Subgroups — C₄×#11719089 #16796733

Order 3328.

1526 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719089	C₂	1	4	yes	maximal
2	C₄×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#88237	C₂	1	1	yes	maximal, characteristic
4	#13130492	C₂	1	4	yes	maximal, complemented
5	C₄×#88100	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×#261	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#88237	C₂²	1	2	yes	—
9	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
11	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊#261	C₄	1	4	yes	—
13	C₄×#6122	C₄	1	4	yes	complemented
14	#11719089	C₄	1	32	yes	direct product factor
15	C₂×#88100	C₂²	1	8	yes	—
16	C₂×C₈×#261	C₂²	1	8	yes	—
17	C₄×C₄⋊#261	C₄	1	4	yes	—
18	#11719057	C₂²	1	2	yes	—
19	C₄²×#261	C₂²	1	2	yes	—
20	C₄×#6118	C₄	1	4	yes	complemented
21	#11719047	C₂²	1	2	yes	—
22	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
24	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
25	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
26	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic, complemented
27	C₄×C₈×D₁₃	C₄	1	2	yes	complemented
28	C₄×C₈×D₁₃	C₄	1	1	yes	characteristic, complemented
29	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₃	C₂²	1	2	yes	—
32	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊#261	C₂×C₄	1	4	yes	—
35	C₂×#6122	C₂×C₄	1	4	yes	—
36	#88237	C₂×C₄	1	16	yes	—
37	C₂×C₄⋊#261	C₂×C₄	1	4	yes	—
38	C₂×C₄×#261	C₂×C₄	1	8	yes	—
39	C₂×C₄×#261	C₂³	1	4	yes	—
40	D₁₃⋊M₁₆	C₂×C₄	1	16	yes	—
41	C₂×#6118	C₂×C₄	1	4	yes	—
42	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
43	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₁₃	C₂×C₄	1	16	yes	—
51	C₂×C₈×D₁₃	C₂³	1	2	yes	—
52	C₂×C₈×D₁₃	C₂³	1	2	yes	—
53	C₄²×D₁₃	C₂×C₄	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₁₃	C₂×C₄	1	2	yes	—
57	C₄²×D₁₃	C₂³	1	1	yes	characteristic
58	C₄²×D₁₃	C₂³	1	1	yes	characteristic
59	(C₂×#261)⋊C₄	C₂×C₄	1	8	yes	—
60	#88110	C₂×C₄	1	8	yes	—
61	C₄×#1057	C₂×C₄	1	8	yes	—
62	#88100	C₂×C₄	1	32	yes	complemented
63	C₈×#261	C₂×C₄	1	32	yes	complemented
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
71	C₄×#1055	C₂×C₄	1	1	yes	characteristic
72	C₄×#1055	C₂×C₄	1	1	yes	characteristic
73	C₄×#1055	C₂×C₄	1	2	yes	—
74	C₄×#405	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×#261	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₄²	1	2	yes	—
80	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₃	—	2	8	no	—
82	C₂×C₄×D₁₃	C₄²	1	2	yes	—
83	C₂×C₄×D₁₃	C₄²	1	2	yes	—
84	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
89	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²⋊#261	C₄²	1	16	yes	—
92	#6122	C₄²	1	16	yes	complemented
93	C₂×#1057	C₂²×C₄	1	8	yes	—
94	C₄⋊#261	C₄²	1	16	yes	—
95	C₄×#261	—	2	16	no	—
96	C₄×#261	C₂²×C₄	1	16	yes	—
97	#6118	C₄²	1	16	yes	complemented
98	D₁₃⋊C₈	C₂²×C₄	1	16	yes	—
99	C₂×C₈×C₁₃	C₄²	1	8	yes	complemented
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	C₄²	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₁₃	C₄²	1	2	yes	—
106	C₄×Dic₁₃	C₄²	1	2	yes	—
107	C₄×Dic₁₃	C₄²	1	2	yes	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
111	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
112	C₂×#1055	C₄²	1	8	yes	complemented
113	C₂×#1055	C₂²×C₄	1	2	yes	—
114	C₂×#1055	C₂²×C₄	1	2	yes	—
115	C₈×D₁₃	C₄²	1	8	yes	complemented
116	C₈×D₁₃	C₄²	1	8	yes	complemented
117	C₈×D₁₃	C₂²×C₄	1	8	yes	—
118	C₈×D₁₃	C₂²×C₄	1	8	yes	—
119	C₂×#405	—	13	4	no	—
120	C₂×C₄×M₁₆	—	13	1	no	—
121	C₂²×C₄×C₈	—	13	1	no	—
122	C₄×#117	—	13	1	no	—
123	#1726	—	13	4	no	—
124	C₄×#97	—	13	2	no	—
125	C₄²×C₈	—	13	2	no	—
126	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
127	C₂²×D₁₃	—	2	4	no	—
128	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
130	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×#261	—	2	16	no	—
132	C₂×#261	C₂×C₄²	1	16	yes	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
137	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	4	no	—
140	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
141	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
143	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
144	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	—	2	4	no	—
147	C₄×D₁₃	—	2	4	no	—
148	C₄×D₁₃	C₂×C₄²	1	4	yes	—
149	C₄×D₁₃	C₂×C₄²	1	4	yes	—
150	C₄×D₁₃	C₂×C₄²	1	4	yes	—
151	C₄×D₁₃	#131	1	8	yes	—
152	C₄×D₁₃	C₂³×C₄	1	4	yes	—
153	#1057	C₂×C₄²	1	16	yes	—
154	C₈×C₁₃	C₂×C₄²	1	4	yes	—
155	C₈×C₁₃	C₂×C₄²	1	4	yes	—
156	#1055	C₂×C₄²	1	4	yes	—
157	#1055	C₂×C₄²	1	4	yes	—
158	C₂²×M₁₆	—	13	2	no	—
159	C₂³×C₈	—	13	2	no	—
160	C₂×#117	—	13	2	no	—
161	C₂²×C₄²	—	13	1	no	—
162	#405	—	13	32	no	—
163	C₄×M₁₆	—	13	4	no	—
164	C₄×M₁₆	—	13	4	no	—
165	C₂×#97	—	13	2	no	—
166	C₂×#97	—	13	8	no	—
167	C₂×C₄×C₈	—	13	2	no	—
168	C₂×C₄×C₈	—	13	8	no	—
169	C₂×C₄×C₈	—	13	2	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	C₂×C₄×C₈	—	13	1	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	C₂×C₄×C₈	—	13	1	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
178	C₄×C₄⋊C₄	—	13	4	no	—
179	C₄×C₂²⋊C₄	—	13	4	no	—
180	#376	—	13	2	no	—
181	C₄³	—	13	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	C₄³	1	2	yes	—
184	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
185	C₂×D₁₃	—	2	2	no	—
186	C₂×D₁₃	—	2	8	no	—
187	C₂×D₁₃	C₄³	1	2	yes	—
188	C₂×D₁₃	C₂²×C₄²	1	2	yes	—
189	C₂×D₁₃	C₂×#131	1	4	yes	—
190	#261	—	2	16	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	C₄³	1	2	yes	—
193	C₄×C₁₃	#405	1	4	yes	complemented
194	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
195	Dic₁₃	—	2	2	no	—
196	Dic₁₃	C₄³	1	2	yes	—
197	Dic₁₃	#405	1	4	yes	complemented
198	Dic₁₃	C₂²×C₄²	1	2	yes	—
199	C₂³×C₄	—	13	1	no	—
200	C₂³×C₄	—	13	2	no	—
201	C₂×M₁₆	—	13	4	no	—
202	C₂×M₁₆	—	13	4	no	—
203	C₂×M₁₆	—	13	16	no	—
204	C₂²×C₈	—	13	4	no	—
205	C₂²×C₈	—	13	2	no	—
206	C₂²×C₈	—	13	16	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	13	2	no	—
211	C₂²×C₈	C₂×#261	1	2	yes	—
212	#117	—	13	16	no	—
213	C₂×C₄⋊C₄	—	13	4	no	—
214	C₂×C₂²⋊C₄	—	13	4	no	—
215	C₂×C₄²	—	13	4	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	1	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	13	2	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	8	no	—
225	C₂×C₄²	C₂×#261	1	1	yes	characteristic
226	#97	—	13	32	no	—
227	#97	—	13	8	no	—
228	C₄×C₈	—	13	32	no	—
229	C₄×C₈	—	13	8	no	—
230	C₄×C₈	—	13	2	no	—
231	C₄×C₈	—	13	2	no	—
232	C₄×C₈	—	13	2	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	2	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	C₂×#261	1	2	yes	—
238	C₄×C₈	C₂×#261	1	1	yes	characteristic
239	C₄×C₈	C₂×#261	1	1	yes	characteristic
240	#95	—	13	8	no	—
241	C₂×C₁₃	—	2	2	no	—
242	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
243	C₂×C₁₃	C₂×#405	1	2	yes	—
244	D₁₃	—	2	2	no	—
245	D₁₃	C₄×#131	1	4	yes	—
246	C₂⁴	—	13	1	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	8	no	—
252	C₂²×C₄	—	13	1	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	8	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	2	no	—
259	C₂²×C₄	—	26	8	no	—
260	C₂²×C₄	C₄×#261	1	2	yes	—
261	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
262	M₁₆	—	13	16	no	—
263	M₁₆	—	13	16	no	—
264	C₂×C₈	—	13	16	no	—
265	C₂×C₈	—	13	8	no	—
266	C₂×C₈	—	13	8	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	8	no	—
270	C₂×C₈	—	13	8	no	—
271	C₂×C₈	—	13	8	no	—
272	C₂×C₈	—	13	8	no	—
273	C₂×C₈	C₄×#261	1	8	yes	complemented
274	C₂×C₈	C₂²×#261	1	2	yes	—
275	C₂×C₈	C₂²×#261	1	2	yes	—
276	C₄⋊C₄	—	13	16	no	—
277	C₂²⋊C₄	—	13	16	no	—
278	C₄²	—	13	16	no	—
279	C₄²	—	13	2	no	—
280	C₄²	—	13	2	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	13	2	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	13	1	no	—
285	C₄²	—	13	1	no	—
286	C₄²	—	26	16	no	—
287	C₄²	C₄×#261	1	2	yes	—
288	C₄²	C₂²×#261	1	1	yes	characteristic
289	C₄²	C₂²×#261	1	1	yes	characteristic
290	C₁₃	C₄×#405	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	13	1	no	—
292	C₂³	—	13	4	no	—
293	C₂³	—	13	1	no	—
294	C₂³	—	26	4	no	—
295	C₂³	C₂×C₄×#261	1	1	yes	characteristic
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	13	4	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	13	4	no	—
303	C₂×C₄	—	13	8	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	16	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	4	no	—
311	C₂×C₄	—	26	16	no	—
312	C₂×C₄	—	26	4	no	—
313	C₂×C₄	C₂×C₄×#261	1	2	yes	—
314	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×#261	1	4	yes	—
316	C₂×C₄	C₂×C₄×#261	1	2	yes	—
317	C₂×C₄	C₂³×#261	1	1	yes	characteristic
318	C₈	—	13	16	no	—
319	C₈	—	13	4	no	—
320	C₈	—	13	4	no	—
321	C₈	C₂×C₄×#261	1	4	yes	—
322	C₈	C₂×C₄×#261	1	4	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	13	4	no	—
325	C₂²	—	13	2	no	—
326	C₂²	—	13	2	no	—
327	C₂²	—	26	8	no	—
328	C₂²	—	26	2	no	—
329	C₂²	C₄²×#261	1	2	yes	—
330	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	13	4	no	—
333	C₄	—	13	2	no	—
334	C₄	—	13	2	no	—
335	C₄	—	26	16	no	—
336	C₄	—	26	2	no	—
337	C₄	C₄²×#261	1	2	yes	—
338	C₄	#11719089	1	4	yes	direct product factor
339	C₄	C₂²×C₄×#261	1	2	yes	—
340	C₂	—	2	2	no	—
341	C₂	—	13	4	no	—
342	C₂	—	26	2	no	—
343	C₂	C₂×C₄²×#261	1	1	yes	characteristic
344	C₂	C₂×#11719089	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₄×#405
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #13130492	C₂	semidirect
5. C₄×#88100	C₂	semidirect
6. C₄×C₈×#261	C₂	semidirect
13. C₄×#6122	C₄	semidirect
14. #11719089	C₄	direct (N × H)
20. C₄×#6118	C₄	semidirect
23. C₂×C₄×C₈×C₁₃	C₄	semidirect
24. C₂×C₈×Dic₁₃	C₄	semidirect
25. C₂×C₄×#1055	C₄	semidirect
26. C₄×C₈×D₁₃	C₄	semidirect
27. C₄×C₈×D₁₃	C₄	semidirect
28. C₄×C₈×D₁₃	C₄	semidirect
62. #88100	C₂×C₄	semidirect
63. C₈×#261	C₂×C₄	semidirect
92. #6122	C₄²	semidirect
97. #6118	C₄²	semidirect
99. C₂×C₈×C₁₃	C₄²	semidirect
112. C₂×#1055	C₄²	semidirect
115. C₈×D₁₃	C₄²	semidirect
116. C₈×D₁₃	C₄²	semidirect
177. C₂×C₄×C₈	#261	semidirect
193. C₄×C₁₃	#405	semidirect
197. Dic₁₃	#405	semidirect
273. C₂×C₈	C₄×#261	semidirect
290. C₁₃	C₄×#405	semidirect
338. C₄	#11719089	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×#11719089	2
C₄×D₁₃⋊M₁₆	2
C₄×#88237	2
#13130492	2
C₄×#88100	2
C₄×C₈×#261	2
C₂×C₄×C₈×D₁₃	2
C₄×#405	13

Subgroups — C4×#11719089 #16796733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719089 #16796733