Subgroups — C₄×#19128134 #35557686

Order 21248.

690 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128134	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581815	C₂	1	2	yes	maximal, complemented
3	Q₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14581735	C₂	1	2	yes	maximal, complemented
5	#24909312	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₃₂₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909309	C₂	1	1	yes	maximal, characteristic, complemented
8	#24909244	C₂	1	1	yes	maximal, characteristic, complemented
9	#24909169	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581815	C₂²	1	2	yes	—
13	C₄×#13050591	—	2	2	no	—
14	#19128134	C₄	1	8	yes	direct product factor
15	C₂×#14581735	C₂²	1	2	yes	—
16	C₂×D₁₃₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
19	Q₁₆×Dic₈₃	C₂²	1	2	yes	—
20	C₄×#13050577	—	2	2	no	—
21	#19127786	—	2	2	no	—
22	C₄×D₆₆₄	C₂²	1	2	yes	—
23	#19127783	C₂²	1	2	yes	—
24	#19127738	C₂²	1	2	yes	—
25	#19127696	—	2	2	no	—
26	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
27	#19127528	C₂²	1	2	yes	—
28	#19127527	C₂²	1	1	yes	characteristic
29	#19127505	C₂²	1	2	yes	—
30	#19127503	C₂²	1	2	yes	—
31	#19127501	C₂²	1	2	yes	—
32	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₈₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₃	C₂²	1	2	yes	—
37	#366×C₈₃	C₂²	1	1	yes	characteristic
38	#358×C₈₃	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
40	C₂×#13050591	—	2	2	no	—
41	#14581815	—	2	4	no	—
42	#14581815	C₂×C₄	1	8	yes	complemented
43	C₂×#13050577	—	2	2	no	—
44	C₂×D₆₆₄	C₂³	1	2	yes	—
45	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₈₃	—	2	2	no	—
48	C₄×D₃₃₂	—	2	2	no	—
49	C₄×D₃₃₂	—	4	2	no	—
50	#14581754	—	2	2	no	—
51	#14581754	—	4	2	no	—
52	#14581750	—	2	2	no	—
53	#14581750	—	4	2	no	—
54	C₄²×D₈₃	—	4	2	no	—
55	C₄²×D₈₃	D₄	1	1	yes	characteristic
56	#14581735	C₂×C₄	1	8	yes	complemented
57	D₁₃₂₈	C₂×C₄	1	4	yes	complemented
58	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₈₃	—	2	2	no	—
61	C₁₆×D₈₃	C₂×C₄	1	4	yes	complemented
62	#14581726	C₂³	1	2	yes	—
63	#14581723	—	2	2	no	—
64	#14581716	—	2	2	no	—
65	#14581714	—	2	2	no	—
66	#14581712	—	2	2	no	—
67	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₈₃	D₄	1	1	yes	characteristic
69	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₈₃	—	2	2	no	—
72	Q₃₂×C₈₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
75	#107×C₈₃	C₂³	1	2	yes	—
76	#103×C₈₃	—	2	2	no	—
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	#13050591	—	2	8	no	—
79	#13050591	—	4	4	no	—
80	C₂×D₃₃₂	—	2	2	no	—
81	C₂×D₃₃₂	—	4	2	no	—
82	C₂×C₄×D₈₃	—	4	2	no	—
83	C₂×C₄×D₈₃	—	4	2	no	—
84	C₂×C₄×D₈₃	—	4	2	no	—
85	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
86	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
87	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
88	#13050577	—	2	8	no	—
89	D₆₆₄	—	2	2	no	—
90	D₆₆₄	C₂²×C₄	1	4	yes	—
91	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₈₃	—	2	2	no	—
94	C₈×D₈₃	C₂²×C₄	1	2	yes	—
95	C₈×D₈₃	#41	1	2	yes	—
96	C₄⋊Dic₈₃	—	2	2	no	—
97	C₄⋊Dic₈₃	—	4	2	no	—
98	(C₂×D₈₃)⋊C₄	—	4	2	no	—
99	(C₂×D₈₃)⋊C₄	—	4	2	no	—
100	(C₂×D₈₃)⋊C₄	—	4	2	no	—
101	C₄×Dic₈₃	—	4	2	no	—
102	C₄×Dic₈₃	—	4	2	no	—
103	C₄×Dic₈₃	D₈	1	1	yes	characteristic
104	C₄×Dic₈₃	D₈	1	1	yes	characteristic
105	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
106	C₈₃⋊C₁₆	—	2	1	no	—
107	C₈₃⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₈₃	—	2	2	no	—
109	Q₁₆×C₈₃	—	2	2	no	—
110	Q₁₆×C₈₃	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₃	—	2	2	no	—
114	C₄⋊C₄×C₈₃	—	4	2	no	—
115	C₄²×C₈₃	—	4	2	no	—
116	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₃	—	2	1	no	—
118	C₁₆×C₈₃	C₂²×C₄	1	2	yes	—
119	C₂²×D₈₃	—	4	2	no	—
120	C₂²×D₈₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₈₃	—	2	4	no	—
122	Q₈×C₈₃	—	4	2	no	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	—	4	2	no	—
125	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	—	4	2	no	—
129	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
132	D₃₃₂	—	2	4	no	—
133	D₃₃₂	—	4	8	no	—
134	D₃₃₂	—	4	2	no	—
135	C₄×D₈₃	—	2	2	no	—
136	C₄×D₈₃	—	2	2	no	—
137	C₄×D₈₃	—	4	8	no	—
138	C₄×D₈₃	—	8	4	no	—
139	C₄×D₈₃	—	8	4	no	—
140	C₄×D₈₃	C₄×D₄	1	2	yes	—
141	C₄×D₈₃	#135	1	2	yes	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
145	C₈₃⋊C₈	—	2	1	no	—
146	C₈₃⋊C₈	C₄×D₄	1	2	yes	—
147	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₃	—	2	1	no	—
149	C₂×D₈₃	—	2	1	no	—
150	C₂×D₈₃	—	4	4	no	—
151	C₂×D₈₃	—	8	2	no	—
152	C₂×D₈₃	—	8	2	no	—
153	C₂×D₈₃	C₄×D₈	1	2	yes	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	4	no	—
156	C₄×C₈₃	—	8	2	no	—
157	C₄×C₈₃	#508	1	2	yes	complemented
158	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	8	2	no	—
162	Dic₈₃	C₄×D₈	1	2	yes	—
163	Dic₈₃	#508	1	2	yes	complemented
164	C₄×#508	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₈₃	C₂×#508	1	1	yes	characteristic
167	C₂×C₈₃	C₂×#508	1	1	yes	characteristic
168	D₈₃	—	2	2	no	—
169	D₈₃	—	8	4	no	—
170	C₂×#508	—	83	1	no	—
171	C₄×#135	—	83	2	no	—
172	C₄×Q₃₂	—	83	1	no	—
173	C₄×Q₃₂	D₈₃	1	1	yes	characteristic, complemented
174	C₄×SD₃₂	—	83	2	no	—
175	C₄×SD₃₂	—	83	2	no	—
176	C₄×D₁₆	—	83	1	no	—
177	C₄×D₁₆	—	83	1	no	—
178	#2158	—	83	1	no	—
179	#2138	—	83	2	no	—
180	C₂×C₄×C₁₆	—	83	1	no	—
181	C₈₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
182	C₂×#135	—	83	2	no	—
183	C₄×#41	—	166	2	no	—
184	#508	—	83	8	no	—
185	C₂×Q₃₂	C₂×D₈₃	1	1	yes	characteristic
186	C₂×SD₃₂	—	83	2	no	—
187	C₂×D₁₆	—	83	1	no	—
188	C₂²×C₁₆	—	83	1	no	—
189	C₂²×C₁₆	—	83	1	no	—
190	C₄×Q₁₆	—	83	2	no	—
191	C₄×Q₁₆	C₂×D₈₃	1	2	yes	—
192	C₄×SD₁₆	—	166	2	no	—
193	C₄×SD₁₆	—	166	2	no	—
194	C₄×D₈	—	83	2	no	—
195	C₄×D₈	—	83	2	no	—
196	#427	—	83	2	no	—
197	#416	—	166	2	no	—
198	C₂×C₄×C₈	—	83	1	no	—
199	#367	—	83	2	no	—
200	#366	—	83	1	no	—
201	#366	C₂×D₈₃	1	1	yes	characteristic
202	#358	—	83	2	no	—
203	#358	C₂×D₈₃	1	2	yes	—
204	#357	—	83	2	no	—
205	#357	—	83	2	no	—
206	C₄×C₁₆	—	83	1	no	—
207	C₄×C₁₆	—	83	1	no	—
208	C₄×C₁₆	—	83	1	no	—
209	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
210	C₂×#41	—	166	2	no	—
211	#135	—	83	8	no	—
212	#135	—	166	4	no	—
213	C₂×Q₁₆	C₂²×D₈₃	1	2	yes	—
214	C₂×SD₁₆	—	166	2	no	—
215	C₂×D₈	—	83	2	no	—
216	C₂²×C₈	—	83	1	no	—
217	C₂²×C₈	—	83	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	166	2	no	—
220	C₄×D₄	—	332	2	no	—
221	C₄×D₄	—	332	2	no	—
222	C₄×D₄	—	166	2	no	—
223	C₄×D₄	—	166	2	no	—
224	#117	—	332	2	no	—
225	#117	—	166	2	no	—
226	C₂×C₄²	—	332	2	no	—
227	C₂×C₄²	—	83	1	no	—
228	Q₃₂	C₄×D₈₃	1	4	yes	complemented
229	SD₃₂	—	83	8	no	—
230	D₁₆	—	83	4	no	—
231	C₂×C₁₆	—	83	1	no	—
232	C₂×C₁₆	—	83	4	no	—
233	C₂×C₁₆	—	83	1	no	—
234	C₂×C₁₆	—	166	2	no	—
235	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
237	#107	—	83	2	no	—
238	#107	C₂²×D₈₃	1	2	yes	—
239	#106	—	166	2	no	—
240	#103	—	2	2	no	—
241	#103	—	166	2	no	—
242	#102	—	166	2	no	—
243	#102	—	166	2	no	—
244	C₄×C₈	—	83	1	no	—
245	C₄×C₈	—	83	1	no	—
246	C₄×C₈	—	83	1	no	—
247	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	#41	—	332	4	no	—
249	#41	—	166	8	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×D₄	—	332	2	no	—
252	C₂×D₄	—	166	2	no	—
253	C₂²×C₄	—	332	2	no	—
254	C₂²×C₄	—	332	2	no	—
255	C₂²×C₄	—	332	2	no	—
256	C₂²×C₄	—	83	1	no	—
257	C₂²×C₄	—	83	1	no	—
258	C₂²×C₄	—	83	1	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	C₂×C₄×D₈₃	1	4	yes	—
261	SD₁₆	—	166	8	no	—
262	D₈	—	83	4	no	—
263	D₈	—	166	2	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	—	83	2	no	—
266	C₂×C₈	—	83	2	no	—
267	C₂×C₈	—	83	1	no	—
268	C₂×C₈	—	166	2	no	—
269	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	332	2	no	—
274	C₄⋊C₄	—	166	2	no	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₂²⋊C₄	—	332	2	no	—
277	C₂²⋊C₄	—	332	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	332	2	no	—
280	C₄²	—	332	2	no	—
281	C₄²	—	83	1	no	—
282	C₄²	—	83	1	no	—
283	C₄²	—	83	1	no	—
284	C₄²	D₄×D₈₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	83	2	no	—
287	C₁₆	—	166	1	no	—
288	C₁₆	C₂×C₄×D₈₃	1	2	yes	—
289	C₂³	—	332	2	no	—
290	C₂³	—	83	1	no	—
291	Q₈	—	2	4	no	—
292	Q₈	—	4	2	no	—
293	D₄	—	332	8	no	—
294	D₄	—	332	2	no	—
295	D₄	—	166	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	332	8	no	—
299	C₂×C₄	—	332	2	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	2	no	—
303	C₂×C₄	—	83	2	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	664	4	no	—
306	C₂×C₄	—	664	4	no	—
307	C₂×C₄	—	166	2	no	—
308	C₂×C₄	—	166	2	no	—
309	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
310	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	83	2	no	—
314	C₈	—	166	1	no	—
315	C₈	C₂²×C₄×D₈₃	1	1	yes	characteristic
316	C₈	#41×D₈₃	1	1	yes	characteristic
317	C₂²	—	332	4	no	—
318	C₂²	—	83	2	no	—
319	C₂²	—	664	2	no	—
320	C₂²	—	664	2	no	—
321	C₂²	—	166	1	no	—
322	C₂²	—	166	1	no	—
323	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	8	2	no	—
327	C₄	—	83	2	no	—
328	C₄	—	83	2	no	—
329	C₄	—	664	2	no	—
330	C₄	—	166	1	no	—
331	C₄	#19128134	1	2	yes	direct product factor
332	C₄	C₄×D₄×D₈₃	1	1	yes	characteristic
333	C₄	#135×D₈₃	1	1	yes	characteristic
334	C₂	—	664	4	no	—
335	C₂	—	166	2	no	—
336	C₂	C₄×D₈×D₈₃	1	1	yes	characteristic
337	C₂	#508×D₈₃	1	1	yes	characteristic
338	C₂	C₂×#19128134	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₃
commutator	C₈×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₄×Q₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂×#19128134	C₂
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₄×Q₃₂	D₈₃
hypocenter	C₈₃	C₄×#508
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. C₄×#14581815	C₂	semidirect
3. Q₃₂×Dic₈₃	C₂	semidirect
4. C₄×#14581735	C₂	semidirect
5. #24909312	C₂	semidirect
6. C₄×D₁₃₂₈	C₂	semidirect
7. #24909309	C₂	semidirect
8. #24909244	C₂	semidirect
9. #24909169	C₂	semidirect
10. C₄×C₁₆×D₈₃	C₂	semidirect
11. C₄×Q₃₂×C₈₃	C₂	semidirect
14. #19128134	C₄	direct (N × H)
42. #14581815	C₂×C₄	semidirect
56. #14581735	C₂×C₄	semidirect
57. D₁₃₂₈	C₂×C₄	semidirect
61. C₁₆×D₈₃	C₂×C₄	semidirect
72. Q₃₂×C₈₃	C₂×C₄	semidirect
157. C₄×C₈₃	#508	semidirect
163. Dic₈₃	#508	semidirect
173. C₄×Q₃₂	D₈₃	semidirect
181. C₈₃	C₄×#508	semidirect
228. Q₃₂	C₄×D₈₃	semidirect
331. C₄	#19128134	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19128134	2
C₄×#14581815	2
Q₃₂×Dic₈₃	2
C₄×#14581735	2
#24909312	2
C₄×D₁₃₂₈	2
#24909309	2
#24909244	2
#24909169	2
C₄×C₁₆×D₈₃	2
C₄×Q₃₂×C₈₃	2
C₄×#508	83

Subgroups — C4×#19128134 #35557686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19128134 #35557686