Subgroups — C₄×#13063346 #19143364

Order 5504.

500 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063346	C₂	1	1	yes	maximal, characteristic
2	C₄×#10621902	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#10621900	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#10621885	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582986	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10621882	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582979	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582934	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582894	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582892	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×#10621900	C₂²	1	1	yes	characteristic
18	#13063346	C₄	1	8	yes	direct product factor
19	C₂×#10621885	C₂²	1	1	yes	characteristic
20	C₂×#10621873	C₂²	1	1	yes	characteristic
21	C₂×#10621882	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
24	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
26	#13063290	C₂²	1	1	yes	characteristic
27	C₄×D₁₇₂	—	2	1	no	—
28	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
29	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	#13063287	—	2	1	no	—
31	#13063287	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₄₃	—	2	1	no	—
33	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
34	#13063283	—	2	1	no	—
35	#13063283	C₂²	1	1	yes	characteristic
36	#13063283	C₂²	1	1	yes	characteristic
37	#13063281	—	2	1	no	—
38	C₂×C₄×Dic₄₃	—	2	1	no	—
39	C₄²×D₄₃	—	2	1	no	—
40	C₄²×D₄₃	C₂²	1	1	yes	characteristic
41	#13063258	C₂²	1	1	yes	characteristic
42	#13063258	C₂²	1	1	yes	characteristic
43	#13063257	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063247	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
56	#106×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	#10621902	—	2	2	no	—
67	#10621902	C₂×C₄	1	4	yes	complemented
68	#10621900	—	2	2	no	—
69	#10621900	C₂×C₄	1	4	yes	complemented
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
76	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
77	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
78	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
79	SD₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	—	2	1	no	—
83	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	—	2	1	no	—
89	#10621885	C₂×C₄	1	4	yes	complemented
90	#10621882	C₂×C₄	1	4	yes	complemented
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	(C₂×D₄₃)⋊C₄	—	2	1	no	—
95	C₄⋊Dic₄₃	—	2	1	no	—
96	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	D₄	1	1	yes	characteristic
106	C₄×Dic₄₃	D₄	1	1	yes	characteristic
107	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
110	#10621873	C₂×C₄	1	4	yes	complemented
111	C₈×D₄₃	—	2	2	no	—
112	C₈×D₄₃	C₂×C₄	1	4	yes	complemented
113	C₂³×C₄₃	—	2	1	no	—
114	C₂²×D₄₃	—	2	1	no	—
115	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₄₃	—	2	1	no	—
117	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
118	D₄×C₄₃	—	2	1	no	—
119	D₄×C₄₃	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₄₃	—	2	4	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	4	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	C₂×Dic₄₃	—	2	1	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
136	D₁₇₂	—	2	4	no	—
137	D₁₇₂	—	2	1	no	—
138	D₁₇₂	C₂²×C₄	1	2	yes	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	2	4	no	—
141	C₄×D₄₃	—	2	2	no	—
142	C₄×D₄₃	—	4	2	no	—
143	C₄×D₄₃	—	4	2	no	—
144	C₄×D₄₃	C₂²×C₄	1	2	yes	—
145	C₄×D₄₃	#41	1	2	yes	—
146	Dic₈₆	—	2	1	no	—
147	Dic₈₆	C₂²×C₄	1	2	yes	—
148	C₈×C₄₃	—	2	1	no	—
149	C₈×C₄₃	C₂²×C₄	1	2	yes	—
150	C₄₃⋊C₈	—	2	1	no	—
151	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	1	no	—
158	C₂×D₄₃	—	2	2	no	—
159	C₂×D₄₃	—	4	1	no	—
160	C₂×D₄₃	—	4	1	no	—
161	C₂×D₄₃	C₄×D₄	1	2	yes	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	4	1	no	—
165	C₄×C₄₃	—	4	1	no	—
166	C₄×C₄₃	#135	1	2	yes	complemented
167	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
169	Dic₄₃	—	2	2	no	—
170	Dic₄₃	—	2	1	no	—
171	Dic₄₃	—	4	1	no	—
172	Dic₄₃	—	4	1	no	—
173	Dic₄₃	C₄×D₄	1	2	yes	—
174	Dic₄₃	#135	1	2	yes	complemented
175	C₄×#135	—	43	1	no	maximal, Sylow
176	C₂×C₄₃	—	4	2	no	—
177	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
179	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
180	D₄₃	—	2	2	no	—
181	D₄₃	—	4	2	no	—
182	C₂×#135	—	43	1	no	—
183	C₄×#41	—	43	1	no	—
184	C₄×#41	—	43	1	no	—
185	C₄×Q₁₆	—	43	1	no	—
186	C₄×Q₁₆	—	43	1	no	—
187	C₄×SD₁₆	—	43	1	no	—
188	C₄×SD₁₆	—	43	1	no	—
189	C₄×SD₁₆	—	43	1	no	—
190	C₄×SD₁₆	D₄₃	1	1	yes	characteristic, complemented
191	C₄×D₈	—	43	1	no	—
192	C₄×D₈	—	43	1	no	—
193	#427	—	43	1	no	—
194	#416	—	43	1	no	—
195	#416	—	43	1	no	—
196	C₂×C₄×C₈	—	43	1	no	—
197	C₄₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	43	1	no	—
199	C₂×#41	—	43	1	no	—
200	#135	—	43	8	no	—
201	C₂×Q₁₆	—	43	1	no	—
202	C₂×SD₁₆	—	43	1	no	—
203	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₂×D₈	—	43	1	no	—
205	C₂²×C₈	—	43	1	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₄×Q₈	—	43	1	no	—
208	C₄×Q₈	—	43	1	no	—
209	C₄×Q₈	—	43	1	no	—
210	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
211	C₄×D₄	—	43	1	no	—
212	C₄×D₄	—	43	1	no	—
213	C₄×D₄	—	43	1	no	—
214	C₄×D₄	—	86	1	no	—
215	C₄×D₄	—	86	1	no	—
216	C₄×D₄	—	86	1	no	—
217	C₄×D₄	—	86	1	no	—
218	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
219	#117	—	43	1	no	—
220	#117	—	43	1	no	—
221	#117	—	86	1	no	—
222	#117	—	86	1	no	—
223	C₂×C₄²	—	43	1	no	—
224	C₂×C₄²	—	86	1	no	—
225	C₂×C₄²	—	86	1	no	—
226	#107	—	43	1	no	—
227	#107	—	43	1	no	—
228	#106	—	43	1	no	—
229	#106	C₂×D₄₃	1	1	yes	characteristic
230	#103	—	43	1	no	—
231	#103	—	43	1	no	—
232	#103	—	43	1	no	—
233	#103	C₂×D₄₃	1	1	yes	characteristic
234	#102	—	43	1	no	—
235	#102	—	43	1	no	—
236	#102	—	43	1	no	—
237	#102	C₂×D₄₃	1	1	yes	characteristic
238	C₄×C₈	—	43	1	no	—
239	C₄×C₈	—	43	1	no	—
240	C₄×C₈	—	43	1	no	—
241	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
242	#41	—	43	4	no	—
243	#41	—	43	4	no	—
244	#41	—	86	2	no	—
245	#41	—	86	2	no	—
246	C₂×Q₈	—	43	1	no	—
247	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₂×D₄	—	43	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	86	1	no	—
258	C₂²×C₄	—	86	1	no	—
259	C₂²×C₄	—	86	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	Q₁₆	—	43	4	no	—
262	SD₁₆	—	43	4	no	—
263	SD₁₆	C₄×D₄₃	1	4	yes	complemented
264	D₈	—	43	4	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	—	43	4	no	—
267	C₂×C₈	—	43	1	no	—
268	C₂×C₈	—	86	2	no	—
269	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	43	1	no	—
273	C₄⋊C₄	—	43	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	—	86	1	no	—
277	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₂²⋊C₄	—	86	1	no	—
282	C₂²⋊C₄	—	86	1	no	—
283	C₂²⋊C₄	—	86	1	no	—
284	C₂²⋊C₄	—	86	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	43	1	no	—
287	C₄²	—	43	1	no	—
288	C₄²	—	43	1	no	—
289	C₄²	—	86	1	no	—
290	C₄²	—	86	1	no	—
291	C₄²	—	86	1	no	—
292	C₄²	—	86	1	no	—
293	C₄²	—	86	1	no	—
294	C₄²	C₂²×D₄₃	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	43	1	no	—
297	C₂³	—	86	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	43	2	no	—
300	Q₈	—	86	1	no	—
301	Q₈	C₂×C₄×D₄₃	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	43	2	no	—
304	D₄	—	86	4	no	—
305	D₄	—	86	4	no	—
306	D₄	—	86	1	no	—
307	D₄	C₂×C₄×D₄₃	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	43	1	no	—
313	C₂×C₄	—	43	1	no	—
314	C₂×C₄	—	43	2	no	—
315	C₂×C₄	—	43	2	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	4	no	—
319	C₂×C₄	—	86	4	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
328	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	43	2	no	—
332	C₈	—	86	1	no	—
333	C₈	C₂×C₄×D₄₃	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	43	2	no	—
338	C₂²	—	86	1	no	—
339	C₂²	—	86	1	no	—
340	C₂²	—	86	2	no	—
341	C₂²	—	172	1	no	—
342	C₂²	—	172	1	no	—
343	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	43	2	no	—
349	C₄	—	43	2	no	—
350	C₄	—	86	2	no	—
351	C₄	—	86	1	no	—
352	C₄	—	172	1	no	—
353	C₄	—	172	1	no	—
354	C₄	#13063346	1	2	yes	direct product factor
355	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
356	C₄	#41×D₄₃	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	86	2	no	—
359	C₂	—	172	2	no	—
360	C₂	C₄×D₄×D₄₃	1	1	yes	characteristic
361	C₂	#135×D₄₃	1	1	yes	characteristic
362	C₂	C₂×#13063346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×SD₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×SD₁₆	D₄₃
hypocenter	C₄₃	C₄×#135
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#10621902	C₂	semidirect
3. C₄×#10621900	C₂	semidirect
4. C₄×#10621885	C₂	semidirect
5. #14582991	C₂	semidirect
6. C₄×#10621873	C₂	semidirect
7. SD₁₆×Dic₄₃	C₂	semidirect
8. #14582986	C₂	semidirect
9. C₄×#10621882	C₂	semidirect
10. #14582979	C₂	semidirect
11. #14582934	C₂	semidirect
12. #14582894	C₂	semidirect
13. #14582892	C₂	semidirect
14. C₄×C₈×D₄₃	C₂	semidirect
15. C₄×SD₁₆×C₄₃	C₂	semidirect
18. #13063346	C₄	direct (N × H)
67. #10621902	C₂×C₄	semidirect
69. #10621900	C₂×C₄	semidirect
79. SD₁₆×C₄₃	C₂×C₄	semidirect
89. #10621885	C₂×C₄	semidirect
90. #10621882	C₂×C₄	semidirect
110. #10621873	C₂×C₄	semidirect
112. C₈×D₄₃	C₂×C₄	semidirect
166. C₄×C₄₃	#135	semidirect
174. Dic₄₃	#135	semidirect
190. C₄×SD₁₆	D₄₃	semidirect
197. C₄₃	C₄×#135	semidirect
263. SD₁₆	C₄×D₄₃	semidirect
354. C₄	#13063346	direct (N × H)

Subgroups — C4×#13063346 #19143364

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13063346 #19143364