Subgroups — C₄×#11718673 #15861302

Order 3328.

1254 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718673	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	2	yes	maximal, complemented
3	#13114705	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114620	C₂	1	4	yes	maximal, complemented
6	C₄×C₂²⋊#1055	C₂	1	2	yes	maximal, complemented
7	C₄²×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	4	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
16	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
19	#11718673	C₄	1	16	yes	direct product factor
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₄×#6099	—	2	8	no	—
22	C₂×#88043	C₂²	1	4	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	2	yes	—
26	C₂×C₄×#1055	C₂²	1	2	yes	—
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718116	C₂²	1	2	yes	—
29	#11718116	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
33	C₄×D₄×C₁₃	—	2	8	no	—
34	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	4	no	—
40	C₂×C₄²×C₁₃	—	2	4	no	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	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	8	no	—
49	C₂×#6099	—	2	8	no	—
50	C₂×#6099	C₂×C₄	1	16	yes	complemented
51	C₂²×#1055	C₂³	1	4	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#95	C₂×C₄	1	2	yes	—
55	C₁₃×#95	C₂×C₄	1	4	yes	—
56	C₂²⋊#1055	C₂×C₄	1	16	yes	complemented
57	C₄⋊#1055	C₂×C₄	1	8	yes	complemented
58	#88043	—	2	8	no	—
59	C₄×#1055	—	2	4	no	—
60	C₄×#1055	—	2	2	no	—
61	C₄×#1055	—	2	4	no	—
62	C₄×#1055	C₂×C₄	1	8	yes	complemented
63	C₄×#1055	D₄	1	4	yes	complemented
64	C₄×#436	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×D₄×C₁₃	—	2	4	no	—
67	C₂×D₄×C₁₃	C₄²	1	4	yes	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
84	C₁₃×C₂²⋊C₄	—	2	8	no	—
85	C₁₃×C₂²⋊C₄	C₄²	1	8	yes	—
86	C₄²×C₁₃	—	2	8	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	M₁₆	1	8	yes	complemented
92	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₁₃	C₂×D₄	1	2	yes	—
94	C₄²×C₁₃	#41	1	2	yes	—
95	#6099	—	2	32	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	C₂²×C₄	1	4	yes	—
101	C₂×#1055	C₂²×C₄	1	8	yes	—
102	C₂×#1055	C₂²×C₄	1	4	yes	—
103	C₂×#1055	C₂×D₄	1	4	yes	—
104	C₂×#1055	#41	1	4	yes	—
105	C₂×#436	—	13	2	no	—
106	C₂×C₄×M₁₆	—	13	2	no	—
107	C₄²×D₄	D₁₃	1	1	yes	characteristic
108	#1770	—	13	2	no	—
109	C₄×C₄⋊C₈	—	13	1	no	—
110	#1753	—	13	4	no	—
111	C₄×C₂²⋊C₈	—	13	2	no	—
112	C₄²×C₈	—	13	1	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	—	2	4	no	—
117	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
118	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	D₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	16	no	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
133	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
136	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₁₃	#131	1	4	yes	—
139	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	#1055	—	2	8	no	—
144	#1055	—	2	4	no	—
145	#1055	C₄×D₄	1	8	yes	complemented
146	C₂²×M₁₆	—	13	4	no	—
147	C₂×C₄×D₄	Dic₁₃	1	2	yes	—
148	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
149	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
150	#436	—	13	16	no	—
151	C₂×C₄⋊C₈	—	13	2	no	—
152	C₂×C₂²⋊C₈	—	13	4	no	—
153	C₄×M₁₆	—	26	8	no	—
154	C₂×#97	—	13	4	no	—
155	C₂×C₄×C₈	—	13	2	no	—
156	C₂×C₄×C₈	—	13	1	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	13	2	no	—
159	C₂×C₄×C₈	—	13	1	no	—
160	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
161	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
162	C₄×C₂²⋊C₄	Dic₁₃	1	4	yes	—
163	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
164	C₄³	C₂×D₁₃	1	1	yes	characteristic
165	#336	—	13	4	no	—
166	#336	—	13	2	no	—
167	C₂²×C₁₃	—	2	4	no	—
168	C₂²×C₁₃	—	2	4	no	—
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	4	no	—
172	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
175	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
176	C₂²×C₁₃	C₂×#131	1	2	yes	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	—	2	4	no	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	C₄×M₁₆	1	4	yes	complemented
182	C₄×C₁₃	#436	1	8	yes	complemented
183	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
184	C₄×C₁₃	C₄×#41	1	2	yes	—
185	C₂²×D₄	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
187	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
188	C₂×M₁₆	—	13	16	no	—
189	C₂×M₁₆	—	26	8	no	—
190	C₂×M₁₆	—	26	8	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₄×D₄	—	2	8	no	—
195	C₄×D₄	C₄×D₁₃	1	8	yes	—
196	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
203	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
204	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
205	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	13	8	no	—
210	C₂²⋊C₈	—	13	16	no	—
211	#97	—	26	8	no	—
212	C₄×C₈	—	13	8	no	—
213	C₄×C₈	—	13	4	no	—
214	C₄×C₈	—	26	4	no	—
215	C₄×C₈	—	26	4	no	—
216	C₄×C₈	—	26	2	no	—
217	#95	C₂×Dic₁₃	1	2	yes	—
218	#95	C₂×Dic₁₃	1	4	yes	—
219	C₂×C₁₃	—	4	4	no	—
220	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
221	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
222	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#436	1	2	yes	—
224	C₂×C₁₃	#2965	1	2	yes	—
225	C₂⁴	C₂²×Dic₁₃	1	2	yes	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	C₄×Dic₁₃	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
235	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
236	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
241	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
242	M₁₆	—	26	32	no	—
243	C₂×C₈	—	13	4	no	—
244	C₂×C₈	—	13	4	no	—
245	C₂×C₈	—	13	4	no	—
246	C₂×C₈	—	13	4	no	—
247	C₂×C₈	—	13	8	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	—	26	2	no	—
251	C₂×C₈	—	26	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
254	C₂²⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	C₄×Dic₁₃	1	8	yes	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	#6099	1	8	yes	complemented
262	C₄²	C₂×C₄×D₁₃	1	4	yes	—
263	C₄²	D₄×D₁₃	1	2	yes	—
264	C₄²	#6129	1	2	yes	—
265	C₁₃	C₄×#436	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	2	4	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₂×C₄×Dic₁₃	1	4	yes	—
271	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
272	D₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	C₂×#6099	1	4	yes	—
285	C₂×C₄	C₂×#6099	1	4	yes	—
286	C₂×C₄	C₂×#6099	1	4	yes	—
287	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
289	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
290	C₂×C₄	D₄×Dic₁₃	1	4	yes	—
291	C₂×C₄	#88202	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×#6129	1	1	yes	characteristic
295	C₂×C₄	#41×D₁₃	1	2	yes	—
296	C₈	—	13	8	no	—
297	C₈	—	26	4	no	—
298	C₈	—	26	8	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	C₂²×#6099	1	2	yes	—
305	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
306	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
307	C₂²	C₂×#88202	1	2	yes	—
308	C₂²	#41×Dic₁₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	4	no	—
313	C₄	C₄×#6099	1	4	yes	complemented
314	C₄	#11718673	1	8	yes	direct product factor
315	C₄	C₄×#6129	1	2	yes	—
316	C₄	C₄×D₄×D₁₃	1	2	yes	—
317	C₂	—	4	4	no	—
318	C₂	C₄×D₄×Dic₁₃	1	1	yes	characteristic
319	C₂	C₂×C₄×#6099	1	1	yes	characteristic
320	C₂	C₄×#88202	1	1	yes	characteristic
321	C₂	C₂×#11718673	1	2	yes	—
322	C₂	#13123803	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×D₄	D₁₃
hypocenter	C₁₃	C₄×#436
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#6099	C₂	semidirect
3. #13114705	C₂	semidirect
4. C₄×C₄⋊#1055	C₂	semidirect
5. #13114620	C₂	semidirect
6. C₄×C₂²⋊#1055	C₂	semidirect
7. C₄²×#1055	C₂	semidirect
19. #11718673	C₄	direct (N × H)
50. C₂×#6099	C₂×C₄	semidirect
56. C₂²⋊#1055	C₂×C₄	semidirect
57. C₄⋊#1055	C₂×C₄	semidirect
62. C₄×#1055	C₂×C₄	semidirect
63. C₄×#1055	D₄	semidirect
91. C₄²×C₁₃	M₁₆	semidirect
145. #1055	C₄×D₄	semidirect
181. C₄×C₁₃	C₄×M₁₆	semidirect
182. C₄×C₁₃	#436	semidirect
261. C₄²	#6099	semidirect
265. C₁₃	C₄×#436	semidirect
313. C₄	C₄×#6099	semidirect
314. C₄	#11718673	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×#11718673	2
C₂×C₄×#6099	2
#13114705	2
C₄×C₄⋊#1055	2
#13114620	2
C₄×C₂²⋊#1055	2
C₄²×#1055	2
C₄²×D₄×C₁₃	2
C₄×#436	13

Subgroups — C4×#11718673 #15861302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718673 #15861302