Subgroups — C₂×C₄×#88059 #15861015

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	2	yes	maximal
3	C₄×#88059	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718144	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×#6099	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
13	C₂×#88059	C₄	1	16	yes	direct product factor
14	C₂×#88059	C₂²	1	8	yes	—
15	C₄×#6099	—	2	4	no	—
16	C₄×#6099	C₂²	1	8	yes	—
17	C₂×#88043	—	2	2	no	—
18	C₂×C₄×#1055	—	2	2	no	—
19	#11718144	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂×#6099	—	2	4	no	—
38	C₂×#6099	—	2	8	no	—
39	C₂×#6099	C₂×C₄	1	16	yes	—
40	C₂×#6099	C₂³	1	8	yes	—
41	C₂²×#1055	—	2	4	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	16	yes	—
43	#88059	C₂×C₄	1	32	yes	direct product factor
44	#88043	—	2	8	no	—
45	C₄×#1055	—	2	8	no	—
46	C₂×C₄×#108	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	8	no	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	C₄²	1	8	yes	—
65	C₂×C₈×C₁₃	C₄²	1	8	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
69	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	2	yes	—
75	#6099	—	2	16	no	—
76	#6099	—	2	8	no	—
77	#6099	C₂²×C₄	1	16	yes	—
78	C₂×#1055	—	2	16	no	—
79	C₂×#1055	—	2	8	no	—
80	C₂²×#108	—	13	2	no	—
81	C₂×C₄×M₁₆	—	13	2	no	—
82	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
83	C₄×#108	—	13	8	no	—
84	C₂×#341	—	13	2	no	—
85	C₂³×C₁₃	—	2	4	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
89	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	8	no	—
93	C₂×C₄×C₁₃	#108	1	8	yes	complemented
94	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₃	#117	1	2	yes	—
101	C₂×C₄×C₁₃	#117	1	2	yes	—
102	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
109	C₈×C₁₃	—	2	4	no	—
110	C₈×C₁₃	C₂×C₄²	1	8	yes	—
111	#1055	—	2	16	no	—
112	C₂²×M₁₆	—	13	4	no	—
113	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
114	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
115	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	C₂×#108	—	13	16	no	—
117	C₂×#108	—	13	8	no	—
118	C₄×M₁₆	—	13	8	no	—
119	C₄×M₁₆	—	26	4	no	—
120	C₂×#97	—	26	2	no	—
121	C₂×C₄×C₈	—	26	2	no	—
122	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
123	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
124	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
125	#341	—	13	8	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	8	no	—
129	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
130	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
131	C₂²×C₁₃	C₂×#108	1	4	yes	—
132	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
136	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
137	C₄×C₁₃	C₂×#108	1	8	yes	complemented
138	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
141	C₂³×C₄	Dic₂₆	1	2	yes	—
142	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×M₁₆	—	13	16	no	—
144	C₂×M₁₆	—	13	8	no	—
145	C₂×M₁₆	—	26	4	no	—
146	C₂×M₁₆	—	26	8	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	26	4	no	—
149	C₂²×C₈	C₄×D₁₃	1	8	yes	—
150	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
151	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
152	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
153	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
155	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
156	C₂×C₄²	Dic₂₆	1	2	yes	—
157	C₂×C₄²	D₅₂	1	1	yes	characteristic
158	C₂×C₄²	D₅₂	1	2	yes	—
159	C₂×C₄²	D₅₂	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
161	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
162	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
163	#108	—	13	32	no	—
164	#97	—	26	8	no	—
165	C₄×C₈	—	26	8	no	—
166	C₄×C₈	C₂×Dic₁₃	1	16	yes	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	2	2	no	—
169	C₂×C₁₃	C₄×#108	1	4	yes	complemented
170	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#108	1	2	yes	—
172	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
176	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
177	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
178	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
180	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
182	C₂²×C₄	#6127	1	4	yes	—
183	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
186	M₁₆	—	13	16	no	—
187	M₁₆	—	26	16	no	—
188	M₁₆	—	26	8	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	26	16	no	—
193	C₂×C₈	—	26	8	no	—
194	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
195	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
196	C₂×C₈	C₂×C₄×D₁₃	1	8	yes	—
197	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
198	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
199	C₄²	C₄⋊Dic₁₃	1	2	yes	—
200	C₄²	C₄⋊Dic₁₃	1	4	yes	—
201	C₄²	C₄⋊Dic₁₃	1	2	yes	—
202	C₄²	C₂×D₅₂	1	2	yes	—
203	C₄²	C₂×D₅₂	1	2	yes	—
204	C₄²	C₂²×Dic₁₃	1	4	yes	—
205	C₁₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	4	no	—
207	C₂³	C₄×Dic₂₆	1	4	yes	—
208	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
209	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
210	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	#88059	1	8	yes	direct product factor
215	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
216	C₂×C₄	C₄×D₅₂	1	4	yes	—
217	C₂×C₄	C₄×D₅₂	1	4	yes	—
218	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
219	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
222	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
224	C₂×C₄	#88180	1	2	yes	—
225	C₂×C₄	#88180	1	2	yes	—
226	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
227	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
228	C₂×C₄	C₂×#6127	1	2	yes	—
229	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
230	C₈	—	2	4	no	—
231	C₈	—	26	16	no	—
232	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	8	no	—
236	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
237	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
238	C₂²	C₂×#88059	1	4	yes	—
239	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
240	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
243	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
244	C₄	C₂×#88059	1	8	yes	direct product factor
245	C₄	C₂×C₄×D₅₂	1	2	yes	—
246	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
247	C₄	C₂×#88180	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₄×#88059	1	4	yes	direct product factor
251	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
252	C₂	C₂²×#88059	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×C₄×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#88059	C₂	direct (N × H)
13. C₂×#88059	C₄	direct (N × H)
43. #88059	C₂×C₄	direct (N × H)
93. C₂×C₄×C₁₃	#108	semidirect
137. C₄×C₁₃	C₂×#108	semidirect
169. C₂×C₁₃	C₄×#108	semidirect
205. C₁₃	C₂×C₄×#108	semidirect
214. C₂×C₄	#88059	direct (N × H)
244. C₄	C₂×#88059	direct (N × H)
250. C₂	C₄×#88059	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×#88059	2
C₂×C₄×#6099	2
C₄×#88059	2
C₂×#11718144	2
C₂²×C₄×C₈×C₁₃	2
C₂×C₄×#108	13

Subgroups — C2×C4×#88059 #15861015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88059 #15861015