Subgroups — #29834593

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840194	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16840178	C₂	1	1	yes	maximal, characteristic, complemented
3	#21455540	C₂	1	2	yes	maximal, complemented
4	#21447950	C₂	1	4	yes	maximal, complemented
5	#21447945	C₂	1	2	yes	maximal, complemented
6	#21447940	C₂	1	2	yes	maximal, complemented
7	#21447566	C₂	1	2	yes	maximal, complemented
8	C₂×C₆₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	#16840562	—	2	2	no	—
10	C₂×#14519799	C₂²	1	2	yes	—
11	C₂×#14519783	C₂²	1	1	yes	characteristic
12	#16840194	—	2	1	no	—
13	#16840194	—	2	1	no	—
14	#16840194	—	2	1	no	—
15	#16840194	—	2	1	no	—
16	C₂×Dic₄₈₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×#14519722	C₂²	1	2	yes	complemented
18	C₂×D₉₇₆	C₂²	1	1	yes	characteristic, complemented
19	#16840178	—	2	1	no	—
20	#16840178	—	2	1	no	—
21	#16840178	C₂²	1	2	yes	—
22	#16840178	C₂²	1	2	yes	—
23	C₂×#14519717	C₂²	1	1	yes	characteristic
24	C₂×#14519716	—	2	1	no	—
25	C₂×C₁₆×D₆₁	—	2	1	no	—
26	#16840038	C₂²	1	4	yes	—
27	#16840033	C₂²	1	4	yes	—
28	#16840026	—	2	4	no	—
29	#16839744	—	2	2	no	—
30	#16839577	C₂²	1	4	yes	complemented
31	#16839574	C₂²	1	4	yes	complemented
32	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#108×C₆₁	C₂²	1	2	yes	—
34	C₆₁×#368	C₂²	1	2	yes	complemented
35	C₆₁×#368	C₂²	1	2	yes	complemented
36	C₂×#11758222	—	2	2	no	—
37	C₂×#11758222	D₄	1	1	yes	characteristic
38	#14519810	—	2	4	no	—
39	#14519810	—	4	2	no	—
40	#14519805	—	2	4	no	—
41	#14519805	—	4	2	no	—
42	#14519799	—	2	2	no	—
43	#14519799	—	2	2	no	—
44	#14519799	—	2	2	no	—
45	#14519799	—	2	2	no	—
46	#14519799	—	4	4	no	—
47	C₂×Dic₂₄₄	C₂³	1	2	yes	—
48	C₂×#11758190	—	2	2	no	—
49	C₂×D₄₈₈	C₂³	1	2	yes	—
50	#14519783	—	2	1	no	—
51	#14519783	—	2	1	no	—
52	#14519783	D₄	1	2	yes	—
53	#14519783	C₂³	1	2	yes	—
54	C₂×#11758194	D₄	1	1	yes	characteristic
55	C₂×#11758189	—	2	1	no	—
56	C₂×C₈×D₆₁	—	2	1	no	—
57	Dic₄₈₈	—	2	1	no	—
58	Dic₄₈₈	—	2	1	no	—
59	#14519722	—	2	2	no	—
60	#14519722	—	2	2	no	—
61	D₉₇₆	—	2	1	no	—
62	D₉₇₆	—	2	1	no	—
63	#14519717	—	2	1	no	—
64	#14519717	C₂³	1	2	yes	—
65	#14519716	—	2	2	no	—
66	#14519716	—	2	2	no	—
67	C₂×#11758185	—	2	1	no	—
68	C₁₆×D₆₁	—	2	2	no	—
69	C₁₆×D₆₁	—	2	2	no	—
70	#14519695	—	2	4	no	—
71	#14519693	—	2	4	no	—
72	C₂×M₁₆×C₆₁	—	2	2	no	—
73	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
76	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
80	#108×C₆₁	—	2	2	no	—
81	#108×C₆₁	C₂³	1	4	yes	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₆₁	—	2	1	no	—
84	C₂×C₂²⋊D₆₁	—	4	2	no	—
85	#11758222	—	2	1	no	—
86	#11758222	—	2	1	no	—
87	#11758222	—	4	2	no	—
88	#11758222	—	4	2	no	—
89	#11758222	—	4	2	no	—
90	#11758222	—	4	2	no	—
91	#11758222	D₈	1	2	yes	—
92	#11758222	C₂×D₄	1	2	yes	—
93	C₂×D₂₄₄	—	2	2	no	—
94	C₂×D₂₄₄	D₈	1	1	yes	characteristic
95	C₂×C₄×D₆₁	—	2	1	no	—
96	C₂×C₄×D₆₁	—	4	2	no	—
97	C₂×Dic₁₂₂	—	2	2	no	—
98	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
99	M₁₆×C₆₁	—	2	4	no	—
100	M₁₆×C₆₁	—	4	2	no	—
101	C₂×C₈×C₆₁	—	4	2	no	—
102	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
106	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
107	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
108	#11758194	—	2	1	no	—
109	#11758194	C₂×D₄	1	2	yes	—
110	C₂×#92418	—	2	1	no	—
111	Dic₂₄₄	—	2	2	no	—
112	Dic₂₄₄	—	2	2	no	—
113	Dic₂₄₄	—	4	4	no	—
114	D₄₈₈	—	2	2	no	—
115	D₄₈₈	—	2	2	no	—
116	D₄₈₈	—	4	4	no	—
117	#11758190	—	4	4	no	—
118	#11758190	—	4	4	no	—
119	#11758190	—	4	2	no	—
120	#11758190	—	4	2	no	—
121	#11758189	—	2	2	no	—
122	#11758189	—	2	2	no	—
123	C₈×D₆₁	—	2	2	no	—
124	C₈×D₆₁	—	2	2	no	—
125	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
129	#11758185	—	2	2	no	—
130	C₂³×C₆₁	#135	1	1	yes	characteristic
131	C₂²×D₆₁	—	2	1	no	—
132	C₂²×D₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₆₁	—	2	2	no	—
140	C₂²⋊D₆₁	—	2	2	no	—
141	C₂²⋊D₆₁	—	8	2	no	—
142	C₂²⋊D₆₁	—	8	2	no	—
143	C₂×Dic₆₁	—	2	1	no	—
144	C₂×Dic₆₁	—	4	2	no	—
145	D₂₄₄	—	2	1	no	—
146	D₂₄₄	—	4	2	no	—
147	D₂₄₄	—	4	2	no	—
148	D₂₄₄	C₂×D₈	1	2	yes	—
149	C₄×D₆₁	—	2	2	no	—
150	C₄×D₆₁	—	2	2	no	—
151	C₄×D₆₁	—	8	2	no	—
152	C₄×D₆₁	—	8	2	no	—
153	Dic₁₂₂	—	2	1	no	—
154	Dic₁₂₂	—	4	2	no	—
155	Dic₁₂₂	—	4	2	no	—
156	Dic₁₂₂	C₂×D₈	1	2	yes	—
157	C₈×C₆₁	—	4	4	no	—
158	C₈×C₆₁	#121	1	1	yes	characteristic
159	C₈×C₆₁	#121	1	1	yes	characteristic
160	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
162	#92418	—	2	2	no	—
163	#33617	—	61	1	no	maximal, Sylow
164	C₂²×C₆₁	—	4	1	no	—
165	C₂²×C₆₁	#466	1	1	yes	characteristic
166	C₂²×C₆₁	#466	1	1	yes	characteristic
167	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
168	C₂×D₆₁	—	2	2	no	—
169	C₂×D₆₁	—	4	2	no	—
170	C₂×D₆₁	—	8	2	no	—
171	C₂×D₆₁	—	16	2	no	—
172	C₄×C₆₁	#466	1	1	yes	characteristic
173	C₄×C₆₁	#466	1	1	yes	characteristic
174	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
175	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
176	Dic₆₁	—	2	2	no	—
177	Dic₆₁	—	8	2	no	—
178	C₂×#508	—	61	1	no	—
179	C₂×#504	—	61	1	no	—
180	#3067	—	61	2	no	—
181	#2223	—	61	4	no	—
182	#2222	—	61	2	no	—
183	#2221	—	61	2	no	—
184	C₂×#368	D₆₁	1	1	yes	characteristic, complemented
185	#2148	—	61	2	no	—
186	C₂×C₆₁	—	2	1	no	—
187	C₂×C₆₁	—	2	1	no	—
188	C₂×C₆₁	—	2	1	no	—
189	C₂×C₆₁	C₂×#466	1	1	yes	characteristic
190	D₆₁	—	4	2	no	—
191	D₆₁	—	16	2	no	—
192	#575	—	122	2	no	—
193	C₂×#135	—	61	2	no	—
194	C₂×#131	—	61	1	no	—
195	#508	—	122	1	no	—
196	#508	—	122	1	no	—
197	#508	—	122	1	no	—
198	#508	—	122	1	no	—
199	C₂×Q₃₂	—	61	1	no	—
200	C₂×SD₃₂	—	61	2	no	—
201	C₂×D₁₆	—	61	1	no	—
202	#504	—	61	2	no	—
203	#504	—	61	2	no	—
204	#504	—	122	1	no	—
205	#504	—	122	1	no	—
206	C₂×M₃₂	—	61	1	no	—
207	C₂×M₃₂	—	122	1	no	—
208	C₂²×C₁₆	—	122	1	no	—
209	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
210	#473	—	61	4	no	—
211	#472	—	61	4	no	—
212	#471	—	122	4	no	—
213	C₂×#108	C₂×D₆₁	1	2	yes	—
214	#413	—	122	2	no	—
215	#368	C₂×D₆₁	1	2	yes	complemented
216	#368	C₂×D₆₁	1	2	yes	complemented
217	#362	—	61	4	no	—
218	#361	—	61	4	no	—
219	C₆₁	#33617	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	61	1	no	—
221	C₂×#41	—	122	2	no	—
222	#137	—	122	4	no	—
223	#137	—	244	2	no	—
224	AGL(1,ℤ/8)	—	122	4	no	—
225	AGL(1,ℤ/8)	—	244	2	no	—
226	#135	—	122	2	no	—
227	#135	—	122	2	no	—
228	#135	—	122	2	no	—
229	#135	—	122	2	no	—
230	#135	—	244	4	no	—
231	C₂×Q₁₆	—	61	2	no	—
232	C₂×SD₁₆	—	122	2	no	—
233	C₂×D₈	—	61	2	no	—
234	#131	—	61	2	no	—
235	#131	—	61	2	no	—
236	#131	—	122	1	no	—
237	#131	—	122	1	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	61	1	no	—
240	C₂×M₁₆	—	122	1	no	—
241	C₂²×C₈	—	122	1	no	—
242	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
243	Q₃₂	—	122	1	no	—
244	Q₃₂	—	122	1	no	—
245	SD₃₂	—	122	2	no	—
246	SD₃₂	—	122	2	no	—
247	D₁₆	—	122	1	no	—
248	D₁₆	—	122	1	no	—
249	M₃₂	—	61	2	no	—
250	M₃₂	—	122	1	no	—
251	M₃₂	—	122	2	no	—
252	M₃₂	—	122	2	no	—
253	C₂×C₁₆	—	122	1	no	—
254	C₂×C₁₆	—	122	2	no	—
255	C₂×C₁₆	—	122	2	no	—
256	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
257	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
258	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
259	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
262	#108	—	2	2	no	—
263	#108	C₂²×D₆₁	1	4	yes	—
264	#101	—	122	4	no	—
265	#100	—	122	4	no	—
266	#41	—	61	2	no	—
267	#41	—	61	2	no	—
268	#41	—	122	1	no	—
269	#41	—	122	1	no	—
270	#41	—	244	2	no	—
271	#41	—	244	2	no	—
272	#41	—	244	2	no	—
273	#41	—	244	2	no	—
274	C₂×Q₈	—	61	1	no	—
275	C₂×Q₈	—	122	2	no	—
276	C₂×D₄	—	61	1	no	—
277	C₂×D₄	—	122	1	no	—
278	C₂×D₄	—	122	2	no	—
279	C₂×D₄	—	244	2	no	—
280	C₂²×C₄	—	122	1	no	—
281	C₂²×C₄	—	244	2	no	—
282	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
283	Q₁₆	—	122	2	no	—
284	Q₁₆	—	122	2	no	—
285	Q₁₆	—	244	4	no	—
286	SD₁₆	—	244	4	no	—
287	SD₁₆	—	244	4	no	—
288	SD₁₆	—	244	2	no	—
289	SD₁₆	—	244	2	no	—
290	D₈	—	122	2	no	—
291	D₈	—	122	2	no	—
292	D₈	—	244	4	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	61	2	no	—
296	M₁₆	—	122	1	no	—
297	M₁₆	—	122	2	no	—
298	M₁₆	—	122	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	122	1	no	—
301	C₂×C₈	—	122	2	no	—
302	C₂×C₈	—	122	2	no	—
303	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
305	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
306	C₂×C₈	#11758226	1	1	yes	characteristic
307	C₂×C₈	#11758226	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
309	C₁₆	—	122	2	no	—
310	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
311	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
312	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
313	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
314	C₂³	—	122	1	no	—
315	C₂³	—	244	2	no	—
316	C₂³	#14519800	1	1	yes	characteristic
317	Q₈	—	61	2	no	—
318	Q₈	—	122	1	no	—
319	Q₈	—	244	2	no	—
320	Q₈	—	244	2	no	—
321	D₄	—	61	2	no	—
322	D₄	—	122	2	no	—
323	D₄	—	122	2	no	—
324	D₄	—	122	1	no	—
325	D₄	—	488	2	no	—
326	D₄	—	488	2	no	—
327	D₄	—	244	2	no	—
328	D₄	—	244	2	no	—
329	C₂×C₄	—	122	1	no	—
330	C₂×C₄	—	122	2	no	—
331	C₂×C₄	—	122	2	no	—
332	C₂×C₄	—	488	2	no	—
333	C₂×C₄	—	488	2	no	—
334	C₂×C₄	—	244	2	no	—
335	C₂×C₄	#14519752	1	1	yes	characteristic
336	C₂×C₄	#14519752	1	1	yes	characteristic
337	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
338	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
339	C₂×C₄	#14519800	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	122	2	no	—
343	C₈	#14519752	1	1	yes	characteristic
344	C₈	#14519752	1	1	yes	characteristic
345	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
346	C₈	C₂×#11758226	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	122	2	no	—
349	C₂²	—	976	2	no	—
350	C₂²	—	488	2	no	—
351	C₂²	—	244	2	no	—
352	C₂²	#16839999	1	1	yes	characteristic
353	C₂²	#16839999	1	1	yes	characteristic
354	C₂²	C₂×#14519800	1	1	yes	characteristic
355	C₄	—	122	2	no	—
356	C₄	—	488	2	no	—
357	C₄	#16839999	1	1	yes	characteristic
358	C₄	#16839999	1	1	yes	characteristic
359	C₄	C₂×#14519752	1	1	yes	characteristic
360	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	976	2	no	—
365	C₂	—	244	2	no	—
366	C₂	C₂×#16839999	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16839999
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#368	C₂
socle	C₂×C₆₁	C₂×#466
hypercenter	C₂×#368	D₆₁
hypocenter	C₆₁	#33617
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#16840194	C₂	semidirect
2. C₂×#16840178	C₂	semidirect
3. #21455540	C₂	semidirect
4. #21447950	C₂	semidirect
5. #21447945	C₂	semidirect
6. #21447940	C₂	semidirect
7. #21447566	C₂	semidirect
8. C₂×C₆₁×#368	C₂	semidirect
16. C₂×Dic₄₈₈	C₂²	semidirect
17. C₂×#14519722	C₂²	semidirect
18. C₂×D₉₇₆	C₂²	semidirect
30. #16839577	C₂²	semidirect
31. #16839574	C₂²	semidirect
34. C₆₁×#368	C₂²	semidirect
35. C₆₁×#368	C₂²	semidirect
184. C₂×#368	D₆₁	semidirect
215. #368	C₂×D₆₁	semidirect
216. #368	C₂×D₆₁	semidirect
219. C₆₁	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840194	2
C₂×#16840178	2
#21455540	2
#21447950	2
#21447945	2
#21447940	2
#21447566	2
C₂×C₆₁×#368	2
#33617	61