Subgroups — C₄×#11718077 #15753860

Order 3328.

708 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88128	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718077	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88128	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂×C₄×D₅₂	—	2	1	no	—
9	C₂×C₄×D₅₂	C₄	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	#11718600	—	2	1	no	—
14	#11718600	—	2	1	no	—
15	#11718600	C₄	1	2	yes	—
16	#11718596	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	C₂×C₄×#1055	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718077	C₄	1	8	yes	direct product factor
22	C₂²×D₅₂	—	2	1	no	—
23	C₂²×D₅₂	—	2	1	no	—
24	C₂²×D₅₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6103	—	2	2	no	—
35	C₂×#6103	—	2	2	no	—
36	C₂×#6103	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂²×#1055	—	2	2	no	—
41	#88128	—	2	2	no	—
42	#88128	C₂×C₄	1	4	yes	—
43	C₄×D₅₂	—	4	4	no	—
44	C₄×D₅₂	—	4	2	no	—
45	C₄×D₅₂	—	4	2	no	—
46	C₄×D₅₂	—	4	2	no	—
47	C₄×D₅₂	—	4	2	no	—
48	C₁₃×C₄⋊C₈	—	2	2	no	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
50	C₄×C₈×C₁₃	—	4	2	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	C₂×C₄	1	4	yes	complemented
53	C₄×#1055	—	4	2	no	—
54	C₄×#331	—	13	1	no	maximal, Sylow
55	C₂³×D₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	2	no	—
64	C₂×D₅₂	—	2	2	no	—
65	C₂×D₅₂	—	2	4	no	—
66	C₂×D₅₂	—	2	4	no	—
67	C₂×D₅₂	—	4	2	no	—
68	C₂×D₅₂	—	4	2	no	—
69	C₂×D₅₂	—	4	1	no	—
70	C₂×D₅₂	—	4	1	no	—
71	C₂×D₅₂	—	4	1	no	—
72	C₂×D₅₂	—	4	1	no	—
73	C₂×D₅₂	—	4	2	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×D₅₂	C₄²	1	4	yes	—
76	C₂×C₄×D₁₃	—	4	4	no	—
77	C₂×C₄×D₁₃	—	4	4	no	—
78	C₂×C₄×D₁₃	—	4	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	C₄²×C₁₃	D₈	1	2	yes	—
87	C₄²×C₁₃	D₈	1	2	yes	—
88	C₄²×C₁₃	SD₁₆	1	2	yes	—
89	C₄²×C₁₃	SD₁₆	1	2	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₃	#41	1	2	yes	—
92	#6103	—	4	4	no	—
93	#6103	—	4	4	no	—
94	#6103	—	4	4	no	—
95	C₄⋊Dic₁₃	—	4	1	no	—
96	C₄⋊Dic₁₃	—	4	1	no	—
97	C₄⋊Dic₁₃	—	4	1	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	C₄⋊Dic₁₃	—	4	2	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×#1055	—	4	2	no	—
103	C₄×#127	—	13	1	no	—
104	C₄×C₄⋊C₈	—	13	1	no	—
105	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
106	C₂×#331	—	13	2	no	—
107	#1296	—	13	2	no	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	4	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	—	4	2	no	—
112	C₂²×D₁₃	—	4	2	no	—
113	C₂²×D₁₃	—	4	2	no	—
114	C₂²×D₁₃	—	4	2	no	—
115	C₂²×D₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	#100	1	2	yes	—
126	C₂×C₄×C₁₃	#102	1	2	yes	—
127	C₂×C₄×C₁₃	#102	1	2	yes	—
128	C₂×C₄×C₁₃	#102	1	2	yes	—
129	C₂×C₄×C₁₃	#102	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#135	1	2	yes	—
139	C₂×C₄×C₁₃	#135	1	2	yes	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	D₅₂	—	4	4	no	—
144	D₅₂	—	4	4	no	—
145	D₅₂	—	4	4	no	—
146	D₅₂	—	4	4	no	—
147	D₅₂	—	4	4	no	—
148	D₅₂	—	4	4	no	—
149	C₄×D₁₃	—	8	8	no	—
150	C₈×C₁₃	—	4	4	no	—
151	#1055	—	4	4	no	—
152	C₂×#127	—	13	1	no	—
153	C₂×C₄×D₄	—	13	2	no	—
154	C₂×C₄×D₄	—	26	1	no	—
155	C₂×C₄×D₄	—	26	1	no	—
156	C₂×C₄⋊C₈	—	13	2	no	—
157	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
158	C₂×C₄×C₈	—	2	1	no	—
159	C₂×C₄×C₈	—	26	1	no	—
160	#390	—	13	2	no	—
161	#390	—	26	1	no	—
162	#390	—	26	1	no	—
163	#384	—	13	1	no	—
164	C₄³	C₂×D₁₃	1	1	yes	characteristic
165	#336	—	2	1	no	—
166	#336	—	26	1	no	—
167	#331	—	13	8	no	—
168	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
170	C₂²×C₁₃	#413	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
173	C₂²×C₁₃	#416	1	1	yes	characteristic
174	C₂²×C₁₃	#416	1	1	yes	characteristic
175	C₂×D₁₃	—	4	4	no	—
176	C₂×D₁₃	—	4	4	no	—
177	C₂×D₁₃	—	4	4	no	—
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	2	no	—
181	C₂×D₁₃	—	8	2	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	4	1	no	—
187	C₄×C₁₃	#331	1	4	yes	complemented
188	C₄×C₁₃	C₄×D₈	1	2	yes	—
189	C₄×C₁₃	C₄×D₈	1	2	yes	—
190	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
191	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
192	Dic₁₃	—	8	2	no	—
193	C₂²×D₄	—	13	2	no	—
194	C₂²×D₄	—	26	1	no	—
195	C₂²×D₄	—	26	1	no	—
196	C₂³×C₄	—	26	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	26	2	no	—
199	#127	—	13	4	no	—
200	#127	—	26	2	no	—
201	C₄×D₄	—	52	4	no	—
202	C₄×D₄	—	52	2	no	—
203	C₄×D₄	—	52	2	no	—
204	C₄×D₄	—	52	2	no	—
205	C₄×D₄	—	52	2	no	—
206	C₂×C₄⋊C₄	—	13	2	no	—
207	C₂×C₄⋊C₄	—	26	1	no	—
208	C₂×C₄⋊C₄	—	26	1	no	—
209	C₂×C₂²⋊C₄	—	26	2	no	—
210	C₂×C₂²⋊C₄	—	26	2	no	—
211	C₂×C₂²⋊C₄	—	26	2	no	—
212	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄²	D₅₂	1	1	yes	characteristic
215	C₂×C₄²	D₅₂	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	13	4	no	—
221	C₄⋊C₈	—	26	2	no	—
222	C₄⋊C₈	C₄×D₁₃	1	4	yes	complemented
223	C₄×C₈	—	4	2	no	—
224	C₄×C₈	—	52	2	no	—
225	C₂×C₁₃	C₂×#331	1	1	yes	characteristic
226	C₂×C₁₃	#1540	1	1	yes	characteristic
227	C₂×C₁₃	#1540	1	1	yes	characteristic
228	C₂×C₁₃	#1543	1	1	yes	characteristic
229	C₂×C₁₃	C₄×#100	1	1	yes	characteristic
230	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
231	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
232	D₁₃	—	8	4	no	—
233	C₂⁴	—	26	2	no	—
234	C₂×D₄	—	13	4	no	—
235	C₂×D₄	—	26	2	no	—
236	C₂×D₄	—	26	4	no	—
237	C₂×D₄	—	26	4	no	—
238	C₂×D₄	—	52	2	no	—
239	C₂×D₄	—	52	2	no	—
240	C₂×D₄	—	52	1	no	—
241	C₂×D₄	—	52	1	no	—
242	C₂×D₄	—	52	2	no	—
243	C₂×D₄	—	52	1	no	—
244	C₂×D₄	—	52	1	no	—
245	C₂×D₄	—	52	2	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	—	52	4	no	—
248	C₂²×C₄	—	52	4	no	—
249	C₂²×C₄	—	52	4	no	—
250	C₂²×C₄	#6103	1	1	yes	characteristic
251	C₂²×C₄	#6103	1	1	yes	characteristic
252	C₂²×C₄	#6103	1	1	yes	characteristic
253	C₂²×C₄	#6103	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	52	2	no	—
262	C₂×C₈	—	52	2	no	—
263	C₄⋊C₄	—	52	2	no	—
264	C₄⋊C₄	—	52	1	no	—
265	C₄⋊C₄	—	52	1	no	—
266	C₄⋊C₄	—	52	1	no	—
267	C₄⋊C₄	—	52	1	no	—
268	C₂²⋊C₄	—	52	4	no	—
269	C₂²⋊C₄	—	52	4	no	—
270	C₂²⋊C₄	—	52	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	#6095	1	2	yes	—
276	C₄²	D₁₀₄	1	2	yes	—
277	C₄²	#6104	1	2	yes	—
278	C₄²	#6106	1	2	yes	—
279	C₄²	C₂×C₄×D₁₃	1	2	yes	—
280	C₄²	#6127	1	2	yes	—
281	C₁₃	C₄×#331	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	26	4	no	—
283	C₂³	—	52	2	no	—
284	C₂³	—	52	2	no	—
285	C₂³	—	52	2	no	—
286	C₂³	—	52	2	no	—
287	C₂³	—	52	2	no	—
288	C₂³	—	52	2	no	—
289	C₂³	C₂×#6103	1	1	yes	characteristic
290	D₄	—	52	4	no	—
291	D₄	—	52	4	no	—
292	D₄	—	52	4	no	—
293	D₄	—	52	4	no	—
294	D₄	—	52	4	no	—
295	D₄	—	52	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	104	8	no	—
309	C₂×C₄	#88049	1	2	yes	—
310	C₂×C₄	#88049	1	2	yes	—
311	C₂×C₄	#88061	1	2	yes	—
312	C₂×C₄	#88061	1	2	yes	—
313	C₂×C₄	#88063	1	2	yes	—
314	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
315	C₂×C₄	#88126	1	1	yes	characteristic
316	C₂×C₄	C₄×D₅₂	1	2	yes	—
317	C₂×C₄	C₂×#6095	1	1	yes	characteristic
318	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
319	C₂×C₄	#88158	1	2	yes	—
320	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
321	C₂×C₄	C₂×#6104	1	1	yes	characteristic
322	C₂×C₄	C₂×#6106	1	1	yes	characteristic
323	C₂×C₄	#88204	1	2	yes	—
324	C₈	—	4	4	no	—
325	C₈	—	52	4	no	—
326	C₂²	—	52	4	no	—
327	C₂²	—	52	4	no	—
328	C₂²	—	52	4	no	—
329	C₂²	—	104	2	no	—
330	C₂²	—	104	2	no	—
331	C₂²	—	104	2	no	—
332	C₂²	—	104	2	no	—
333	C₂²	C₄×#6103	1	1	yes	characteristic
334	C₂²	C₂×#88049	1	1	yes	characteristic
335	C₂²	#11718663	1	1	yes	characteristic
336	C₂²	C₂×#88061	1	1	yes	characteristic
337	C₂²	#11718774	1	1	yes	characteristic
338	C₂²	C₂×#88063	1	1	yes	characteristic
339	C₂²	#11718793	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	104	2	no	—
346	C₄	#11718077	1	4	yes	direct product factor
347	C₄	C₄×#6095	1	2	yes	—
348	C₄	C₄×D₁₀₄	1	2	yes	—
349	C₄	C₄×#6104	1	2	yes	—
350	C₄	C₄×#6106	1	2	yes	—
351	C₂	—	104	4	no	—
352	C₂	C₂×#11718077	1	1	yes	characteristic
353	C₂	#13113483	1	1	yes	characteristic
354	C₂	#13113486	1	1	yes	characteristic
355	C₂	#13113496	1	1	yes	characteristic
356	C₂	C₄×#88063	1	1	yes	characteristic
357	C₂	C₄×#88061	1	1	yes	characteristic
358	C₂	C₄×#88049	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	C₄×#331
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊#1055	C₂	semidirect
4. #13112720	C₂	semidirect
5. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
21. #11718077	C₄	direct (N × H)
49. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
52. C₄⋊#1055	C₂×C₄	semidirect
105. C₄×C₄⋊C₈	D₁₃	semidirect
187. C₄×C₁₃	#331	semidirect
222. C₄⋊C₈	C₄×D₁₃	semidirect
281. C₁₃	C₄×#331	semidirect
346. C₄	#11718077	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#88128	2
C₄×C₄⋊#1055	2
C₂×#11718077	2
#13112720	2
C₄×C₁₃×C₄⋊C₈	2
C₄×#331	13

Subgroups — C4×#11718077 #15753860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718077 #15753860