Subgroups — C₄×#88196 #13123655

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88196	C₂	1	1	yes	maximal, characteristic
3	C₄×#6127	C₂	1	1	yes	maximal, characteristic
4	#11718886	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718691	C₂	1	2	yes	maximal, complemented
6	C₄×#6107	C₂	1	2	yes	maximal, complemented
7	#11718687	C₂	1	2	yes	maximal, complemented
8	C₄×#6106	C₂	1	2	yes	maximal, complemented
9	#11718627	C₂	1	1	yes	maximal, characteristic
10	#11718625	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6127	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	—	2	1	no	—
15	C₄×Q₈×C₁₃	—	2	1	no	—
16	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
17	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6107	C₂²	1	2	yes	—
23	#88196	C₄	1	8	yes	direct product factor
24	C₂×#6106	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₁₃	—	2	1	no	—
26	C₄×C₂²⋊D₁₃	—	2	1	no	—
27	#88180	C₂²	1	1	yes	characteristic
28	C₂×#6099	C₂²	1	1	yes	characteristic
29	C₂×#6099	C₂²	1	1	yes	characteristic
30	C₄×D₅₂	C₂²	1	1	yes	characteristic
31	C₄×D₅₂	C₂²	1	1	yes	characteristic
32	#88126	—	2	1	no	—
33	C₄²×D₁₃	—	2	1	no	—
34	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
36	#88075	C₂²	1	2	yes	—
37	#88075	C₂²	1	2	yes	—
38	#88050	C₂²	1	2	yes	—
39	#88049	C₂²	1	2	yes	—
40	#88048	C₂²	1	2	yes	—
41	#88047	C₂²	1	2	yes	—
42	#88043	C₂²	1	2	yes	—
43	C₄×#1055	C₂²	1	2	yes	—
44	C₂×Q₈×C₁₃	—	2	1	no	—
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₃	—	2	1	no	—
54	#6127	—	2	2	no	—
55	#6127	C₂×C₄	1	4	yes	—
56	C₂×D₅₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₃	—	2	1	no	—
58	C₂×C₄×D₁₃	—	2	1	no	—
59	C₂×C₄×D₁₃	—	2	1	no	—
60	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	D₄	1	1	yes	characteristic
68	C₄²×C₁₃	D₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂³	1	1	yes	characteristic
70	C₄²×C₁₃	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₁₃	—	2	1	no	—
72	#6107	C₂×C₄	1	8	yes	complemented
73	#6106	C₂×C₄	1	8	yes	complemented
74	#6103	—	2	1	no	—
75	#6103	—	2	1	no	—
76	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
78	#6101	—	2	1	no	—
79	#6101	—	2	1	no	—
80	C₄×Dic₁₃	—	2	1	no	—
81	C₄×Dic₁₃	—	2	1	no	—
82	C₄×Dic₁₃	—	2	1	no	—
83	#6099	—	2	2	no	—
84	#6099	C₂×C₄	1	4	yes	complemented
85	C₂×#1055	C₂³	1	2	yes	—
86	C₂×#1055	C₂³	1	2	yes	—
87	C₄×#137	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₃	—	2	1	no	—
90	Q₈×C₁₃	—	2	4	no	—
91	Q₈×C₁₃	—	2	2	no	—
92	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#41	1	2	yes	—
106	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₁₃	—	2	4	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	D₅₂	—	2	1	no	—
112	D₅₂	C₂²×C₄	1	2	yes	—
113	C₄×D₁₃	—	2	4	no	—
114	C₄×D₁₃	—	4	2	no	—
115	C₄×D₁₃	—	4	2	no	—
116	Dic₂₆	—	2	1	no	—
117	Dic₂₆	C₂²×C₄	1	2	yes	—
118	#1055	—	2	2	no	—
119	#1055	C₂²×C₄	1	4	yes	—
120	C₂×#137	—	13	1	no	—
121	C₄×#41	—	13	1	no	—
122	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
123	#441	—	13	2	no	—
124	#440	—	13	2	no	—
125	C₄×Q₁₆	—	13	2	no	—
126	C₄×SD₁₆	—	13	2	no	—
127	#428	—	13	1	no	—
128	#419	—	13	1	no	—
129	#417	—	13	1	no	—
130	C₄×M₁₆	—	13	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	C₄×D₄	1	2	yes	—
134	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	4	1	no	—
137	C₂×D₁₃	—	4	1	no	—
138	C₄×C₁₃	—	2	4	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	C₄×D₄	1	2	yes	—
144	C₄×C₁₃	#137	1	2	yes	complemented
145	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	—	4	1	no	—
149	Dic₁₃	—	4	1	no	—
150	C₂×#41	—	13	1	no	—
151	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
152	#137	—	13	8	no	—
153	C₂×Q₁₆	—	13	2	no	—
154	C₂×SD₁₆	—	13	2	no	—
155	C₂×M₁₆	—	13	1	no	—
156	C₂×M₁₆	—	13	1	no	—
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	—	2	1	no	—
159	C₄×Q₈	—	13	1	no	—
160	C₄×Q₈	—	13	1	no	—
161	C₄×Q₈	C₂×D₁₃	1	2	yes	—
162	C₄×Q₈	C₂×D₁₃	1	2	yes	—
163	C₄×D₄	—	13	1	no	—
164	C₄×D₄	—	13	1	no	—
165	C₄×D₄	—	26	1	no	—
166	C₄×D₄	—	26	1	no	—
167	#117	—	13	1	no	—
168	#117	—	26	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	—	26	1	no	—
173	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
174	#107	—	13	2	no	—
175	#106	—	13	2	no	—
176	#103	—	13	2	no	—
177	#103	—	13	2	no	—
178	#103	—	13	2	no	—
179	#102	—	13	2	no	—
180	#97	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
185	C₂×C₁₃	#575	1	1	yes	characteristic
186	D₁₃	—	4	2	no	—
187	#41	—	13	4	no	—
188	#41	—	26	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	13	1	no	—
192	C₂×Q₈	C₄×D₁₃	1	4	yes	complemented
193	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
194	C₂×D₄	—	13	1	no	—
195	C₂×D₄	—	26	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	26	1	no	—
199	C₂²×C₄	—	26	1	no	—
200	C₂²×C₄	—	26	1	no	—
201	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
204	Q₁₆	—	13	8	no	—
205	SD₁₆	—	13	8	no	—
206	M₁₆	—	13	4	no	—
207	M₁₆	—	26	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	13	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	13	1	no	—
214	C₄⋊C₄	—	13	1	no	—
215	C₄⋊C₄	—	26	1	no	—
216	C₄⋊C₄	—	26	1	no	—
217	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
218	C₂²⋊C₄	—	26	1	no	—
219	C₂²⋊C₄	—	26	1	no	—
220	C₂²⋊C₄	—	26	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	26	1	no	—
224	C₄²	—	26	1	no	—
225	C₄²	—	26	1	no	—
226	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
228	C₄²	C₂²×D₁₃	1	1	yes	characteristic
229	C₄²	C₂²×D₁₃	1	1	yes	characteristic
230	C₁₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	26	1	no	—
232	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	13	2	no	—
236	Q₈	—	26	1	no	—
237	Q₈	C₂×C₄×D₁₃	1	4	yes	—
238	D₄	—	13	2	no	—
239	D₄	—	26	4	no	—
240	D₄	—	26	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	—	26	4	no	—
249	C₂×C₄	—	26	1	no	—
250	C₂×C₄	—	26	1	no	—
251	C₂×C₄	—	52	2	no	—
252	C₂×C₄	—	52	2	no	—
253	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
254	C₂×C₄	#6127	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
261	C₈	—	13	4	no	—
262	C₈	—	26	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	26	2	no	—
266	C₂²	—	52	1	no	—
267	C₂²	—	52	1	no	—
268	C₂²	C₄×C₂²⋊D₁₃	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	26	2	no	—
276	C₄	—	52	1	no	—
277	C₄	—	52	1	no	—
278	C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
279	C₄	#88196	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
281	C₄	C₂×#6127	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	52	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
285	C₂	C₂×#88196	1	1	yes	characteristic
286	C₂	#11719527	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#6127	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₄×#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
4. #11718886	C₂	semidirect
5. #11718691	C₂	semidirect
6. C₄×#6107	C₂	semidirect
7. #11718687	C₂	semidirect
8. C₄×#6106	C₂	semidirect
10. #11718625	C₂	semidirect
11. C₄×#6099	C₂	semidirect
23. #88196	C₄	direct (N × H)
46. C₂×Q₈×C₁₃	C₂×C₄	semidirect
72. #6107	C₂×C₄	semidirect
73. #6106	C₂×C₄	semidirect
84. #6099	C₂×C₄	semidirect
122. C₂×C₄×Q₈	D₁₃	semidirect
144. C₄×C₁₃	#137	semidirect
192. C₂×Q₈	C₄×D₁₃	semidirect
230. C₁₃	C₄×#137	semidirect
279. C₄	#88196	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×Q₈×C₁₃	2
C₂×#88196	2
C₄×#6127	2
#11718886	2
#11718691	2
C₄×#6107	2
#11718687	2
C₄×#6106	2
#11718627	2
#11718625	2
C₄×#6099	2
C₄×#137	13

Subgroups — C4×#88196 #13123655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88196 #13123655