Subgroups — C₄×#11718151 #15768129

Order 3328.

906 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
2	C₄×#88180	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718151	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88180	C₂²	1	2	yes	—
8	C₂²×#6099	C₂²	1	2	yes	—
9	C₄×C₁₃×M₁₆	—	2	2	no	—
10	C₄×C₁₃×M₁₆	C₄	1	4	yes	complemented
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₄×C₂²⋊Dic₁₃	C₄	1	4	yes	—
14	C₄×C₄⋊Dic₁₃	—	2	2	no	—
15	#11718593	—	2	1	no	—
16	C₄²×Dic₁₃	—	2	1	no	—
17	C₄×#6099	—	2	1	no	—
18	C₄×#6099	—	2	1	no	—
19	C₄×#6099	—	2	1	no	—
20	C₄×#6099	—	2	1	no	—
21	C₂×#88043	C₄	1	2	yes	complemented
22	C₂×C₄×#1055	C₄	1	2	yes	complemented
23	#11718151	C₄	1	16	yes	direct product factor
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
40	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	#88180	—	2	4	no	—
42	#88180	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	4	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	4	no	—
49	C₂×#6099	C₂×C₄	1	8	yes	—
50	C₂²×#1055	C₂×C₄	1	4	yes	—
51	C₁₃×#97	—	2	4	no	—
52	C₄×C₈×C₁₃	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88043	—	2	2	no	—
55	#88043	—	2	2	no	—
56	C₄×#1055	—	2	2	no	—
57	C₄×#1055	—	2	2	no	—
58	C₄×#344	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	#41	1	4	yes	—
75	C₂²×Dic₁₃	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	2	8	no	—
78	C₁₃×M₁₆	C₄²	1	8	yes	complemented
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	8	no	—
81	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
91	C₂²⋊Dic₁₃	—	2	4	no	—
92	C₂²⋊Dic₁₃	C₄²	1	8	yes	—
93	C₄⋊Dic₁₃	—	2	8	no	—
94	C₄×Dic₁₃	—	2	8	no	—
95	C₄×Dic₁₃	—	4	8	no	—
96	#6099	—	2	8	no	—
97	#6099	—	2	8	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	C₄²	1	8	yes	complemented
101	C₂×C₄×M₁₆	—	13	1	no	—
102	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
103	C₄×#117	—	13	1	no	—
104	C₂×#344	—	13	4	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	#95	1	2	yes	—
118	C₂×C₄×C₁₃	#95	1	2	yes	—
119	C₂×C₄×C₁₃	#95	1	2	yes	—
120	C₂×C₄×C₁₃	#95	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	2	yes	—
128	C₂×C₄×C₁₃	#117	1	2	yes	—
129	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
133	C₂×Dic₁₃	—	2	8	no	—
134	C₂×Dic₁₃	—	4	8	no	—
135	C₈×C₁₃	—	2	8	no	—
136	#1055	—	2	8	no	—
137	C₂²×M₁₆	—	13	2	no	—
138	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
139	C₂×#117	—	13	2	no	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	C₄×M₁₆	—	2	2	no	—
142	C₄×M₁₆	—	26	1	no	—
143	C₄×M₁₆	—	26	1	no	—
144	C₄×M₁₆	—	26	1	no	—
145	C₄×M₁₆	—	26	1	no	—
146	C₄×M₁₆	Dic₁₃	1	4	yes	complemented
147	C₂×#97	—	2	1	no	—
148	C₂×#97	—	13	2	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₄×C₄⋊C₄	—	26	2	no	—
152	C₄×C₂²⋊C₄	—	13	4	no	—
153	#376	—	26	1	no	—
154	C₄³	—	26	1	no	—
155	#344	—	13	16	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	C₄³	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
163	C₂²×C₁₃	#376	1	1	yes	characteristic
164	C₂²×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
165	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	#344	1	4	yes	complemented
170	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
171	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
172	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
173	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
174	Dic₁₃	—	4	8	no	—
175	C₂³×C₄	C₄×D₁₃	1	2	yes	—
176	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	13	8	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	4	no	—
182	C₂×M₁₆	—	26	2	no	—
183	C₂×M₁₆	C₄×D₁₃	1	8	yes	—
184	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	13	4	no	—
187	#117	—	13	8	no	—
188	#117	—	26	4	no	—
189	C₂×C₄⋊C₄	—	26	2	no	—
190	C₂×C₂²⋊C₄	—	13	4	no	—
191	C₂×C₄²	—	26	2	no	—
192	C₂×C₄²	—	26	4	no	—
193	C₂×C₄²	Dic₂₆	1	2	yes	—
194	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
196	C₂×C₄²	D₅₂	1	1	yes	characteristic
197	C₂×C₄²	D₅₂	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
202	#97	—	2	4	no	—
203	#97	—	26	2	no	—
204	#97	—	26	2	no	—
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	—	26	2	no	—
207	C₄×C₈	—	26	2	no	—
208	#95	—	26	4	no	—
209	C₂×C₁₃	—	2	2	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#344	1	2	yes	—
215	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	26	4	no	—
218	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
219	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
220	C₂²×C₄	#6101	1	2	yes	—
221	C₂²×C₄	#6101	1	2	yes	—
222	C₂²×C₄	#6103	1	2	yes	—
223	C₂²×C₄	#6103	1	2	yes	—
224	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
228	C₂²×C₄	#6127	1	4	yes	—
229	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	8	no	—
234	M₁₆	—	26	8	no	—
235	M₁₆	—	26	8	no	—
236	M₁₆	C₄×Dic₁₃	1	8	yes	complemented
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	13	8	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	—	26	4	no	—
242	C₄⋊C₄	—	26	8	no	—
243	C₂²⋊C₄	—	13	8	no	—
244	C₂²⋊C₄	—	26	4	no	—
245	C₄²	—	26	8	no	—
246	C₄²	—	52	8	no	—
247	C₄²	#6101	1	2	yes	—
248	C₄²	#6101	1	2	yes	—
249	C₄²	C₄⋊Dic₁₃	1	2	yes	—
250	C₄²	C₄⋊Dic₁₃	1	2	yes	—
251	C₄²	#6103	1	1	yes	characteristic
252	C₄²	#6103	1	1	yes	characteristic
253	C₄²	#6103	1	1	yes	characteristic
254	C₄²	#6103	1	1	yes	characteristic
255	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
256	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
257	C₁₃	C₄×#344	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	4	2	no	—
259	C₂³	C₄²×D₁₃	1	2	yes	—
260	C₂³	#88126	1	2	yes	—
261	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
262	C₂³	C₂×#6101	1	1	yes	characteristic
263	C₂³	C₂×#6103	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	26	8	no	—
271	C₂×C₄	—	52	8	no	—
272	C₂×C₄	#88071	1	2	yes	—
273	C₂×C₄	#88071	1	2	yes	—
274	C₂×C₄	#88071	1	2	yes	—
275	C₂×C₄	#88071	1	2	yes	—
276	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
277	C₂×C₄	C₄²×D₁₃	1	2	yes	—
278	C₂×C₄	#88126	1	2	yes	—
279	C₂×C₄	C₄×D₅₂	1	4	yes	—
280	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
281	C₂×C₄	C₂×#6101	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
283	C₂×C₄	#88180	1	2	yes	—
284	C₂×C₄	C₂×#6103	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
286	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
287	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
288	C₈	—	2	8	no	—
289	C₈	—	26	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₄²×Dic₁₃	1	1	yes	characteristic
296	C₂²	C₄×#6101	1	2	yes	—
297	C₂²	#11718593	1	1	yes	characteristic
298	C₂²	C₄×#6103	1	2	yes	—
299	C₂²	C₂×#88071	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	52	8	no	—
304	C₄	#11718151	1	4	yes	direct product factor
305	C₄	C₄×#6101	1	2	yes	—
306	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
307	C₄	C₄×#6103	1	2	yes	—
308	C₄	C₄×C₂²⋊Dic₁₃	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	2	no	—
312	C₂	—	2	1	no	—
313	C₂	C₄×#88071	1	1	yes	characteristic
314	C₂	C₂×#11718151	1	2	yes	—

	subgroup	G/H
center	C₄²	#6103
commutator	C₂²×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₄×#344
LCS	G ▷ C₂²×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
10. C₄×C₁₃×M₁₆	C₄	semidirect
21. C₂×#88043	C₄	semidirect
22. C₂×C₄×#1055	C₄	semidirect
23. #11718151	C₄	direct (N × H)
78. C₁₃×M₁₆	C₄²	semidirect
100. C₂×#1055	C₄²	semidirect
146. C₄×M₁₆	Dic₁₃	semidirect
169. C₄×C₁₃	#344	semidirect
236. M₁₆	C₄×Dic₁₃	semidirect
257. C₁₃	C₄×#344	semidirect
304. C₄	#11718151	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×C₄×#6099	2
C₄×#88180	2
C₂×#11718151	2
C₂×C₄×C₁₃×M₁₆	2
C₄×#344	13

Subgroups — C4×#11718151 #15768129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718151 #15768129