Subgroups — C₂×C₄×#88058 #15860963

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718143	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₁₃	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
13	C₂×#88071	—	2	2	no	—
14	C₂×#88058	C₄	1	16	yes	direct product factor
15	C₂×#88058	C₂²	1	8	yes	—
16	C₄×C₄⋊Dic₁₃	—	2	4	no	—
17	C₄×C₄⋊Dic₁₃	C₂²	1	8	yes	—
18	#11718143	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	—	2	4	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	—	2	4	no	—
34	C₂×C₄⋊Dic₁₃	—	2	8	no	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
37	C₂×C₄×Dic₁₃	—	2	8	no	—
38	C₂×C₄×Dic₁₃	—	4	8	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
40	#88071	—	2	8	no	—
41	#88058	C₂×C₄	1	32	yes	direct product factor
42	C₂×C₄×#107	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	D₈	1	4	yes	—
48	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₁₃	#41	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	16	no	—
58	C₂²×Dic₁₃	—	2	8	no	—
59	C₂²×Dic₁₃	—	4	8	no	—
60	C₂²×Dic₁₃	—	4	8	no	—
61	C₂×C₈×C₁₃	—	2	8	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
64	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₁₃	—	2	16	no	—
71	C₄⋊Dic₁₃	—	2	8	no	—
72	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
73	C₄×Dic₁₃	—	4	16	no	—
74	C₂²×#107	—	13	2	no	—
75	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
76	C₂×C₄×C₄⋊C₄	—	13	2	no	—
77	C₄×#107	—	13	8	no	—
78	C₂×#340	—	13	2	no	—
79	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₁₃	C₄×D₄	1	4	yes	—
81	C₂³×C₁₃	C₂×D₈	1	2	yes	—
82	C₂³×C₁₃	C₂×Q₁₆	1	2	yes	—
83	C₂³×C₁₃	#135	1	4	yes	—
84	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	#107	1	16	yes	complemented
88	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₁₃	#117	1	4	yes	—
93	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
95	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
96	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
101	C₂×Dic₁₃	—	2	16	no	—
102	C₂×Dic₁₃	—	4	8	no	—
103	C₂×Dic₁₃	—	4	8	no	—
104	C₂×Dic₁₃	—	4	32	no	—
105	C₈×C₁₃	—	2	4	no	—
106	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
107	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	—	13	4	no	—
110	C₂²×C₄²	—	26	2	no	—
111	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
112	C₂×#107	—	13	16	no	—
113	C₂×#107	—	13	8	no	—
114	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
115	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
116	C₄×C₄⋊C₄	—	13	8	no	—
117	C₄×C₄⋊C₄	—	26	4	no	—
118	C₂×#95	—	26	2	no	—
119	#340	—	13	8	no	—
120	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
121	C₂²×C₁₃	C₂×#107	1	4	yes	—
122	C₂²×C₁₃	#427	1	4	yes	—
123	C₂²×C₁₃	C₄×D₈	1	8	yes	—
124	C₂²×C₁₃	C₄×Q₁₆	1	8	yes	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
127	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×#135	1	2	yes	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
132	C₄×C₁₃	C₂×#107	1	8	yes	complemented
133	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
135	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
136	Dic₁₃	—	4	16	no	—
137	C₂³×C₄	—	26	4	no	—
138	C₂³×C₄	D₅₂	1	1	yes	characteristic
139	C₂³×C₄	D₅₂	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₄×D₁₃	1	8	yes	—
143	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
144	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
145	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
146	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
147	C₂×C₄⋊C₄	—	13	16	no	—
148	C₂×C₄⋊C₄	—	13	8	no	—
149	C₂×C₄⋊C₄	—	26	4	no	—
150	C₂×C₄⋊C₄	—	26	8	no	—
151	C₂×C₄²	—	26	8	no	—
152	C₂×C₄²	—	52	8	no	—
153	C₂×C₄²	Dic₂₆	1	4	yes	—
154	C₂×C₄²	D₅₂	1	2	yes	—
155	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
156	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
157	#107	—	13	32	no	—
158	C₄×C₈	C₂×Dic₁₃	1	16	yes	complemented
159	#95	—	26	8	no	—
160	C₂×C₁₃	C₄×#107	1	8	yes	complemented
161	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
164	C₂×C₁₃	C₂×C₄×D₈	1	2	yes	—
165	C₂×C₁₃	C₂×C₄×Q₁₆	1	2	yes	—
166	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	26	16	no	—
169	C₂²×C₄	—	26	8	no	—
170	C₂²×C₄	—	52	8	no	—
171	C₂²×C₄	—	52	8	no	—
172	C₂²×C₄	D₁₀₄	1	4	yes	—
173	C₂²×C₄	Dic₅₂	1	4	yes	—
174	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
175	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
176	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
178	C₂²×C₄	C₂×D₅₂	1	2	yes	—
179	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₅₂	1	2	yes	—
181	C₂²×C₄	#6127	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
187	C₂×C₈	C₂×C₄×D₁₃	1	8	yes	—
188	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
189	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
190	C₄⋊C₄	—	13	16	no	—
191	C₄⋊C₄	—	26	16	no	—
192	C₄⋊C₄	—	26	8	no	—
193	C₄²	—	52	16	no	—
194	C₄²	C₄⋊Dic₁₃	1	8	yes	—
195	C₄²	C₂×Dic₂₆	1	4	yes	—
196	C₄²	C₂²×Dic₁₃	1	4	yes	—
197	C₁₃	C₂×C₄×#107	1	1	yes	characteristic, complemented, Sylow
198	C₂³	C₄×D₅₂	1	4	yes	—
199	C₂³	C₂×D₁₀₄	1	2	yes	—
200	C₂³	#88158	1	4	yes	—
201	C₂³	C₂×Dic₅₂	1	2	yes	—
202	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
203	C₂³	C₂²×D₅₂	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	26	16	no	—
207	C₂×C₄	—	52	8	no	—
208	C₂×C₄	—	52	8	no	—
209	C₂×C₄	—	52	32	no	—
210	C₂×C₄	#88058	1	16	yes	direct product factor
211	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
212	C₂×C₄	C₄×D₅₂	1	4	yes	—
213	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
214	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
215	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
216	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
219	C₂×C₄	#88180	1	4	yes	—
220	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
222	C₂×C₄	C₂×#6127	1	2	yes	—
223	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×C₄×Dic₁₃	1	8	yes	complemented
226	C₂²	C₄×D₁₀₄	1	8	yes	—
227	C₂²	C₄×Dic₅₂	1	8	yes	—
228	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
229	C₂²	C₂×#88058	1	4	yes	—
230	C₂²	#11718767	1	4	yes	—
231	C₂²	C₂×C₄×D₅₂	1	2	yes	—
232	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
233	C₂²	C₂×#88158	1	2	yes	—
234	C₂²	C₂²×Dic₅₂	1	1	yes	characteristic
235	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
236	C₄	—	2	4	no	—
237	C₄	—	52	16	no	—
238	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
239	C₄	C₂×#88058	1	8	yes	direct product factor
240	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
241	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
242	C₄	C₂×#88180	1	1	yes	characteristic
243	C₂	C₄×#88058	1	8	yes	direct product factor
244	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
245	C₂	C₂²×#88058	1	1	yes	characteristic
246	C₂	C₂×#11718767	1	1	yes	characteristic
247	C₂	C₂×C₄×D₁₀₄	1	2	yes	—
248	C₂	C₂×C₄×Dic₅₂	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×C₄×#107
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#88058	C₂	direct (N × H)
11. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×#88058	C₄	direct (N × H)
39. C₄×C₈×C₁₃	C₂×C₄	semidirect
41. #88058	C₂×C₄	direct (N × H)
63. C₂×C₈×C₁₃	C₄²	semidirect
87. C₂×C₄×C₁₃	#107	semidirect
106. C₈×C₁₃	C₂×C₄²	semidirect
114. C₂×C₄×C₈	Dic₁₃	semidirect
132. C₄×C₁₃	C₂×#107	semidirect
158. C₄×C₈	C₂×Dic₁₃	semidirect
160. C₂×C₁₃	C₄×#107	semidirect
186. C₂×C₈	C₄×Dic₁₃	semidirect
197. C₁₃	C₂×C₄×#107	semidirect
210. C₂×C₄	#88058	direct (N × H)
225. C₈	C₂×C₄×Dic₁₃	semidirect
239. C₄	C₂×#88058	direct (N × H)
243. C₂	C₄×#88058	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88058	2
C₂×C₄×C₄⋊Dic₁₃	2
C₄×#88058	2
C₂×#11718143	2
C₂²×C₄×C₈×C₁₃	2
C₂×C₄×#107	13

Subgroups — C2×C4×#88058 #15860963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88058 #15860963