Subgroups — C₂×#5830 #87266

Order 384.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#5830	C₂	1	16	yes	maximal, direct product factor
4	C₂×#947	C₂	1	2	yes	maximal, complemented
5	C₂×#940	C₂	1	2	yes	maximal, complemented
6	C₂×#937	C₂	1	4	yes	maximal, complemented
7	C₂×#930	C₂	1	4	yes	maximal, complemented
8	C₂×#926	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#535	—	3	1	no	maximal, Sylow
10	C₂⁴×D₃	C₂²	1	2	yes	—
11	C₂×C₃×#41	—	2	8	no	—
12	C₂×C₃×#41	C₂²	1	8	yes	complemented
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₂²	1	2	yes	complemented
18	C₂²×#67	C₂²	1	8	yes	complemented
19	C₂²×D₁₂	—	2	8	no	—
20	C₂²×D₁₂	C₂²	1	2	yes	—
21	C₂²×D₁₂	C₂²	1	2	yes	—
22	C₂²×D₁₂	C₂²	1	8	yes	complemented
23	C₂×#217	C₂²	1	2	yes	—
24	#947	C₂²	1	16	yes	complemented
25	#940	C₂²	1	16	yes	complemented
26	#937	C₂²	1	32	yes	complemented
27	#930	C₂²	1	32	yes	complemented
28	C₂×(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
29	#926	C₂²	1	8	yes	complemented
30	C₂×C₄⋊Dic₃	C₂²	1	2	yes	complemented
31	C₂×C₄×Dic₃	C₂²	1	2	yes	complemented
32	C₂²×#41	D₃	1	1	yes	characteristic, complemented
33	C₂³×D₄	—	3	1	no	—
34	#535	—	3	16	no	—
35	C₂×#127	—	3	2	no	—
36	C₂×#124	—	3	2	no	—
37	C₂×#121	—	3	4	no	—
38	C₂×#120	—	3	4	no	—
39	C₂×#117	—	3	1	no	—
40	C₂⁴×C₃	C₂³	1	2	yes	—
41	C₂⁴×C₃	C₂³	1	1	yes	characteristic
42	C₂³×D₃	—	2	8	no	—
43	C₂³×D₃	—	4	8	no	—
44	C₂³×D₃	C₂³	1	4	yes	—
45	C₂³×D₃	C₂³	1	8	yes	—
46	C₃×#41	—	2	32	no	—
47	C₂×C₃×Q₈	—	2	4	no	—
48	C₂×C₃×Q₈	C₂³	1	4	yes	complemented
49	C₂×C₃×D₄	—	2	8	no	—
50	C₂×C₃×D₄	—	2	4	no	—
51	C₂×C₃×D₄	C₂³	1	8	yes	—
52	C₂×C₃×D₄	C₂³	1	4	yes	complemented
53	C₂²×C₃×C₄	—	2	8	no	—
54	C₂²×C₃×C₄	D₄	1	8	yes	complemented
55	C₂²×C₃×C₄	C₂³	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	4	yes	—
59	C₂²×C₃×C₄	C₂³	1	8	yes	complemented
60	C₂×#67	—	4	16	no	—
61	C₂×#67	C₂³	1	32	yes	complemented
62	C₂²×Dic₃	C₂³	1	4	yes	—
63	C₂×D₁₂	—	2	8	no	—
64	C₂×D₁₂	—	2	8	no	—
65	C₂×D₁₂	—	2	32	no	—
66	C₂×D₁₂	C₂³	1	8	yes	—
67	C₂×D₁₂	C₂³	1	8	yes	—
68	#217	C₂³	1	8	yes	—
69	(C₂×D₃)⋊C₄	C₂³	1	32	yes	complemented
70	C₄⋊Dic₃	C₂³	1	8	yes	complemented
71	C₄×Dic₃	C₂³	1	8	yes	complemented
72	C₂⁵	—	3	2	no	—
73	C₂×#41	—	2	8	no	—
74	C₂×#41	C₂×D₃	1	8	yes	complemented
75	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
76	C₂²×D₄	—	3	8	no	—
77	C₂²×D₄	—	3	2	no	—
78	C₂²×D₄	—	3	2	no	—
79	C₂²×D₄	—	3	8	no	—
80	C₂²×D₄	—	6	8	no	—
81	C₂²×D₄	C₂×D₃	1	2	yes	—
82	C₂²×D₄	C₂×D₃	1	1	yes	characteristic, complemented
83	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₃	1	2	yes	complemented
85	#127	—	3	16	no	—
86	#124	—	3	16	no	—
87	#121	—	3	32	no	—
88	#120	—	3	32	no	—
89	#117	—	3	8	no	—
90	C₂×C₄⋊C₄	—	3	2	no	—
91	C₂×C₂²⋊C₄	—	3	2	no	—
92	C₂×C₂²⋊C₄	—	3	8	no	—
93	C₂×C₄²	—	3	2	no	—
94	C₂³×C₃	—	2	4	no	—
95	C₂³×C₃	—	2	4	no	—
96	C₂³×C₃	—	2	8	no	—
97	C₂³×C₃	C₂×D₄	1	4	yes	—
98	C₂³×C₃	C₂⁴	1	4	yes	—
99	C₂³×C₃	C₂⁴	1	2	yes	—
100	C₂³×C₃	C₂⁴	1	1	yes	characteristic
101	C₂²×D₃	—	2	16	no	—
102	C₂²×D₃	—	2	8	no	—
103	C₂²×D₃	—	2	16	no	—
104	C₂²×D₃	—	4	16	no	—
105	C₂²×D₃	—	4	32	no	—
106	C₂²×D₃	C₂⁴	1	8	yes	—
107	C₃×Q₈	—	2	8	no	—
108	C₃×D₄	—	2	16	no	—
109	C₃×D₄	—	2	8	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	16	no	—
112	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
113	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	16	yes	complemented
115	C₂×C₃×C₄	C₂⁴	1	4	yes	—
116	C₂×C₃×C₄	C₂⁴	1	2	yes	—
117	C₂×C₃×C₄	C₂⁴	1	2	yes	—
118	#67	—	4	32	no	—
119	C₂×Dic₃	—	2	8	no	—
120	C₂×Dic₃	C₂⁴	1	8	yes	—
121	D₁₂	—	2	16	no	—
122	D₁₂	—	2	16	no	—
123	C₂⁴	—	3	4	no	—
124	C₂⁴	—	3	8	no	—
125	C₂⁴	—	6	8	no	—
126	C₂⁴	—	12	8	no	—
127	C₂⁴	C₂²×D₃	1	2	yes	—
128	C₂⁴	C₂²×D₃	1	1	yes	characteristic
129	#41	—	2	32	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	C₂²×D₃	1	4	yes	complemented
132	C₂×D₄	—	2	8	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	3	32	no	—
135	C₂×D₄	—	3	8	no	—
136	C₂×D₄	—	3	8	no	—
137	C₂×D₄	—	6	8	no	—
138	C₂×D₄	—	6	8	no	—
139	C₂×D₄	—	6	32	no	—
140	C₂×D₄	—	12	16	no	—
141	C₂×D₄	C₂²×D₃	1	8	yes	—
142	C₂×D₄	C₂²×D₃	1	4	yes	complemented
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	3	4	no	—
145	C₂²×C₄	#67	1	8	yes	complemented
146	C₂²×C₄	C₂²×D₃	1	2	yes	—
147	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₃	1	4	yes	—
150	C₂²×C₄	C₂²×D₃	1	8	yes	complemented
151	C₄⋊C₄	—	3	8	no	—
152	C₂²⋊C₄	—	3	8	no	—
153	C₂²⋊C₄	—	3	32	no	—
154	C₄²	—	3	8	no	—
155	C₂²×C₃	—	2	8	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	2	8	no	—
159	C₂²×C₃	—	4	8	no	—
160	C₂²×C₃	C₂²×D₄	1	4	yes	—
161	C₂²×C₃	C₂²×D₄	1	2	yes	—
162	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
163	C₂²×C₃	C₂⁵	1	1	yes	characteristic
164	C₂×D₃	—	2	8	no	—
165	C₂×D₃	—	2	16	no	—
166	C₂×D₃	—	4	32	no	—
167	C₂×D₃	—	4	16	no	—
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	C₂²×D₄	1	4	yes	—
170	C₃×C₄	C₂²×D₄	1	4	yes	—
171	Dic₃	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	8	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	3	8	no	—
176	C₂³	—	6	16	no	—
177	C₂³	—	6	16	no	—
178	C₂³	—	6	8	no	—
179	C₂³	—	12	16	no	—
180	C₂³	—	12	32	no	—
181	C₂³	C₂×#67	1	4	yes	—
182	C₂³	C₂³×D₃	1	4	yes	—
183	C₂³	C₂³×D₃	1	2	yes	—
184	C₂³	C₂³×D₃	1	1	yes	characteristic
185	Q₈	—	2	8	no	—
186	D₄	—	2	16	no	—
187	D₄	—	2	8	no	—
188	D₄	—	6	16	no	—
189	D₄	—	6	16	no	—
190	D₄	—	12	32	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	3	8	no	—
194	C₂×C₄	—	6	8	no	—
195	C₂×C₄	C₂×#67	1	4	yes	—
196	C₂×C₄	C₂×#67	1	4	yes	—
197	C₂×C₄	C₂×#67	1	16	yes	complemented
198	C₂×C₄	C₂³×D₃	1	4	yes	—
199	C₂×C₄	C₂³×D₃	1	2	yes	—
200	C₂×C₄	C₂³×D₃	1	2	yes	—
201	C₂×C₃	—	2	4	no	—
202	C₂×C₃	—	4	4	no	—
203	C₂×C₃	#535	1	4	yes	complemented
204	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
205	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
206	D₃	—	4	8	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	8	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	6	16	no	—
213	C₂²	—	6	8	no	—
214	C₂²	—	12	32	no	—
215	C₂²	—	12	16	no	—
216	C₂²	#1009	1	4	yes	—
217	C₂²	C₂²×#67	1	4	yes	—
218	C₂²	C₂²×#67	1	2	yes	—
219	C₂²	C₂⁴×D₃	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	6	8	no	—
222	C₄	C₂²×#67	1	4	yes	—
223	C₄	C₂²×#67	1	4	yes	—
224	C₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	—	12	8	no	—
228	C₂	#5830	1	4	yes	direct product factor
229	C₂	C₂×#1009	1	2	yes	—
230	C₂	C₂³×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×#41	C₂
socle	C₂³×C₃	C₂⁴
hypercenter	C₂²×#41	D₃
hypocenter	C₃	C₂×#535
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#41	C₂	semidirect
2. C₂³×D₁₂	C₂	semidirect
3. #5830	C₂	direct (N × H)
4. C₂×#947	C₂	semidirect
5. C₂×#940	C₂	semidirect
6. C₂×#937	C₂	semidirect
7. C₂×#930	C₂	semidirect
8. C₂×#926	C₂	semidirect
12. C₂×C₃×#41	C₂²	semidirect
13. C₂²×C₃×Q₈	C₂²	semidirect
15. C₂²×C₃×D₄	C₂²	semidirect
17. C₂³×C₃×C₄	C₂²	semidirect
18. C₂²×#67	C₂²	semidirect
22. C₂²×D₁₂	C₂²	semidirect
24. #947	C₂²	semidirect
25. #940	C₂²	semidirect
26. #937	C₂²	semidirect
27. #930	C₂²	semidirect
28. C₂×(C₂×D₃)⋊C₄	C₂²	semidirect
29. #926	C₂²	semidirect
30. C₂×C₄⋊Dic₃	C₂²	semidirect
31. C₂×C₄×Dic₃	C₂²	semidirect
32. C₂²×#41	D₃	semidirect
48. C₂×C₃×Q₈	C₂³	semidirect
52. C₂×C₃×D₄	C₂³	semidirect
54. C₂²×C₃×C₄	D₄	semidirect
59. C₂²×C₃×C₄	C₂³	semidirect
61. C₂×#67	C₂³	semidirect
69. (C₂×D₃)⋊C₄	C₂³	semidirect
70. C₄⋊Dic₃	C₂³	semidirect
71. C₄×Dic₃	C₂³	semidirect
74. C₂×#41	C₂×D₃	semidirect
75. C₂²×Q₈	C₂×D₃	semidirect
82. C₂²×D₄	C₂×D₃	semidirect
84. C₂³×C₄	C₂×D₃	semidirect
114. C₂×C₃×C₄	C₂×D₄	semidirect
131. C₂×Q₈	C₂²×D₃	semidirect
142. C₂×D₄	C₂²×D₃	semidirect
145. C₂²×C₄	#67	semidirect
150. C₂²×C₄	C₂²×D₃	semidirect
197. C₂×C₄	C₂×#67	semidirect
203. C₂×C₃	#535	semidirect
224. C₃	C₂×#535	semidirect
228. C₂	#5830	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×C₃×#41	2
C₂³×D₁₂	2
#5830	2
C₂×#947	2
C₂×#940	2
C₂×#937	2
C₂×#930	2
C₂×#926	2
C₂×#535	3

Subgroups — C2×#5830 #87266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5830 #87266