Subgroups — C₄×#11718057 #15753838

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88124	C₂	1	1	yes	maximal, characteristic
2	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718057	C₂	1	2	yes	maximal
4	#13112718	C₂	1	2	yes	maximal
5	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×#88124	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#97	—	2	1	no	—
8	C₂×C₁₃×#97	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	1	no	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718597	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
14	#11718594	—	2	1	no	—
15	#11718594	—	2	1	no	—
16	C₄×#6101	—	2	1	no	—
17	C₄×#6101	—	2	1	no	—
18	#11718589	C₄	1	1	yes	characteristic
19	#11718589	C₄	1	1	yes	characteristic
20	C₂×#88043	—	2	1	no	—
21	C₂×#88043	C₂²	1	2	yes	—
22	C₂×C₄×#1055	—	2	1	no	—
23	#11718057	C₄	1	8	yes	direct product factor
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	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	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×#6101	—	2	1	no	—
39	C₂×#6101	—	2	1	no	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂×C₄×Dic₁₃	—	2	1	no	—
42	C₂²×#1055	—	2	2	no	—
43	#88124	—	2	2	no	—
44	#88124	C₂×C₄	1	4	yes	—
45	C₁₃×#97	—	2	4	no	—
46	C₁₃×#97	—	2	2	no	—
47	C₁₃×#97	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₃	—	2	4	no	—
49	#88071	—	2	1	no	—
50	#88071	—	2	1	no	—
51	#88071	—	2	1	no	—
52	#88071	—	2	1	no	—
53	#88071	—	2	1	no	—
54	#88071	—	2	1	no	—
55	#88043	—	2	4	no	—
56	#88043	—	2	2	no	—
57	#88043	C₂×C₄	1	4	yes	—
58	C₄×#1055	—	2	4	no	—
59	C₄×#330	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	1	no	—
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂²×Dic₁₃	—	2	1	no	—
70	C₂²×Dic₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	4	1	no	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	#41	1	2	yes	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	1	no	—
84	C₄⋊Dic₁₃	—	4	1	no	—
85	C₄⋊Dic₁₃	—	4	1	no	—
86	C₄⋊Dic₁₃	C₄²	1	2	yes	—
87	C₄⋊Dic₁₃	C₄²	1	2	yes	—
88	#6101	—	2	4	no	—
89	#6101	—	2	4	no	—
90	#6101	—	4	2	no	—
91	#6101	—	4	2	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	C₄×Dic₁₃	—	4	2	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₄×#125	—	13	1	no	—
98	C₄×#97	—	13	1	no	—
99	C₄×#97	D₁₃	1	1	yes	characteristic
100	C₂×#330	—	13	2	no	—
101	#1295	—	13	2	no	—
102	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	#101	1	2	yes	—
117	C₂×C₄×C₁₃	#104	1	2	yes	—
118	C₂×C₄×C₁₃	#104	1	2	yes	—
119	C₂×C₄×C₁₃	#104	1	2	yes	—
120	C₂×C₄×C₁₃	#104	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	1	no	—
129	C₂×Dic₁₃	—	4	1	no	—
130	C₂×Dic₁₃	—	4	1	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	1	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	C₈×C₁₃	—	2	8	no	—
138	#1055	—	2	8	no	—
139	C₂×#125	—	13	1	no	—
140	C₂×#97	—	2	1	no	—
141	C₂×#97	—	13	2	no	—
142	C₂×#97	—	26	1	no	—
143	C₂×#97	C₂×D₁₃	1	2	yes	—
144	C₂×C₄×C₈	—	2	1	no	—
145	C₂×C₄×C₈	—	26	1	no	—
146	#389	—	13	1	no	—
147	#389	—	13	1	no	—
148	#387	—	26	1	no	—
149	#387	—	26	1	no	—
150	#382	—	13	1	no	—
151	C₄×C₄⋊C₄	—	13	1	no	—
152	C₄×C₄⋊C₄	—	13	1	no	—
153	C₄×C₄⋊C₄	—	26	1	no	—
154	C₄×C₄⋊C₄	—	26	1	no	—
155	C₄³	C₂×D₁₃	1	1	yes	characteristic
156	#330	—	13	8	no	—
157	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
159	C₂²×C₁₃	#413	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	#330	1	4	yes	complemented
172	Dic₁₃	—	4	2	no	—
173	Dic₁₃	—	4	2	no	—
174	Dic₁₃	—	8	1	no	—
175	Dic₁₃	—	8	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	26	2	no	—
178	#125	—	13	4	no	—
179	#125	—	26	2	no	—
180	C₂×C₄⋊C₄	—	13	1	no	—
181	C₂×C₄⋊C₄	—	13	1	no	—
182	C₂×C₄⋊C₄	—	13	1	no	—
183	C₂×C₄⋊C₄	—	13	1	no	—
184	C₂×C₄⋊C₄	—	26	1	no	—
185	C₂×C₄⋊C₄	—	26	1	no	—
186	C₂×C₄⋊C₄	—	26	1	no	—
187	C₂×C₄⋊C₄	—	26	1	no	—
188	C₂×C₄²	—	26	1	no	—
189	C₂×C₄²	—	26	1	no	—
190	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
192	C₂×C₄²	D₅₂	1	1	yes	characteristic
193	C₂×C₄²	D₅₂	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	#97	—	2	4	no	—
198	#97	—	2	2	no	—
199	#97	—	13	4	no	—
200	#97	—	26	4	no	—
201	#97	—	26	2	no	—
202	#97	C₄×D₁₃	1	4	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	—	26	4	no	—
205	#95	—	26	1	no	—
206	#95	—	26	1	no	—
207	#95	—	26	1	no	—
208	#95	—	26	1	no	—
209	#95	—	26	1	no	—
210	#95	—	26	1	no	—
211	C₂×C₁₃	C₂×#330	1	1	yes	characteristic
212	C₂×C₁₃	#1524	1	1	yes	characteristic
213	C₂×C₁₃	#1524	1	1	yes	characteristic
214	C₂×C₁₃	#1526	1	1	yes	characteristic
215	C₂×C₁₃	C₄×#101	1	1	yes	characteristic
216	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
217	C₂×C₁₃	C₄×#104	1	1	yes	characteristic
218	C₂²×C₄	—	26	1	no	—
219	C₂²×C₄	—	26	1	no	—
220	C₂²×C₄	—	26	1	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	#6103	1	1	yes	characteristic
223	C₂²×C₄	#6103	1	1	yes	characteristic
224	C₂²×C₄	#6103	1	1	yes	characteristic
225	C₂²×C₄	#6103	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	—	26	4	no	—
241	C₄⋊C₄	—	52	1	no	—
242	C₄⋊C₄	—	52	1	no	—
243	C₄⋊C₄	—	52	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	52	2	no	—
252	C₄²	—	52	2	no	—
253	C₄²	C₂×C₄×D₁₃	1	2	yes	—
254	C₄²	#6127	1	2	yes	—
255	C₁₃	C₄×#330	1	1	yes	characteristic, complemented, Sylow
256	C₂³	C₂×#6103	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	26	2	no	—
272	C₂×C₄	—	52	1	no	—
273	C₂×C₄	—	52	1	no	—
274	C₂×C₄	—	52	1	no	—
275	C₂×C₄	—	52	1	no	—
276	C₂×C₄	—	52	1	no	—
277	C₂×C₄	—	52	1	no	—
278	C₂×C₄	—	52	1	no	—
279	C₂×C₄	—	52	1	no	—
280	C₂×C₄	—	52	1	no	—
281	C₂×C₄	—	52	1	no	—
282	C₂×C₄	#88045	1	2	yes	—
283	C₂×C₄	#88045	1	2	yes	—
284	C₂×C₄	#88064	1	2	yes	—
285	C₂×C₄	#88065	1	2	yes	—
286	C₂×C₄	#88065	1	2	yes	—
287	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
288	C₂×C₄	#88126	1	1	yes	characteristic
289	C₂×C₄	C₄×D₅₂	1	2	yes	—
290	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
291	C₈	—	2	8	no	—
292	C₈	—	26	8	no	—
293	C₂²	C₂×#88045	1	1	yes	characteristic
294	C₂²	C₂×#88045	1	1	yes	characteristic
295	C₂²	C₄×#6103	1	1	yes	characteristic
296	C₂²	#11718793	1	1	yes	characteristic
297	C₂²	C₂×#88064	1	1	yes	characteristic
298	C₂²	C₂×#88065	1	1	yes	characteristic
299	C₂²	C₂×#88065	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	1	no	—
307	C₄	—	52	2	no	—
308	C₄	—	52	2	no	—
309	C₄	—	104	1	no	—
310	C₄	—	104	1	no	—
311	C₄	#11718057	1	4	yes	direct product factor
312	C₂	C₂×#11718057	1	1	yes	characteristic
313	C₂	#13113422	1	1	yes	characteristic
314	C₂	#13113425	1	1	yes	characteristic
315	C₂	#13113430	1	1	yes	characteristic
316	C₂	C₄×#88064	1	1	yes	characteristic
317	C₂	C₄×#88045	1	1	yes	characteristic
318	C₂	C₄×#88065	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×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	C₄×#330
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
23. #11718057	C₄	direct (N × H)
171. C₄×C₁₃	#330	semidirect
255. C₁₃	C₄×#330	semidirect
311. C₄	#11718057	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×#88124	2
C₄×#88043	2
C₂×#11718057	2
#13112718	2
C₄×C₁₃×#97	2
C₄×#330	13

Subgroups — C4×#11718057 #15753838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718057 #15753838