Subgroups — C₄×#11718785 #15861027

Order 3328.

1034 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718785	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
4	#13114743	C₂	1	4	yes	maximal, complemented
5	C₄×#88062	C₂	1	4	yes	maximal, complemented
6	#13114562	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₂²×#6099	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×M₁₆	—	2	1	no	—
14	C₄×C₁₃×M₁₆	—	2	1	no	—
15	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	—	2	1	no	—
19	#11718785	C₄	1	16	yes	direct product factor
20	C₂×#88062	C₂²	1	8	yes	—
21	C₄×#6099	—	2	1	no	—
22	C₄×#6099	—	2	1	no	—
23	C₄×#6099	C₄	1	4	yes	—
24	C₄×#6099	C₄	1	4	yes	—
25	C₄×#6099	C₂²	1	2	yes	—
26	C₄×#6099	C₂²	1	2	yes	—
27	C₂×#88043	—	2	1	no	—
28	C₂×#88043	C₂²	1	2	yes	—
29	C₂×C₄×#1055	—	2	1	no	—
30	C₂×C₄×#1055	C₂²	1	2	yes	—
31	#11718150	C₂²	1	8	yes	—
32	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	—	2	4	no	—
36	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	4	no	—
50	C₂×#6099	—	2	4	no	—
51	C₂×#6099	C₂×C₄	1	8	yes	—
52	C₂×#6099	C₂×C₄	1	2	yes	—
53	C₂×#6099	C₂×C₄	1	2	yes	—
54	C₂×#6099	C₂×C₄	1	8	yes	—
55	C₂×#6099	C₂×C₄	1	2	yes	—
56	C₂×#6099	C₂×C₄	1	2	yes	—
57	C₂×#6099	C₂³	1	4	yes	—
58	C₂²×#1055	—	2	2	no	—
59	C₂²×#1055	C₂³	1	2	yes	—
60	C₂²×#1055	C₂³	1	2	yes	—
61	C₁₃×#97	—	2	2	no	—
62	C₁₃×#97	—	2	2	no	—
63	C₄×C₈×C₁₃	—	2	2	no	—
64	C₄×C₈×C₁₃	—	2	2	no	—
65	#88062	C₂×C₄	1	32	yes	complemented
66	#88043	—	2	2	no	—
67	#88043	—	2	2	no	—
68	#88043	C₂×C₄	1	8	yes	—
69	C₄×#1055	—	2	2	no	—
70	C₄×#1055	—	2	2	no	—
71	C₄×#1055	C₂×C₄	1	8	yes	—
72	C₄×#430	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	—	4	4	no	—
75	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	#41	1	4	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×M₁₆	—	2	8	no	—
90	C₁₃×M₁₆	—	2	4	no	—
91	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
92	C₂×C₈×C₁₃	—	2	8	no	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
96	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
97	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	#6099	—	2	4	no	—
105	#6099	—	2	4	no	—
106	#6099	—	2	8	no	—
107	#6099	—	2	4	no	—
108	#6099	C₄²	1	8	yes	—
109	#6099	C₄²	1	8	yes	—
110	#6099	C₂²×C₄	1	8	yes	—
111	C₂×#1055	—	2	8	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	C₂²×C₄	1	4	yes	—
115	C₂×#1055	C₂²×C₄	1	4	yes	—
116	C₂×#1055	C₂²×C₄	1	8	yes	—
117	C₂×#430	—	13	2	no	—
118	C₂×C₄×M₁₆	—	13	1	no	—
119	C₂×C₄×M₁₆	—	13	1	no	—
120	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
121	#1778	—	13	4	no	—
122	C₄×#108	—	13	4	no	—
123	#1738	—	13	2	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
128	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₁₃	#117	1	2	yes	—
144	C₂×C₄×C₁₃	#117	1	2	yes	—
145	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
146	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
147	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
152	C₈×C₁₃	—	2	8	no	—
153	#1055	—	2	4	no	—
154	#1055	—	2	8	no	—
155	#1055	C₂×C₄²	1	8	yes	—
156	C₂²×M₁₆	—	13	1	no	—
157	C₂²×M₁₆	—	13	1	no	—
158	C₂²×M₁₆	—	13	2	no	—
159	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
160	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
161	#430	—	13	16	no	—
162	C₂×#108	—	13	8	no	—
163	C₄×M₁₆	—	2	1	no	—
164	C₄×M₁₆	—	2	1	no	—
165	C₄×M₁₆	—	13	4	no	—
166	C₄×M₁₆	—	13	4	no	—
167	C₄×M₁₆	—	13	2	no	—
168	C₄×M₁₆	—	13	2	no	—
169	C₄×M₁₆	—	26	1	no	—
170	C₄×M₁₆	—	26	1	no	—
171	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
172	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
173	C₂×#97	—	2	1	no	—
174	C₂×#97	—	13	2	no	—
175	C₂×#97	—	26	1	no	—
176	C₂×C₄×C₈	—	2	1	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	C₂×C₄×C₈	—	26	1	no	—
179	#341	—	13	8	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	4	4	no	—
184	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
185	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
186	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
192	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
193	C₄×C₁₃	#430	1	4	yes	complemented
194	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
195	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
196	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
197	C₂³×C₄	Dic₂₆	1	2	yes	—
198	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	13	2	no	—
202	C₂×M₁₆	—	13	2	no	—
203	C₂×M₁₆	—	13	8	no	—
204	C₂×M₁₆	—	13	2	no	—
205	C₂×M₁₆	—	13	2	no	—
206	C₂×M₁₆	—	13	8	no	—
207	C₂×M₁₆	—	13	4	no	—
208	C₂×M₁₆	—	26	4	no	—
209	C₂×M₁₆	—	26	4	no	—
210	C₂×M₁₆	—	26	2	no	—
211	C₂×M₁₆	C₄×D₁₃	1	8	yes	—
212	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	13	2	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	26	2	no	—
217	C₂×C₄²	Dic₂₆	1	2	yes	—
218	C₂×C₄²	C₄×D₁₃	1	2	yes	—
219	C₂×C₄²	C₄×D₁₃	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
226	#108	—	13	32	no	—
227	#97	—	2	2	no	—
228	#97	—	2	2	no	—
229	#97	—	13	8	no	—
230	#97	—	26	2	no	—
231	#97	—	26	2	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	—	2	2	no	—
234	C₄×C₈	—	13	8	no	—
235	C₄×C₈	—	26	2	no	—
236	C₄×C₈	—	26	2	no	—
237	C₂×C₁₃	—	2	2	no	—
238	C₂×C₁₃	—	2	2	no	—
239	C₂×C₁₃	—	2	2	no	—
240	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#430	1	2	yes	—
242	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	#6101	1	2	yes	—
245	C₂²×C₄	#6101	1	2	yes	—
246	C₂²×C₄	#6101	1	2	yes	—
247	C₂²×C₄	#6101	1	2	yes	—
248	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
249	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
250	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
252	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	#6127	1	4	yes	—
255	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
258	M₁₆	—	2	8	no	—
259	M₁₆	—	2	4	no	—
260	M₁₆	—	13	8	no	—
261	M₁₆	—	13	8	no	—
262	M₁₆	—	13	8	no	—
263	M₁₆	—	26	4	no	—
264	M₁₆	—	26	4	no	—
265	M₁₆	—	26	8	no	—
266	M₁₆	—	26	4	no	—
267	M₁₆	C₂×C₄×D₁₃	1	8	yes	—
268	C₂×C₈	—	2	8	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	13	4	no	—
271	C₂×C₈	—	13	8	no	—
272	C₂×C₈	—	13	4	no	—
273	C₂×C₈	—	26	8	no	—
274	C₂×C₈	—	26	4	no	—
275	C₂×C₈	—	26	4	no	—
276	C₄²	#6101	1	2	yes	—
277	C₄²	#6101	1	2	yes	—
278	C₄²	#6101	1	2	yes	—
279	C₄²	#6101	1	2	yes	—
280	C₄²	C₂×C₄×D₁₃	1	2	yes	—
281	C₄²	C₂×C₄×D₁₃	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
286	C₁₃	C₄×#430	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	4	2	no	—
288	C₂³	C₄×Dic₂₆	1	4	yes	—
289	C₂³	C₂×#6101	1	1	yes	characteristic
290	C₂³	C₂×#6101	1	1	yes	characteristic
291	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
299	C₂×C₄	C₄²×D₁₃	1	2	yes	—
300	C₂×C₄	C₄²×D₁₃	1	2	yes	—
301	C₂×C₄	#88126	1	2	yes	—
302	C₂×C₄	#88126	1	2	yes	—
303	C₂×C₄	C₂×#6101	1	2	yes	—
304	C₂×C₄	C₂×#6101	1	2	yes	—
305	C₂×C₄	C₂×#6101	1	2	yes	—
306	C₂×C₄	C₂×#6101	1	2	yes	—
307	C₂×C₄	C₂×#6101	1	1	yes	characteristic
308	C₂×C₄	C₂×#6101	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
310	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
311	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
313	C₂×C₄	C₂×#6127	1	2	yes	—
314	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
315	C₈	—	2	8	no	—
316	C₈	—	13	8	no	—
317	C₈	—	26	4	no	—
318	C₈	—	26	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	4	no	—
323	C₂²	C₄×#6101	1	2	yes	—
324	C₂²	C₄×#6101	1	2	yes	—
325	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
326	C₂²	C₂²×#6101	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	C₄×#6101	1	2	yes	—
331	C₄	C₄×#6101	1	2	yes	—
332	C₄	#11718785	1	4	yes	direct product factor
333	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
334	C₄	C₂×#88126	1	1	yes	characteristic
335	C₄	C₂×C₄×C₂²⋊D₁₃	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	C₂×C₄×#6101	1	1	yes	characteristic
340	C₂	C₂×#11718785	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₄×#430
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13114743	C₂	semidirect
5. C₄×#88062	C₂	semidirect
19. #11718785	C₄	direct (N × H)
65. #88062	C₂×C₄	semidirect
193. C₄×C₁₃	#430	semidirect
286. C₁₃	C₄×#430	semidirect
332. C₄	#11718785	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×#11718785	2
C₂×C₄×#6099	2
#13114743	2
C₄×#88062	2
#13114562	2
C₂×C₄×C₁₃×M₁₆	2
C₄×#430	13

Subgroups — C4×#11718785 #15861027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718785 #15861027