yes	characteristic
374	C₄	#16840434	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	488	2	no	—
378	C₂	#21448766	1	1	yes	characteristic
379	C₂	C₂×#16839885	1	1	yes	characteristic
380	C₂	#21454737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1770	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1770	D₆₁
hypocenter	C₆₁	#22803
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #21454484	C₂	semidirect
4. #21446468	C₂	semidirect
5. #21445991	C₂	semidirect
6. #21445840	C₂	semidirect
7. #21445703	C₂	semidirect
8. C₆₁×#1770	C₂	semidirect
11. C₄×Dic₂₄₄	C₄	semidirect
12. C₄×Dic₂₄₄	C₄	semidirect
13. C₄×#11758190	C₄	semidirect
14. C₄×#11758190	C₄	semidirect
15. C₄×D₄₈₈	C₄	semidirect
16. C₄×D₄₈₈	C₄	semidirect
192. #1770	D₆₁	semidirect
224. C₆₁	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519799	2
#21454484	2
#21446712	2
#21446468	2
#21445991	2
#21445840	2
#21445703	2
C₆₁×#1770	2
#22803	61