Subgroups — C₄×#16813597 #28393089

Order 14464.

824 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×#16813597	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₅₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#20298499	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298497	C₂	1	1	yes	maximal, characteristic, complemented
7	#20298490	C₂	1	2	yes	maximal, complemented
8	#20298476	C₂	1	2	yes	maximal, complemented
9	#20298455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14268624	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
13	C₂²×D₄₅₂	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
18	#16813597	C₄	1	8	yes	direct product factor
19	C₄×D₄₅₂	—	2	8	no	—
20	C₄×D₄₅₂	—	2	4	no	—
21	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
23	C₂×#14268624	C₂²	1	2	yes	—
24	C₂×#14268624	C₂²	1	2	yes	—
25	C₂×#14268624	C₂²	1	2	yes	—
26	C₂×#14268624	C₂²	1	2	yes	—
27	C₂×#14268624	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
31	C₄²×D₁₁₃	—	2	4	no	—
32	#16813529	C₂²	1	2	yes	—
33	#16813529	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
37	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
38	#95×C₁₁₃	C₂²	1	2	yes	—
39	C₂³×D₁₁₃	C₂³	1	2	yes	—
40	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
41	C₂×D₄₅₂	—	2	4	no	—
42	C₂×D₄₅₂	—	2	4	no	—
43	C₂×D₄₅₂	—	2	2	no	—
44	C₂×D₄₅₂	—	2	2	no	—
45	C₂×D₄₅₂	C₂×C₄	1	8	yes	complemented
46	C₂×D₄₅₂	C₂×C₄	1	4	yes	complemented
47	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
48	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₁₃	—	2	2	no	—
52	C₂×C₄×D₁₁₃	—	2	2	no	—
53	C₂×C₄×D₁₁₃	—	2	2	no	—
54	C₂×C₄×D₁₁₃	—	2	2	no	—
55	C₂×C₄×D₁₁₃	—	2	2	no	—
56	C₂×C₄×D₁₁₃	—	2	2	no	—
57	C₂×C₄×D₁₁₃	—	2	2	no	—
58	C₂×C₄×D₁₁₃	—	2	4	no	—
59	C₂×C₄×D₁₁₃	—	2	4	no	—
60	C₂×C₄×D₁₁₃	—	2	4	no	—
61	C₂×C₄×D₁₁₃	C₂×C₄	1	4	yes	complemented
62	C₂×C₄×D₁₁₃	D₄	1	4	yes	complemented
63	C₄⋊Dic₁₁₃	—	2	2	no	—
64	C₄⋊Dic₁₁₃	—	2	4	no	—
65	#14268624	—	2	4	no	—
66	#14268624	—	2	4	no	—
67	#14268624	—	2	4	no	—
68	#14268624	—	2	4	no	—
69	#14268624	C₂×C₄	1	8	yes	complemented
70	C₄×Dic₁₁₃	—	2	2	no	—
71	C₄×Dic₁₁₃	—	2	2	no	—
72	C₄×Dic₁₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₁₃	—	2	2	no	—
79	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	complemented
80	C₄²×C₁₁₃	—	2	4	no	—
81	C₄²×C₁₁₃	D₄	1	4	yes	complemented
82	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₁₃	—	2	1	no	—
84	C₂²×D₁₁₃	—	2	1	no	—
85	C₂²×D₁₁₃	—	2	1	no	—
86	C₂²×D₁₁₃	—	2	1	no	—
87	C₂²×D₁₁₃	—	2	2	no	—
88	C₂²×D₁₁₃	—	2	2	no	—
89	C₂²×D₁₁₃	—	2	2	no	—
90	C₂²×D₁₁₃	—	2	2	no	—
91	C₂²×D₁₁₃	—	2	2	no	—
92	C₂²×D₁₁₃	—	2	2	no	—
93	C₂²×D₁₁₃	C₂²×C₄	1	4	yes	—
94	C₂²×D₁₁₃	C₂²×C₄	1	2	yes	—
95	C₂²×D₁₁₃	C₂×D₄	1	2	yes	—
96	C₂²×D₁₁₃	#41	1	2	yes	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	1	no	—
99	C₂×C₄×C₁₁₃	—	2	1	no	—
100	C₂×C₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	2	2	no	—
102	C₂×C₄×C₁₁₃	—	2	2	no	—
103	C₂×C₄×C₁₁₃	—	2	2	no	—
104	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₁₃	#41	1	2	yes	—
111	C₂×C₄×C₁₁₃	#41	1	2	yes	—
112	C₂×Dic₁₁₃	—	2	1	no	—
113	C₂×Dic₁₁₃	—	2	1	no	—
114	C₂×Dic₁₁₃	—	2	1	no	—
115	C₂×Dic₁₁₃	—	2	1	no	—
116	C₂×Dic₁₁₃	—	2	1	no	—
117	C₂×Dic₁₁₃	—	2	1	no	—
118	C₂×Dic₁₁₃	—	2	2	no	—
119	C₂×Dic₁₁₃	—	2	2	no	—
120	C₂×Dic₁₁₃	—	2	2	no	—
121	C₂×Dic₁₁₃	C₂²×C₄	1	2	yes	—
122	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
123	C₂×Dic₁₁₃	#41	1	2	yes	—
124	D₄₅₂	—	2	16	no	—
125	D₄₅₂	—	2	8	no	—
126	C₄×D₁₁₃	—	2	8	no	—
127	C₄×D₁₁₃	—	2	8	no	—
128	C₄×D₁₁₃	—	2	8	no	—
129	C₄×D₁₁₃	—	4	8	no	—
130	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
134	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×D₁₁₃	—	2	2	no	—
138	C₂×D₁₁₃	—	2	2	no	—
139	C₂×D₁₁₃	—	2	2	no	—
140	C₂×D₁₁₃	—	2	2	no	—
141	C₂×D₁₁₃	—	2	2	no	—
142	C₂×D₁₁₃	—	2	2	no	—
143	C₂×D₁₁₃	—	2	4	no	—
144	C₂×D₁₁₃	—	2	4	no	—
145	C₂×D₁₁₃	—	2	4	no	—
146	C₂×D₁₁₃	—	4	2	no	—
147	C₂×D₁₁₃	—	4	2	no	—
148	C₂×D₁₁₃	—	4	2	no	—
149	C₂×D₁₁₃	—	4	2	no	—
150	C₂×D₁₁₃	C₄×D₄	1	4	yes	complemented
151	C₄×C₁₁₃	—	2	4	no	—
152	C₄×C₁₁₃	—	2	2	no	—
153	C₄×C₁₁₃	—	4	2	no	—
154	C₄×C₁₁₃	C₄×D₄	1	4	yes	complemented
155	C₄×C₁₁₃	#121	1	4	yes	complemented
156	Dic₁₁₃	—	2	2	no	—
157	Dic₁₁₃	—	2	2	no	—
158	Dic₁₁₃	—	2	2	no	—
159	Dic₁₁₃	—	4	2	no	—
160	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
163	C₂×C₁₁₃	C₂×#121	1	1	yes	characteristic
164	C₂×C₁₁₃	#525	1	1	yes	characteristic
165	C₂×C₁₁₃	#529	1	1	yes	characteristic
166	C₂×C₁₁₃	#532	1	1	yes	characteristic
167	D₁₁₃	—	2	4	no	—
168	D₁₁₃	—	4	4	no	—
169	C₄×#121	—	113	1	no	maximal, Sylow
170	C₁₁₃	C₄×#121	1	1	yes	characteristic, complemented, Sylow
171	C₂×#121	—	113	1	no	—
172	C₂×C₄×D₄	—	113	2	no	—
173	C₂×C₄×D₄	—	113	1	no	—
174	C₂²×C₄²	—	113	1	no	—
175	#390	—	113	1	no	—
176	#389	—	113	1	no	—
177	#388	—	113	2	no	—
178	#386	—	113	2	no	—
179	#380	—	113	1	no	—
180	C₄×C₄⋊C₄	D₁₁₃	1	1	yes	characteristic, complemented
181	C₄×C₂²⋊C₄	—	113	2	no	—
182	C₂²×D₄	—	113	2	no	—
183	C₂²×D₄	—	113	1	no	—
184	C₂³×C₄	—	113	1	no	—
185	C₂³×C₄	—	113	1	no	—
186	C₂³×C₄	—	113	1	no	—
187	C₂³×C₄	—	113	2	no	—
188	#121	—	113	8	no	—
189	C₄×D₄	—	226	8	no	—
190	C₄×D₄	—	226	4	no	—
191	C₂×C₄⋊C₄	—	113	1	no	—
192	C₂×C₄⋊C₄	—	113	2	no	—
193	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	113	2	no	—
196	C₂×C₂²⋊C₄	—	113	2	no	—
197	C₂×C₂²⋊C₄	—	113	2	no	—
198	C₂×C₂²⋊C₄	—	113	2	no	—
199	C₂×C₂²⋊C₄	—	113	2	no	—
200	C₂×C₄²	—	113	1	no	—
201	C₂×C₄²	—	113	1	no	—
202	C₂×C₄²	—	113	1	no	—
203	C₂×C₄²	—	226	4	no	—
204	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
205	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
206	#95	—	113	2	no	—
207	#95	—	113	2	no	—
208	#95	C₂×D₁₁₃	1	2	yes	—
209	C₂⁴	—	113	1	no	—
210	C₂⁴	—	113	2	no	—
211	C₂×D₄	—	113	8	no	—
212	C₂×D₄	—	113	4	no	—
213	C₂×D₄	—	226	4	no	—
214	C₂×D₄	—	226	4	no	—
215	C₂×D₄	—	226	2	no	—
216	C₂×D₄	—	226	2	no	—
217	C₂²×C₄	—	113	1	no	—
218	C₂²×C₄	—	113	1	no	—
219	C₂²×C₄	—	113	1	no	—
220	C₂²×C₄	—	113	4	no	—
221	C₂²×C₄	—	113	4	no	—
222	C₂²×C₄	—	113	2	no	—
223	C₂²×C₄	—	226	2	no	—
224	C₂²×C₄	—	226	2	no	—
225	C₂²×C₄	—	226	4	no	—
226	C₂²×C₄	—	226	4	no	—
227	C₂²×C₄	—	226	4	no	—
228	C₂²×C₄	—	226	2	no	—
229	C₂²×C₄	—	226	2	no	—
230	C₂²×C₄	—	226	2	no	—
231	C₂²×C₄	—	226	2	no	—
232	C₂²×C₄	—	226	2	no	—
233	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
235	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	226	4	no	—
240	C₄⋊C₄	—	226	2	no	—
241	C₄⋊C₄	C₄×D₁₁₃	1	4	yes	complemented
242	C₂²⋊C₄	—	113	8	no	—
243	C₂²⋊C₄	—	226	4	no	—
244	C₂²⋊C₄	—	226	4	no	—
245	C₂²⋊C₄	—	226	4	no	—
246	C₂²⋊C₄	—	226	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	226	2	no	—
249	C₄²	—	226	2	no	—
250	C₄²	—	226	2	no	—
251	C₄²	D₄₅₂	1	4	yes	complemented
252	C₂³	—	113	2	no	—
253	C₂³	—	113	2	no	—
254	C₂³	—	113	2	no	—
255	C₂³	—	113	4	no	—
256	C₂³	—	226	2	no	—
257	C₂³	—	226	2	no	—
258	C₂³	—	226	2	no	—
259	C₂³	—	226	2	no	—
260	C₂³	—	226	2	no	—
261	C₂³	—	226	2	no	—
262	C₂³	—	226	1	no	—
263	C₂³	—	226	1	no	—
264	C₂³	—	226	1	no	—
265	C₂³	—	226	1	no	—
266	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
267	D₄	—	226	16	no	—
268	D₄	—	226	8	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	113	2	no	—
277	C₂×C₄	—	113	2	no	—
278	C₂×C₄	—	113	2	no	—
279	C₂×C₄	—	452	8	no	—
280	C₂×C₄	—	226	8	no	—
281	C₂×C₄	—	226	2	no	—
282	C₂×C₄	—	226	2	no	—
283	C₂×C₄	—	226	2	no	—
284	C₂×C₄	—	226	1	no	—
285	C₂×C₄	—	226	1	no	—
286	C₂×C₄	—	226	1	no	—
287	C₂×C₄	—	226	1	no	—
288	C₂×C₄	—	226	1	no	—
289	C₂×C₄	—	226	1	no	—
290	C₂×C₄	—	226	8	no	—
291	C₂×C₄	—	226	8	no	—
292	C₂×C₄	C₂×C₄×D₁₁₃	1	4	yes	—
293	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
294	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
295	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
296	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
297	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
298	C₂×C₄	#14268647	1	2	yes	—
299	C₂×C₄	#14268648	1	2	yes	—
300	C₂²	—	113	4	no	—
301	C₂²	—	452	2	no	—
302	C₂²	—	452	2	no	—
303	C₂²	—	452	2	no	—
304	C₂²	—	452	2	no	—
305	C₂²	—	226	4	no	—
306	C₂²	—	226	4	no	—
307	C₂²	—	226	4	no	—
308	C₂²	—	226	2	no	—
309	C₂²	—	226	2	no	—
310	C₂²	—	226	2	no	—
311	C₂²	—	226	2	no	—
312	C₂²	—	226	2	no	—
313	C₂²	—	226	2	no	—
314	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
315	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
316	C₂²	C₂²×D₄₅₂	1	1	yes	characteristic
317	C₂²	#41×D₁₁₃	1	1	yes	characteristic
318	C₂²	#41×D₁₁₃	1	1	yes	characteristic
319	C₂²	C₂×#14268647	1	1	yes	characteristic
320	C₂²	C₂×#14268648	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	452	2	no	—
325	C₄	—	226	2	no	—
326	C₄	—	226	2	no	—
327	C₄	—	226	2	no	—
328	C₄	C₄×D₄₅₂	1	4	yes	complemented
329	C₄	#16813597	1	4	yes	direct product factor
330	C₂	—	452	4	no	—
331	C₂	—	226	4	no	—
332	C₂	C₄×D₄×D₁₁₃	1	1	yes	characteristic
333	C₂	C₂×C₄×D₄₅₂	1	1	yes	characteristic
334	C₂	C₄×#14268648	1	1	yes	characteristic
335	C₂	C₂×#16813597	1	1	yes	characteristic
336	C₂	#20299110	1	1	yes	characteristic
337	C₂	#20299135	1	1	yes	characteristic
338	C₂	#20299145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂²×C₁₁₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₁₃
Fitting	C₄×C₄⋊C₄×C₁₁₃	C₂
socle, cosocle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂×#16813597	C₂
hypercenter	C₄×C₄⋊C₄	D₁₁₃
hypocenter	C₁₁₃	C₄×#121
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₅₂	C₂	semidirect
3. C₂×C₄×D₄₅₂	C₂	semidirect
4. C₂×C₄²×D₁₁₃	C₂	semidirect
5. #20298499	C₂	semidirect
6. #20298497	C₂	semidirect
7. #20298490	C₂	semidirect
8. #20298476	C₂	semidirect
9. #20298455	C₂	semidirect
10. C₄×#14268624	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₁₃	C₂	semidirect
18. #16813597	C₄	direct (N × H)
45. C₂×D₄₅₂	C₂×C₄	semidirect
46. C₂×D₄₅₂	C₂×C₄	semidirect
61. C₂×C₄×D₁₁₃	C₂×C₄	semidirect
62. C₂×C₄×D₁₁₃	D₄	semidirect
69. #14268624	C₂×C₄	semidirect
79. C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
81. C₄²×C₁₁₃	D₄	semidirect
150. C₂×D₁₁₃	C₄×D₄	semidirect
154. C₄×C₁₁₃	C₄×D₄	semidirect
155. C₄×C₁₁₃	#121	semidirect
170. C₁₁₃	C₄×#121	semidirect
180. C₄×C₄⋊C₄	D₁₁₃	semidirect
241. C₄⋊C₄	C₄×D₁₁₃	semidirect
251. C₄²	D₄₅₂	semidirect
328. C₄	C₄×D₄₅₂	semidirect
329. C₄	#16813597	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×#16813597	2
C₂×C₄×D₄₅₂	2
C₂×C₄²×D₁₁₃	2
#20298499	2
#20298497	2
#20298490	2
#20298476	2
#20298455	2
C₄×#14268624	2
C₄×C₄⋊C₄×C₁₁₃	2
C₄×#121	113

Subgroups — C4×#16813597 #28393089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16813597 #28393089