Subgroups — C₄×#11718153 #15777967

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718153	C₂	1	2	yes	maximal
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112925	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
9	C₁₃×#405	—	2	1	no	—
10	C₁₃×#405	—	2	1	no	—
11	C₁₃×#405	C₄	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×M₁₆	—	2	1	no	—
14	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88052	C₂²	1	2	yes	—
18	C₂²×#6090	C₂²	1	2	yes	—
19	#11718153	C₄	1	8	yes	direct product factor
20	#11718053	—	2	1	no	—
21	C₄×#6090	—	2	1	no	—
22	C₄×#6090	—	2	1	no	—
23	C₄×#6090	—	2	1	no	—
24	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	—	2	2	no	—
27	C₁₃×#131	C₈	1	4	yes	—
28	C₁₃×#131	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₈	1	2	yes	—
38	C₄×D₄×C₁₃	—	2	2	no	—
39	C₄×D₄×C₁₃	C₈	1	2	yes	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	—	2	1	no	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₁₃×#97	—	2	1	no	—
45	C₁₃×#97	—	2	1	no	—
46	C₁₃×#97	—	2	1	no	—
47	C₄×C₈×C₁₃	—	2	1	no	—
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	#88052	—	2	2	no	—
55	#88052	C₂×C₄	1	4	yes	complemented
56	C₂×#6090	—	2	2	no	—
57	C₂×#6090	—	2	2	no	—
58	C₂×#6090	—	2	2	no	—
59	C₂×#6090	C₂×C₄	1	4	yes	complemented
60	C₄×#350	—	13	1	no	maximal, Sylow
61	C₁₃×#41	—	2	1	no	—
62	C₁₃×#41	—	2	1	no	—
63	C₁₃×#41	C₄²	1	2	yes	—
64	C₁₃×#41	C₂×C₈	1	2	yes	—
65	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₁₃×M₁₆	—	2	1	no	—
73	C₁₃×M₁₆	—	2	1	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	C₄²	1	2	yes	—
77	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₁₃×C₂²⋊C₄	—	2	1	no	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	1	no	—
96	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×C₈	1	2	yes	—
99	#6090	—	2	4	no	—
100	#6090	—	2	4	no	—
101	C₄×#131	D₁₃	1	1	yes	characteristic
102	C₂×#350	—	13	2	no	—
103	C₄×M₃₂	—	13	1	no	—
104	C₂×C₄×C₁₆	—	13	1	no	—
105	#1379	—	13	2	no	—
106	C₂³×C₁₃	—	2	1	no	—
107	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	Q₈×C₁₃	—	2	1	no	—
109	Q₈×C₁₃	C₄×C₈	1	2	yes	—
110	D₄×C₁₃	—	2	2	no	—
111	D₄×C₁₃	—	2	2	no	—
112	D₄×C₁₃	—	2	1	no	—
113	D₄×C₁₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
131	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
132	C₈×C₁₃	C₄×D₄	1	2	yes	—
133	C₈×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×#131	Dic₁₃	1	1	yes	characteristic
135	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
136	C₄×#41	Dic₁₃	1	1	yes	characteristic
137	C₂×M₃₂	—	13	2	no	—
138	C₂²×C₁₆	—	13	2	no	—
139	#405	—	2	1	no	—
140	#405	—	2	1	no	—
141	#405	Dic₁₃	1	1	yes	characteristic
142	#405	C₂×D₁₃	1	1	yes	characteristic
143	C₄×M₁₆	—	2	1	no	—
144	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
145	C₂×C₄×C₈	—	2	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#350	—	13	8	no	—
148	#346	—	26	1	no	—
149	C₄×C₁₆	—	26	1	no	—
150	C₄×C₁₆	—	26	1	no	—
151	C₄×C₁₆	—	26	1	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₁₃	#431	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	#350	1	4	yes	complemented
162	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₁₃	#431	1	1	yes	characteristic
166	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
167	#131	—	2	2	no	—
168	#131	—	2	2	no	—
169	#131	#1055	1	4	yes	—
170	#131	C₄×D₁₃	1	4	yes	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₄×Q₈	#1055	1	2	yes	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	#1055	1	2	yes	—
182	#117	—	2	1	no	—
183	#117	C₂×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
186	M₃₂	—	13	4	no	—
187	M₃₂	—	26	2	no	—
188	C₂×C₁₆	—	13	4	no	—
189	C₂×C₁₆	—	26	2	no	—
190	C₂×C₁₆	—	26	2	no	—
191	C₂×C₁₆	—	26	2	no	—
192	#97	—	2	1	no	—
193	#97	—	2	1	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	—	4	2	no	—
205	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#350	1	2	yes	—
207	#41	—	2	1	no	—
208	#41	—	2	1	no	—
209	#41	C₂×#1055	1	2	yes	—
210	#41	C₄×Dic₁₃	1	2	yes	—
211	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₂×#1055	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	#6099	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	C₂×#1055	1	2	yes	—
223	M₁₆	C₄×Dic₁₃	1	2	yes	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	C₂×#1055	1	2	yes	—
229	C₂×C₈	#6099	1	2	yes	—
230	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
233	C₂×C₈	#6127	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂×#1055	1	2	yes	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	C₂×#1055	1	2	yes	—
243	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
244	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
245	C₁₆	—	26	4	no	—
246	C₁₆	—	26	4	no	—
247	C₁₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#6099	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#1055	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#1055	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊#1055	1	2	yes	—
263	C₂×C₄	C₂²⋊#1055	1	2	yes	—
264	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
265	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
266	C₂×C₄	C₂×#6099	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
268	C₂×C₄	#88180	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂²⋊#1055	1	2	yes	—
273	C₈	C₂²⋊#1055	1	2	yes	—
274	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
275	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#6099	1	1	yes	characteristic
280	C₂²	#11718586	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#11718153	1	4	yes	direct product factor
286	C₄	C₂×C₄×#1055	1	1	yes	characteristic
287	C₄	#11718586	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
289	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#1055	1	1	yes	characteristic
294	C₂	C₂×#11718153	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	C₄×#350
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#88052	C₂	semidirect
3. C₂×C₄×#6090	C₂	semidirect
4. #13112925	C₂	semidirect
19. #11718153	C₄	direct (N × H)
55. #88052	C₂×C₄	semidirect
59. C₂×#6090	C₂×C₄	semidirect
161. C₄×C₁₃	#350	semidirect
247. C₁₃	C₄×#350	semidirect
285. C₄	#11718153	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×#11718153	2
C₄×#88052	2
C₂×C₄×#6090	2
#13112925	2
C₄×C₁₃×#131	2
C₄×#350	13

Subgroups — C4×#11718153 #15777967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718153 #15777967