Subgroups — C₂×#5757 #87172

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×#1004	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#932	C₂	1	2	yes	maximal, complemented
4	#5757	C₂	1	16	yes	maximal, direct product factor
5	C₂×#912	C₂	1	2	yes	maximal, complemented
6	C₂×#911	C₂	1	4	yes	maximal, complemented
7	C₂×D₃×D₈	C₂	1	2	yes	maximal, complemented
8	C₂×#904	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	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
12	C₂²×#67	—	2	4	no	—
13	#1004	—	2	8	no	—
14	#1004	C₂²	1	16	yes	complemented
15	C₂×#237	—	2	2	no	—
16	C₂×#237	C₂²	1	4	yes	complemented
17	C₂×D₃×D₄	—	2	2	no	—
18	C₂×D₃×D₄	C₂²	1	4	yes	complemented
19	C₂×#235	C₂²	1	1	yes	characteristic
20	C₂×#235	C₂²	1	2	yes	complemented
21	C₂×C₃×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃×D₈	C₂²	1	2	yes	complemented
23	C₂×C₃×D₈	C₂²	1	8	yes	complemented
24	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×#214	C₂²	1	4	yes	complemented
26	#932	C₂²	1	16	yes	complemented
27	C₂×#213	C₂²	1	4	yes	complemented
28	C₂×#208	C₂²	1	1	yes	characteristic, complemented
29	#912	C₂²	1	16	yes	complemented
30	#911	C₂²	1	32	yes	complemented
31	D₃×D₈	C₂²	1	16	yes	complemented
32	C₂×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
33	#904	C₂²	1	8	yes	complemented
34	C₂×D₂₄	C₂²	1	1	yes	characteristic, complemented
35	C₂×#204	C₂²	1	2	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	—	3	1	no	—
45	C₂²×D₈	—	3	2	no	—
46	C₂²×D₈	D₃	1	1	yes	characteristic, complemented
47	C₂×#131	—	3	1	no	—
48	C₂⁴×C₃	—	2	2	no	—
49	C₂³×D₃	—	2	4	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂×C₃×D₄	—	2	4	no	—
52	C₂×C₃×D₄	C₂³	1	4	yes	—
53	C₂×C₃×D₄	C₂³	1	8	yes	complemented
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₂×#67	—	2	2	no	—
56	C₂×#67	—	2	4	no	—
57	C₂×#67	—	2	32	no	—
58	C₂×#67	—	2	4	no	—
59	C₂×#67	D₄	1	2	yes	—
60	C₂²×Dic₃	—	2	4	no	—
61	#237	—	2	8	no	—
62	#237	—	2	16	no	—
63	#237	C₂³	1	16	yes	complemented
64	D₃×D₄	—	2	16	no	—
65	D₃×D₄	—	2	8	no	—
66	D₃×D₄	C₂³	1	16	yes	complemented
67	#235	—	2	4	no	—
68	#235	D₄	1	4	yes	—
69	#235	C₂³	1	4	yes	—
70	#235	C₂³	1	8	yes	complemented
71	C₂×D₁₂	D₄	1	1	yes	characteristic
72	C₂×D₁₂	C₂³	1	2	yes	—
73	C₂×C₄×D₃	—	2	2	no	—
74	C₂×C₄×D₃	C₂³	1	2	yes	—
75	C₂×Dic₆	D₄	1	1	yes	characteristic
76	C₂×Dic₆	C₂³	1	2	yes	—
77	C₃×D₈	C₂³	1	8	yes	complemented
78	C₃×D₈	C₂³	1	8	yes	complemented
79	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂³	1	4	yes	complemented
82	#214	C₂³	1	16	yes	complemented
83	#213	C₂³	1	16	yes	complemented
84	#208	C₂³	1	4	yes	complemented
85	C₂×C₃⋊C₈	C₂³	1	2	yes	—
86	Dic₁₂	C₂³	1	4	yes	complemented
87	D₂₄	C₂³	1	4	yes	complemented
88	#204	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	—	3	2	no	—
96	C₂×#41	—	6	2	no	—
97	C₂²×D₄	—	3	4	no	—
98	C₂²×D₄	—	6	2	no	—
99	C₂²×D₄	—	6	4	no	—
100	C₂²×D₄	C₂×D₃	1	2	yes	complemented
101	AGL(1,ℤ/8)	—	3	16	no	—
102	AGL(1,ℤ/8)	—	3	32	no	—
103	#135	—	3	16	no	—
104	#135	—	3	8	no	—
105	C₂×Q₁₆	—	3	1	no	—
106	C₂×SD₁₆	—	3	4	no	—
107	C₂×SD₁₆	—	3	2	no	—
108	C₂×D₈	—	3	4	no	—
109	C₂×D₈	—	3	16	no	—
110	C₂×D₈	—	3	1	no	—
111	C₂×D₈	C₂×D₃	1	2	yes	complemented
112	C₂×D₈	C₂×D₃	1	2	yes	complemented
113	C₂×D₈	C₂×D₃	1	8	yes	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	4	no	—
120	C₂³×C₃	—	2	8	no	—
121	C₂³×C₃	—	8	4	no	—
122	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×D₃	—	2	2	no	—
124	C₂²×D₃	—	2	16	no	—
125	C₂²×D₃	—	8	8	no	—
126	C₂²×D₃	C₂×D₄	1	2	yes	—
127	C₃×D₄	—	2	4	no	—
128	C₃×D₄	—	2	8	no	—
129	C₃×D₄	C₂⁴	1	8	yes	—
130	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂⁴	1	2	yes	—
134	#67	—	2	8	no	—
135	#67	—	2	16	no	—
136	#67	—	2	16	no	—
137	#67	C₂×D₄	1	8	yes	—
138	C₂×Dic₃	—	2	2	no	—
139	C₂×Dic₃	—	2	16	no	—
140	C₂×Dic₃	C₂×D₄	1	2	yes	—
141	D₁₂	—	2	2	no	—
142	D₁₂	C₂×D₄	1	4	yes	—
143	D₁₂	C₂⁴	1	4	yes	—
144	C₄×D₃	—	2	8	no	—
145	C₄×D₃	C₂×D₄	1	4	yes	—
146	C₄×D₃	C₂⁴	1	4	yes	—
147	Dic₆	—	2	2	no	—
148	Dic₆	C₂×D₄	1	4	yes	—
149	Dic₆	C₂⁴	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	—	2	2	no	—
154	C₂⁴	—	6	4	no	—
155	#41	—	3	4	no	—
156	#41	—	3	4	no	—
157	#41	—	3	16	no	—
158	#41	—	3	8	no	—
159	#41	—	6	8	no	—
160	#41	—	6	16	no	—
161	#41	—	6	4	no	—
162	C₂×Q₈	—	3	1	no	—
163	C₂×Q₈	—	3	2	no	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	3	2	no	—
167	C₂×D₄	—	3	16	no	—
168	C₂×D₄	—	3	1	no	—
169	C₂×D₄	—	3	2	no	—
170	C₂×D₄	—	6	8	no	—
171	C₂×D₄	—	6	4	no	—
172	C₂×D₄	—	6	16	no	—
173	C₂×D₄	—	6	32	no	—
174	C₂×D₄	—	6	4	no	—
175	C₂×D₄	—	6	2	no	—
176	C₂×D₄	C₂²×D₃	1	4	yes	—
177	C₂×D₄	C₂²×D₃	1	8	yes	complemented
178	C₂²×C₄	—	3	2	no	—
179	C₂²×C₄	—	6	4	no	—
180	C₂²×C₄	—	6	2	no	—
181	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
182	Q₁₆	—	3	4	no	—
183	SD₁₆	—	3	16	no	—
184	SD₁₆	—	3	8	no	—
185	D₈	—	3	16	no	—
186	D₈	—	3	4	no	—
187	D₈	C₂²×D₃	1	8	yes	complemented
188	D₈	C₂²×D₃	1	8	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	8	no	—
198	C₂²×C₃	—	4	8	no	—
199	C₂²×C₃	—	8	8	no	—
200	C₂²×C₃	C₂²×D₄	1	2	yes	—
201	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×D₃	—	2	4	no	—
203	C₂×D₃	—	2	4	no	—
204	C₂×D₃	—	4	4	no	—
205	C₂×D₃	—	8	16	no	—
206	C₂×D₃	C₂²×D₄	1	4	yes	—
207	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
208	C₃×C₄	C₂²×D₄	1	2	yes	—
209	C₃×C₄	C₂⁵	1	1	yes	characteristic
210	Dic₃	—	2	4	no	—
211	Dic₃	C₂²×D₄	1	4	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	3	2	no	—
215	C₂³	—	6	16	no	—
216	C₂³	—	6	2	no	—
217	C₂³	—	8	4	no	—
218	C₂³	—	24	8	no	—
219	C₂³	D₃×D₄	1	1	yes	characteristic
220	Q₈	—	3	4	no	—
221	Q₈	—	3	4	no	—
222	Q₈	—	6	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	8	no	—
225	D₄	—	3	8	no	—
226	D₄	—	3	4	no	—
227	D₄	—	3	4	no	—
228	D₄	—	6	16	no	—
229	D₄	—	6	16	no	—
230	D₄	—	6	8	no	—
231	D₄	—	6	2	no	—
232	D₄	C₂³×D₃	1	8	yes	—
233	C₂×C₄	—	3	2	no	—
234	C₂×C₄	—	3	4	no	—
235	C₂×C₄	—	3	4	no	—
236	C₂×C₄	—	6	16	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	—	6	8	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	8	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	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	3	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	6	4	no	—
257	C₂²	—	6	4	no	—
258	C₂²	—	8	8	no	—
259	C₂²	—	12	4	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	—	3	4	no	—
264	C₄	—	6	4	no	—
265	C₄	C₂×D₃×D₄	1	1	yes	characteristic
266	C₄	C₂×D₃×D₄	1	2	yes	—
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	8	no	—
271	C₂	—	6	4	no	—
272	C₂	—	12	4	no	—
273	C₂	#5757	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	C₂²×C₃×D₈	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₂²×D₈	D₃
hypocenter	C₃	C₂×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#1004	C₂	semidirect
2. C₂²×C₃×D₈	C₂	semidirect
3. C₂×#932	C₂	semidirect
4. #5757	C₂	direct (N × H)
5. C₂×#912	C₂	semidirect
6. C₂×#911	C₂	semidirect
7. C₂×D₃×D₈	C₂	semidirect
8. C₂×#904	C₂	semidirect
9. C₂×#901	C₂	semidirect
11. C₂²×C₃×D₄	C₂²	semidirect
14. #1004	C₂²	semidirect
16. C₂×#237	C₂²	semidirect
18. C₂×D₃×D₄	C₂²	semidirect
20. C₂×#235	C₂²	semidirect
21. C₂×C₃×D₈	C₂²	semidirect
22. C₂×C₃×D₈	C₂²	semidirect
23. C₂×C₃×D₈	C₂²	semidirect
24. C₂²×C₃×C₈	C₂²	semidirect
25. C₂×#214	C₂²	semidirect
26. #932	C₂²	semidirect
27. C₂×#213	C₂²	semidirect
28. C₂×#208	C₂²	semidirect
29. #912	C₂²	semidirect
30. #911	C₂²	semidirect
31. D₃×D₈	C₂²	semidirect
32. C₂×Dic₁₂	C₂²	semidirect
33. #904	C₂²	semidirect
34. C₂×D₂₄	C₂²	semidirect
35. C₂×#204	C₂²	semidirect
36. #901	C₂²	semidirect
37. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
38. C₂×D₃×C₈	C₂²	semidirect
46. C₂²×D₈	D₃	semidirect
53. C₂×C₃×D₄	C₂³	semidirect
63. #237	C₂³	semidirect
66. D₃×D₄	C₂³	semidirect
70. #235	C₂³	semidirect
77. C₃×D₈	C₂³	semidirect
78. C₃×D₈	C₂³	semidirect
81. C₂×C₃×C₈	C₂³	semidirect
82. #214	C₂³	semidirect
83. #213	C₂³	semidirect
84. #208	C₂³	semidirect
86. Dic₁₂	C₂³	semidirect
87. D₂₄	C₂³	semidirect
88. #204	C₂³	semidirect
89. ΓL(1,𝔽₂₅)	C₂³	semidirect
90. D₃×C₈	C₂³	semidirect
100. C₂²×D₄	C₂×D₃	semidirect
111. C₂×D₈	C₂×D₃	semidirect
112. C₂×D₈	C₂×D₃	semidirect
113. C₂×D₈	C₂×D₃	semidirect
118. C₂²×C₈	C₂×D₃	semidirect
177. C₂×D₄	C₂²×D₃	semidirect
187. D₈	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	#5757	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×#1004	2
C₂²×C₃×D₈	2
C₂×#932	2
#5757	2
C₂×#912	2
C₂×#911	2
C₂×D₃×D₈	2
C₂×#904	2
C₂×#901	2
C₂×#576	3

Subgroups — C2×#5757 #87172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5757 #87172