Subgroups — C₄×#11718585 #15860821

Order 3328.

1382 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	2	yes	maximal
3	#13114689	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊#1055	C₂	1	4	yes	maximal, complemented
5	#13114530	C₂	1	2	yes	maximal
6	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₄	1	4	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	4	yes	—
12	C₄³×C₁₃	C₄	1	2	yes	—
13	C₄³×C₁₃	C₄	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	2	yes	—
16	#11718585	C₄	1	16	yes	direct product factor
17	C₂×C₄⋊#1055	C₂²	1	8	yes	—
18	C₄×#6099	—	2	8	no	—
19	C₂×#88043	C₂²	1	4	yes	—
20	C₂×C₄×#1055	C₂²	1	4	yes	—
21	#11718116	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	—	2	4	no	—
27	C₂×C₄²×C₁₃	—	2	16	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃	D₄	1	4	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6099	—	2	8	no	—
44	C₂×#6099	—	2	8	no	—
45	C₂×#6099	C₂×C₄	1	16	yes	—
46	C₂²×#1055	C₂³	1	8	yes	—
47	C₄⋊#1055	C₂×C₄	1	32	yes	complemented
48	#88043	—	2	8	no	—
49	C₄×#1055	—	2	8	no	—
50	C₄×#423	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₄²×C₁₃	—	2	4	no	—
70	C₄²×C₁₃	—	2	8	no	—
71	C₄²×C₁₃	—	2	8	no	—
72	C₄²×C₁₃	—	2	8	no	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	C₄²	1	4	yes	—
75	C₄²×C₁₃	C₄²	1	4	yes	—
76	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
77	C₄²×C₁₃	M₁₆	1	16	yes	complemented
78	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
83	#6099	—	2	32	no	—
84	C₂×#1055	—	2	8	no	—
85	C₂×#1055	—	2	8	no	—
86	C₂×#1055	C₂²×C₄	1	16	yes	—
87	C₂×#423	—	13	2	no	—
88	C₂×C₄×M₁₆	—	13	2	no	—
89	C₂×C₄³	D₁₃	1	1	yes	characteristic
90	#1767	—	13	4	no	—
91	C₄×C₄⋊C₈	—	13	4	no	—
92	#1728	—	13	2	no	—
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃	#117	1	2	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	16	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	16	no	—
104	C₂×C₄×C₁₃	—	2	16	no	—
105	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₁₃	#117	1	2	yes	—
112	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
113	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
115	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
121	#1055	—	2	16	no	—
122	C₂²×M₁₆	—	13	4	no	—
123	C₂²×C₄²	Dic₁₃	1	4	yes	—
124	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
126	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
127	#423	—	13	16	no	—
128	C₂×C₄⋊C₈	—	13	8	no	—
129	C₄×M₁₆	—	26	8	no	—
130	C₂×#97	—	13	4	no	—
131	C₂×C₄×C₈	—	13	4	no	—
132	C₄³	Dic₁₃	1	2	yes	—
133	C₄³	Dic₁₃	1	2	yes	—
134	C₄³	C₂×D₁₃	1	2	yes	—
135	C₄³	C₂×D₁₃	1	2	yes	—
136	#336	—	13	8	no	—
137	C₂²×C₁₃	—	2	4	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
141	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
142	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
144	C₂²×C₁₃	C₂²×M₁₆	1	4	yes	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	2	8	no	—
148	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
149	C₄×C₁₃	C₄×M₁₆	1	8	yes	complemented
150	C₄×C₁₃	#423	1	8	yes	complemented
151	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
152	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
153	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
154	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
155	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
156	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×M₁₆	—	13	16	no	—
158	C₂×M₁₆	—	26	8	no	—
159	C₂×M₁₆	—	26	8	no	—
160	C₂²×C₈	—	13	8	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	—	2	16	no	—
163	C₂×C₄²	Dic₂₆	1	4	yes	—
164	C₂×C₄²	C₄×D₁₃	1	8	yes	—
165	C₂×C₄²	D₅₂	1	4	yes	—
166	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
168	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
169	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
170	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
171	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
175	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₈	—	13	32	no	—
179	#97	—	26	8	no	—
180	C₄×C₈	—	26	8	no	—
181	C₂×C₁₃	—	2	4	no	—
182	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
184	C₂×C₁₃	C₂×#423	1	2	yes	—
185	C₂×C₁₃	#2920	1	2	yes	—
186	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	C₄×Dic₁₃	1	8	yes	—
192	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
193	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
195	C₂²×C₄	C₂×D₅₂	1	2	yes	—
196	C₂²×C₄	#6127	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
204	M₁₆	—	26	32	no	—
205	C₂×C₈	—	13	16	no	—
206	C₂×C₈	—	26	8	no	—
207	C₂×C₈	—	26	8	no	—
208	C₄²	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	#6099	1	16	yes	complemented
214	C₄²	C₄×Dic₁₃	1	4	yes	—
215	C₄²	C₄×Dic₁₃	1	4	yes	—
216	C₄²	C₄⋊Dic₁₃	1	8	yes	—
217	C₄²	C₂×Dic₂₆	1	2	yes	—
218	C₄²	C₂×Dic₂₆	1	2	yes	—
219	C₄²	C₂×C₄×D₁₃	1	8	yes	—
220	C₄²	C₂×D₅₂	1	2	yes	—
221	C₄²	C₂×D₅₂	1	2	yes	—
222	C₁₃	C₄×#423	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	4	no	—
224	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
225	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
226	C₂³	#88180	1	2	yes	—
227	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
236	C₂×C₄	C₄×D₅₂	1	8	yes	—
237	C₂×C₄	C₂×#6099	1	8	yes	—
238	C₂×C₄	C₂×#6099	1	8	yes	—
239	C₂×C₄	C₂×#6099	1	8	yes	—
240	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
245	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
246	C₂×C₄	#88180	1	2	yes	—
247	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
249	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
250	C₂×C₄	C₂×#6127	1	2	yes	—
251	C₈	—	26	16	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
256	C₂²	C₂²×#6099	1	4	yes	—
257	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
258	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
259	C₂²	C₂×#88180	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	8	no	—
262	C₄	—	2	4	no	—
263	C₄	C₄×#6099	1	8	yes	complemented
264	C₄	#11718585	1	8	yes	direct product factor
265	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
266	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
267	C₄	C₂×C₄×D₅₂	1	2	yes	—
268	C₂	—	2	4	no	—
269	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
270	C₂	C₂×C₄×#6099	1	2	yes	—
271	C₂	C₂×#11718585	1	2	yes	—
272	C₂	#13123469	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	C₄×#423
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #13114689	C₂	semidirect
4. C₄×C₄⋊#1055	C₂	semidirect
16. #11718585	C₄	direct (N × H)
47. C₄⋊#1055	C₂×C₄	semidirect
77. C₄²×C₁₃	M₁₆	semidirect
149. C₄×C₁₃	C₄×M₁₆	semidirect
150. C₄×C₁₃	#423	semidirect
213. C₄²	#6099	semidirect
222. C₁₃	C₄×#423	semidirect
263. C₄	C₄×#6099	semidirect
264. C₄	#11718585	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×#11718585	2
C₂×C₄×#6099	2
#13114689	2
C₄×C₄⋊#1055	2
#13114530	2
C₂×C₄³×C₁₃	2
C₄×#423	13

Subgroups — C4×#11718585 #15860821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718585 #15860821