Subgroups — C₂×#5769 #87184

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1009	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#949	C₂	1	2	yes	maximal, complemented
4	#5769	C₂	1	16	yes	maximal, direct product factor
5	C₂×#919	C₂	1	2	yes	maximal, complemented
6	C₂×#914	C₂	1	4	yes	maximal, complemented
7	C₂×D₃×D₈	C₂	1	2	yes	maximal, complemented
8	C₂²×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#901	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#576	—	3	1	no	maximal, Sylow
11	C₂×C₃×#41	C₂²	1	2	yes	complemented
12	#1009	—	2	8	no	—
13	#1009	C₂²	1	16	yes	complemented
14	C₂×#239	C₂²	1	4	yes	complemented
15	C₂×D₃×D₄	—	2	4	no	—
16	C₂×D₃×D₄	C₂²	1	4	yes	complemented
17	C₂×#235	—	2	2	no	—
18	C₂×#235	C₂²	1	1	yes	characteristic
19	C₂²×D₁₂	—	2	2	no	—
20	C₂²×D₁₂	C₂²	1	2	yes	complemented
21	C₃×#135	C₂²	1	8	yes	complemented
22	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
24	C₂×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
26	#949	C₂²	1	16	yes	complemented
27	C₂×#215	C₂²	1	4	yes	complemented
28	C₂×#213	C₂²	1	4	yes	complemented
29	C₂×#208	C₂²	1	1	yes	characteristic, complemented
30	#919	C₂²	1	16	yes	complemented
31	#914	C₂²	1	32	yes	complemented
32	D₃×D₈	C₂²	1	16	yes	complemented
33	C₂×D₂₄	C₂²	1	2	yes	complemented
34	C₂×D₂₄	C₂²	1	2	yes	complemented
35	C₂×D₂₄	C₂²	1	8	yes	complemented
36	#901	C₂²	1	8	yes	complemented
37	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
38	C₂×D₃×C₈	C₂²	1	2	yes	complemented
39	C₂×H(5,𝔽₂)	—	3	2	no	—
40	#576	—	3	16	no	—
41	C₂×AGL(1,ℤ/8)	—	3	2	no	—
42	C₂×AGL(1,ℤ/8)	—	3	4	no	—
43	C₂×#135	—	3	2	no	—
44	C₂×#135	D₃	1	1	yes	characteristic, complemented
45	C₂²×D₈	—	3	2	no	—
46	C₂²×D₈	—	3	1	no	—
47	C₂×#131	—	3	1	no	—
48	C₂³×D₃	—	2	2	no	—
49	C₂³×D₃	—	2	4	no	—
50	C₃×#41	—	2	4	no	—
51	C₃×#41	C₂³	1	8	yes	complemented
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂×#67	—	2	4	no	—
58	C₂×#67	D₄	1	2	yes	—
59	#239	—	2	8	no	—
60	#239	C₂³	1	16	yes	complemented
61	D₃×D₄	—	2	32	no	—
62	D₃×D₄	—	2	8	no	—
63	D₃×D₄	C₂³	1	16	yes	complemented
64	#235	—	2	16	no	—
65	#235	D₄	1	4	yes	—
66	#235	C₂³	1	4	yes	—
67	C₂×D₁₂	—	2	4	no	—
68	C₂×D₁₂	—	2	2	no	—
69	C₂×D₁₂	—	2	4	no	—
70	C₂×D₁₂	—	2	16	no	—
71	C₂×D₁₂	D₄	1	1	yes	characteristic
72	C₂×D₁₂	C₂³	1	4	yes	—
73	C₂×D₁₂	C₂³	1	8	yes	complemented
74	C₂×C₄×D₃	—	2	4	no	—
75	C₂×C₄×D₃	C₂³	1	2	yes	—
76	C₂×Dic₆	D₄	1	1	yes	characteristic
77	C₃×Q₁₆	C₂³	1	4	yes	complemented
78	C₃×SD₁₆	C₂³	1	8	yes	complemented
79	C₃×D₈	C₂³	1	4	yes	complemented
80	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂³	1	4	yes	complemented
83	#215	C₂³	1	16	yes	complemented
84	#213	C₂³	1	16	yes	complemented
85	#208	C₂³	1	4	yes	complemented
86	C₂×C₃⋊C₈	C₂³	1	2	yes	—
87	D₂₄	C₂³	1	8	yes	complemented
88	D₂₄	C₂³	1	8	yes	complemented
89	ΓL(1,𝔽₂₅)	C₂³	1	8	yes	complemented
90	D₃×C₈	C₂³	1	8	yes	complemented
91	H(5,𝔽₂)	—	3	16	no	—
92	H(5,𝔽₂)	—	6	8	no	—
93	C₂×#41	—	3	1	no	—
94	C₂×#41	—	3	4	no	—
95	C₂×#41	—	6	2	no	—
96	C₂×#41	C₂×D₃	1	2	yes	complemented
97	C₂²×D₄	—	3	4	no	—
98	C₂²×D₄	—	3	2	no	—
99	C₂²×D₄	—	6	2	no	—
100	C₂²×D₄	—	6	4	no	—
101	AGL(1,ℤ/8)	—	3	16	no	—
102	AGL(1,ℤ/8)	—	3	32	no	—
103	#135	—	3	16	no	—
104	#135	C₂×D₃	1	8	yes	complemented
105	C₂×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
106	C₂×SD₁₆	—	3	4	no	—
107	C₂×SD₁₆	C₂×D₃	1	2	yes	complemented
108	C₂×D₈	—	3	4	no	—
109	C₂×D₈	—	3	16	no	—
110	C₂×D₈	—	3	2	no	—
111	C₂×D₈	—	3	2	no	—
112	C₂×D₈	—	3	8	no	—
113	C₂×D₈	C₂×D₃	1	1	yes	characteristic, complemented
114	#131	—	3	8	no	—
115	C₂×M₁₆	—	3	1	no	—
116	C₂×M₁₆	—	3	2	no	—
117	C₂²×C₈	—	3	2	no	—
118	C₂²×C₈	C₂×D₃	1	1	yes	characteristic, complemented
119	C₂³×C₃	—	2	2	no	—
120	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×D₃	—	2	4	no	—
122	C₂²×D₃	—	2	8	no	—
123	C₂²×D₃	—	2	16	no	—
124	C₂²×D₃	—	8	8	no	—
125	C₂²×D₃	—	8	4	no	—
126	C₂²×D₃	C₂×D₄	1	2	yes	—
127	C₃×Q₈	—	2	2	no	—
128	C₃×Q₈	C₂⁴	1	4	yes	—
129	C₃×D₄	—	2	8	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	C₂⁴	1	4	yes	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
136	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂⁴	1	2	yes	—
138	#67	—	2	16	no	—
139	#67	C₂×D₄	1	8	yes	—
140	C₂×Dic₃	C₂×D₄	1	2	yes	—
141	D₁₂	—	2	16	no	—
142	D₁₂	—	2	8	no	—
143	D₁₂	—	2	4	no	—
144	D₁₂	C₂×D₄	1	4	yes	—
145	D₁₂	C₂⁴	1	8	yes	—
146	C₄×D₃	—	2	16	no	—
147	C₄×D₃	C₂×D₄	1	4	yes	—
148	C₄×D₃	C₂⁴	1	4	yes	—
149	Dic₆	C₂×D₄	1	4	yes	—
150	C₃×C₈	C₂⁴	1	2	yes	—
151	C₃×C₈	C₂⁴	1	2	yes	—
152	C₃⋊C₈	C₂⁴	1	4	yes	—
153	C₂⁴	—	6	4	no	—
154	C₂⁴	—	6	2	no	—
155	#41	—	2	4	no	—
156	#41	—	3	4	no	—
157	#41	—	3	4	no	—
158	#41	—	3	16	no	—
159	#41	—	6	16	no	—
160	#41	—	6	8	no	—
161	#41	C₂²×D₃	1	8	yes	complemented
162	C₂×Q₈	—	3	1	no	—
163	C₂×Q₈	C₂²×D₃	1	2	yes	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	3	2	no	—
166	C₂×D₄	—	3	16	no	—
167	C₂×D₄	—	3	1	no	—
168	C₂×D₄	—	3	4	no	—
169	C₂×D₄	—	3	8	no	—
170	C₂×D₄	—	6	16	no	—
171	C₂×D₄	—	6	4	no	—
172	C₂×D₄	—	6	8	no	—
173	C₂×D₄	—	6	4	no	—
174	C₂×D₄	—	6	32	no	—
175	C₂×D₄	—	6	2	no	—
176	C₂×D₄	—	6	4	no	—
177	C₂×D₄	C₂²×D₃	1	2	yes	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	3	2	no	—
180	C₂²×C₄	—	6	4	no	—
181	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
182	Q₁₆	C₂²×D₃	1	4	yes	complemented
183	SD₁₆	—	3	16	no	—
184	SD₁₆	C₂²×D₃	1	8	yes	complemented
185	D₈	—	3	16	no	—
186	D₈	—	3	8	no	—
187	D₈	—	3	8	no	—
188	D₈	C₂²×D₃	1	4	yes	complemented
189	M₁₆	—	3	4	no	—
190	M₁₆	—	3	8	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	8	no	—
193	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
195	C₂×C₈	C₂²×D₃	1	4	yes	complemented
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	4	4	no	—
199	C₂²×C₃	C₂²×D₄	1	2	yes	—
200	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
201	C₂×D₃	—	2	4	no	—
202	C₂×D₃	—	2	8	no	—
203	C₂×D₃	—	4	8	no	—
204	C₂×D₃	—	8	16	no	—
205	C₂×D₃	—	8	8	no	—
206	C₂×D₃	C₂²×D₄	1	4	yes	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	C₂²×D₄	1	2	yes	—
209	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
210	C₃×C₄	C₂⁵	1	1	yes	characteristic
211	Dic₃	C₂²×D₄	1	4	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	3	2	no	—
214	C₂³	—	6	16	no	—
215	C₂³	—	6	4	no	—
216	C₂³	—	6	8	no	—
217	C₂³	—	24	4	no	—
218	C₂³	—	24	8	no	—
219	C₂³	D₃×D₄	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	3	4	no	—
222	Q₈	C₂³×D₃	1	4	yes	—
223	D₄	—	2	8	no	—
224	D₄	—	2	2	no	—
225	D₄	—	3	8	no	—
226	D₄	—	3	4	no	—
227	D₄	—	3	8	no	—
228	D₄	—	6	16	no	—
229	D₄	—	6	16	no	—
230	D₄	—	6	8	no	—
231	D₄	—	6	4	no	—
232	D₄	C₂³×D₃	1	4	yes	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	3	2	no	—
236	C₂×C₄	—	3	4	no	—
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	6	16	no	—
239	C₂×C₄	D₃×D₄	1	2	yes	—
240	C₂×C₄	D₃×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₃	1	2	yes	—
242	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
243	C₈	—	3	4	no	—
244	C₈	C₂³×D₃	1	2	yes	—
245	C₈	C₂³×D₃	1	2	yes	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	4	4	no	—
248	C₂×C₃	#576	1	2	yes	complemented
249	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
250	D₃	—	2	4	no	—
251	D₃	—	4	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	3	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	6	4	no	—
257	C₂²	—	6	8	no	—
258	C₂²	—	12	8	no	—
259	C₂²	—	24	8	no	—
260	C₂²	—	24	16	no	—
261	C₂²	C₂×D₃×D₄	1	2	yes	—
262	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	3	4	no	—
265	C₄	C₂×D₃×D₄	1	2	yes	—
266	C₄	C₂×D₃×D₄	1	1	yes	characteristic
267	C₄	C₂⁴×D₃	1	1	yes	characteristic
268	C₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	6	4	no	—
272	C₂	—	12	8	no	—
273	C₂	#5769	1	2	yes	direct product factor
274	C₂	C₂²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂×C₃×#135	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₂×#135	D₃
hypocenter	C₃	C₂×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#1009	C₂	semidirect
2. C₂×C₃×#135	C₂	semidirect
3. C₂×#949	C₂	semidirect
4. #5769	C₂	direct (N × H)
5. C₂×#919	C₂	semidirect
6. C₂×#914	C₂	semidirect
7. C₂×D₃×D₈	C₂	semidirect
8. C₂²×D₂₄	C₂	semidirect
9. C₂×#901	C₂	semidirect
11. C₂×C₃×#41	C₂²	semidirect
13. #1009	C₂²	semidirect
14. C₂×#239	C₂²	semidirect
16. C₂×D₃×D₄	C₂²	semidirect
20. C₂²×D₁₂	C₂²	semidirect
21. C₃×#135	C₂²	semidirect
22. C₂×C₃×Q₁₆	C₂²	semidirect
23. C₂×C₃×SD₁₆	C₂²	semidirect
24. C₂×C₃×D₈	C₂²	semidirect
25. C₂²×C₃×C₈	C₂²	semidirect
26. #949	C₂²	semidirect
27. C₂×#215	C₂²	semidirect
28. C₂×#213	C₂²	semidirect
29. C₂×#208	C₂²	semidirect
30. #919	C₂²	semidirect
31. #914	C₂²	semidirect
32. D₃×D₈	C₂²	semidirect
33. C₂×D₂₄	C₂²	semidirect
34. C₂×D₂₄	C₂²	semidirect
35. C₂×D₂₄	C₂²	semidirect
36. #901	C₂²	semidirect
37. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
38. C₂×D₃×C₈	C₂²	semidirect
44. C₂×#135	D₃	semidirect
51. C₃×#41	C₂³	semidirect
60. #239	C₂³	semidirect
63. D₃×D₄	C₂³	semidirect
73. C₂×D₁₂	C₂³	semidirect
77. C₃×Q₁₆	C₂³	semidirect
78. C₃×SD₁₆	C₂³	semidirect
79. C₃×D₈	C₂³	semidirect
82. C₂×C₃×C₈	C₂³	semidirect
83. #215	C₂³	semidirect
84. #213	C₂³	semidirect
85. #208	C₂³	semidirect
87. D₂₄	C₂³	semidirect
88. D₂₄	C₂³	semidirect
89. ΓL(1,𝔽₂₅)	C₂³	semidirect
90. D₃×C₈	C₂³	semidirect
96. C₂×#41	C₂×D₃	semidirect
104. #135	C₂×D₃	semidirect
105. C₂×Q₁₆	C₂×D₃	semidirect
107. C₂×SD₁₆	C₂×D₃	semidirect
113. C₂×D₈	C₂×D₃	semidirect
118. C₂²×C₈	C₂×D₃	semidirect
161. #41	C₂²×D₃	semidirect
182. Q₁₆	C₂²×D₃	semidirect
184. SD₁₆	C₂²×D₃	semidirect
188. D₈	C₂²×D₃	semidirect
195. C₂×C₈	C₂²×D₃	semidirect
248. C₂×C₃	#576	semidirect
268. C₃	C₂×#576	semidirect
273. C₂	#5769	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×#1009	2
C₂×C₃×#135	2
C₂×#949	2
#5769	2
C₂×#919	2
C₂×#914	2
C₂×D₃×D₈	2
C₂²×D₂₄	2
C₂×#901	2
C₂×#576	3

Subgroups — C2×#5769 #87184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5769 #87184