Subgroups — C₂×H(3,𝔽₄)×D₇₁ #32904323

Order 18176.

10246 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986742	C₂	1	6	yes	maximal, complemented
2	C₂×#17986741	C₂	1	9	yes	maximal, complemented
3	H(3,𝔽₄)×D₇₁	C₂	1	32	yes	maximal, direct product factor
4	C₂×#124×D₇₁	C₂	1	9	yes	maximal, complemented
5	C₂×#120×D₇₁	C₂	1	6	yes	maximal, complemented
6	C₂×H(3,𝔽₄)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₁	C₂²	1	2	yes	complemented
8	C₂²×Q₈×D₇₁	C₂²	1	3	yes	complemented
9	C₂²×D₄×D₇₁	C₂²	1	9	yes	—
10	#17986742	C₂²	1	96	yes	complemented
11	#17986741	C₂²	1	144	yes	complemented
12	C₂×#14556117	C₂²	1	9	yes	complemented
13	C₂×#14556116	C₂²	1	9	yes	complemented
14	C₂×#14556115	C₂²	1	9	yes	complemented
15	C₂×#14556114	C₂²	1	36	yes	complemented
16	#124×D₇₁	C₂²	1	144	yes	complemented
17	C₂×#14556098	C₂²	1	18	yes	complemented
18	C₂×#14556097	C₂²	1	6	yes	complemented
19	C₂×#14556096	C₂²	1	18	yes	complemented
20	#120×D₇₁	C₂²	1	96	yes	complemented
21	C₂×C₂²⋊C₄×D₇₁	C₂²	1	18	yes	complemented
22	C₂×C₄²×D₇₁	C₂²	1	3	yes	complemented
23	H(3,𝔽₄)×C₇₁	C₂²	1	16	yes	complemented
24	C₂×#124×C₇₁	C₂²	1	9	yes	complemented
25	C₂×#120×C₇₁	C₂²	1	6	yes	complemented
26	C₂⁴×D₇₁	—	4	24	no	—
27	C₂⁴×D₇₁	C₂³	1	6	yes	—
28	C₂⁴×D₇₁	C₂³	1	6	yes	complemented
29	C₂⁴×D₇₁	C₂³	1	16	yes	complemented
30	C₂²×Dic₁₄₂	C₂³	1	9	yes	complemented
31	C₂×Q₈×D₇₁	—	4	12	no	—
32	C₂×Q₈×D₇₁	C₂³	1	24	yes	complemented
33	C₂²×D₂₈₄	C₂³	1	9	yes	—
34	C₂²×C₂²⋊D₇₁	C₂³	1	18	yes	—
35	C₂×D₄×D₇₁	—	4	36	no	—
36	C₂×D₄×D₇₁	C₂³	1	72	yes	—
37	C₂²×C₄×D₇₁	C₂³	1	9	yes	—
38	#14556117	C₂³	1	72	yes	complemented
39	#14556116	C₂³	1	72	yes	complemented
40	#14556115	C₂³	1	72	yes	complemented
41	#14556114	C₂³	1	288	yes	complemented
42	#14556098	C₂³	1	144	yes	complemented
43	#14556097	C₂³	1	48	yes	complemented
44	#14556096	C₂³	1	144	yes	complemented
45	C₂×C₂²⋊Dic₇₁	C₂³	1	18	yes	complemented
46	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	36	yes	complemented
47	C₂²⋊C₄×D₇₁	C₂³	1	144	yes	complemented
48	C₂×C₄×Dic₇₁	C₂³	1	9	yes	complemented
49	C₄²×D₇₁	C₂³	1	24	yes	complemented
50	C₂⁵×C₇₁	C₂³	1	2	yes	complemented
51	C₂²×Q₈×C₇₁	C₂³	1	3	yes	complemented
52	C₂²×D₄×C₇₁	C₂³	1	9	yes	—
53	#124×C₇₁	C₂³	1	72	yes	complemented
54	#120×C₇₁	C₂³	1	48	yes	complemented
55	C₂×C₂²⋊C₄×C₇₁	C₂³	1	18	yes	complemented
56	C₂×C₄²×C₇₁	C₂³	1	3	yes	complemented
57	C₂³×D₇₁	—	2	72	no	—
58	C₂³×D₇₁	—	4	96	no	—
59	C₂³×D₇₁	—	4	36	no	—
60	C₂³×D₇₁	—	4	128	no	—
61	C₂³×D₇₁	C₂⁴	1	24	yes	—
62	C₂³×D₇₁	C₂⁴	1	6	yes	—
63	C₂³×D₇₁	C₂⁴	1	1	yes	characteristic
64	C₂³×D₇₁	C₂⁴	1	24	yes	complemented
65	C₂×Dic₁₄₂	—	4	18	no	—
66	C₂×Dic₁₄₂	C₂⁴	1	36	yes	complemented
67	Q₈×D₇₁	—	4	48	no	—
68	C₂×D₂₈₄	—	4	18	no	—
69	C₂×D₂₈₄	C₂⁴	1	36	yes	—
70	C₂×C₂²⋊D₇₁	—	4	36	no	—
71	C₂×C₂²⋊D₇₁	C₂⁴	1	72	yes	—
72	D₄×D₇₁	—	4	144	no	—
73	C₂²×Dic₇₁	C₂⁴	1	9	yes	—
74	C₂×C₄×D₇₁	—	2	36	no	—
75	C₂×C₄×D₇₁	C₂⁴	1	36	yes	—
76	C₂²⋊Dic₇₁	C₂⁴	1	72	yes	complemented
77	(C₂×D₇₁)⋊C₄	C₂⁴	1	144	yes	complemented
78	C₄×Dic₇₁	C₂⁴	1	36	yes	complemented
79	C₂⁴×C₇₁	—	4	12	no	—
80	C₂⁴×C₇₁	C₂⁴	1	6	yes	—
81	C₂⁴×C₇₁	C₂⁴	1	8	yes	complemented
82	C₂×Q₈×C₇₁	—	4	6	no	—
83	C₂×Q₈×C₇₁	C₂⁴	1	12	yes	complemented
84	C₂×D₄×C₇₁	—	4	18	no	—
85	C₂×D₄×C₇₁	C₂⁴	1	36	yes	—
86	C₂²×C₄×C₇₁	C₂⁴	1	9	yes	—
87	C₂²⋊C₄×C₇₁	C₂⁴	1	72	yes	complemented
88	C₄²×C₇₁	C₂⁴	1	12	yes	complemented
89	C₂³×C₇₁	—	2	36	no	—
90	C₂³×C₇₁	—	4	24	no	—
91	C₂³×C₇₁	—	4	32	no	—
92	C₂³×C₇₁	C₂⁵	1	12	yes	—
93	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
94	C₂²×D₇₁	—	2	36	no	—
95	C₂²×D₇₁	—	2	144	no	—
96	C₂²×D₇₁	—	4	72	no	—
97	C₂²×D₇₁	—	4	96	no	—
98	C₂²×D₇₁	—	4	256	no	—
99	C₂²×D₇₁	—	4	96	no	—
100	C₂²×D₇₁	H(5,𝔽₂)	1	12	yes	—
101	C₂²×D₇₁	C₂⁵	1	12	yes	—
102	C₂²×D₇₁	C₂⁵	1	2	yes	—
103	Q₈×C₇₁	—	4	12	no	—
104	D₄×C₇₁	—	4	36	no	—
105	C₂×C₄×C₇₁	—	2	18	no	—
106	C₂×C₄×C₇₁	C₂⁵	1	18	yes	—
107	C₂²⋊D₇₁	—	4	72	no	—
108	C₂×Dic₇₁	—	2	18	no	—
109	C₂×Dic₇₁	C₂⁵	1	18	yes	—
110	D₂₈₄	—	4	36	no	—
111	C₄×D₇₁	—	2	72	no	—
112	Dic₁₄₂	—	4	36	no	—
113	C₂²×C₇₁	—	2	36	no	—
114	C₂²×C₇₁	—	4	32	no	—
115	C₂²×C₇₁	—	4	24	no	—
116	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	6	yes	—
117	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
118	C₂×D₇₁	—	2	36	no	—
119	C₂×D₇₁	—	4	96	no	—
120	C₂×D₇₁	—	4	24	no	—
121	C₂×D₇₁	—	4	96	no	—
122	C₂×D₇₁	H(3,𝔽₄)	1	16	yes	direct product factor
123	C₂×D₇₁	C₂×H(5,𝔽₂)	1	12	yes	—
124	C₄×C₇₁	—	2	18	no	—
125	Dic₇₁	—	2	18	no	—
126	C₂²×H(3,𝔽₄)	—	71	1	no	maximal, Sylow
127	C₂×C₇₁	—	4	12	no	—
128	C₂×C₇₁	C₂×H(3,𝔽₄)	1	4	yes	complemented
129	C₂×C₇₁	C₂²×H(5,𝔽₂)	1	3	yes	—
130	D₇₁	—	4	12	no	—
131	D₇₁	C₂×H(3,𝔽₄)	1	8	yes	direct product factor
132	C₂×H(3,𝔽₄)	—	71	32	no	—
133	C₂×H(3,𝔽₄)	—	71	9	no	—
134	C₂×H(3,𝔽₄)	—	71	6	no	—
135	C₂×H(3,𝔽₄)	D₇₁	1	1	yes	characteristic, direct product factor
136	C₂²×#124	—	71	9	no	—
137	C₂²×#120	—	71	6	no	—
138	C₇₁	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
139	C₂⁶	—	71	2	no	—
140	C₂³×Q₈	—	71	3	no	—
141	C₂³×D₄	—	71	9	no	—
142	H(3,𝔽₄)	—	71	144	no	—
143	H(3,𝔽₄)	—	71	96	no	—
144	H(3,𝔽₄)	C₂×D₇₁	1	16	yes	direct product factor
145	C₂×#124	—	71	9	no	—
146	C₂×#124	—	71	9	no	—
147	C₂×#124	—	71	144	no	—
148	C₂×#124	—	71	36	no	—
149	C₂×#124	—	71	9	no	—
150	C₂×#124	C₂×D₇₁	1	9	yes	complemented
151	C₂×#120	—	71	6	no	—
152	C₂×#120	—	71	18	no	—
153	C₂×#120	—	71	96	no	—
154	C₂×#120	—	71	18	no	—
155	C₂×#120	C₂×D₇₁	1	6	yes	complemented
156	C₂²×C₂²⋊C₄	—	71	18	no	—
157	C₂²×C₄²	—	71	3	no	—
158	C₂⁵	—	284	24	no	—
159	C₂⁵	—	71	6	no	—
160	C₂⁵	—	71	16	no	—
161	C₂⁵	—	71	6	no	—
162	C₂⁵	C₂²×D₇₁	1	2	yes	complemented
163	C₂²×Q₈	—	284	12	no	—
164	C₂²×Q₈	—	71	24	no	—
165	C₂²×Q₈	—	71	9	no	—
166	C₂²×Q₈	C₂²×D₇₁	1	3	yes	complemented
167	C₂²×D₄	—	284	36	no	—
168	C₂²×D₄	—	71	18	no	—
169	C₂²×D₄	—	71	72	no	—
170	C₂²×D₄	—	71	9	no	—
171	C₂²×D₄	C₂²×D₇₁	1	9	yes	—
172	C₂³×C₄	—	71	9	no	—
173	#124	—	71	72	no	—
174	#124	—	71	72	no	—
175	#124	—	71	288	no	—
176	#124	—	71	72	no	—
177	#124	C₂²×D₇₁	1	72	yes	complemented
178	#120	—	71	48	no	—
179	#120	—	71	144	no	—
180	#120	—	71	144	no	—
181	#120	C₂²×D₇₁	1	48	yes	complemented
182	C₂×C₂²⋊C₄	—	71	18	no	—
183	C₂×C₂²⋊C₄	—	71	36	no	—
184	C₂×C₂²⋊C₄	—	71	144	no	—
185	C₂×C₂²⋊C₄	C₂²×D₇₁	1	18	yes	complemented
186	C₂×C₄²	—	71	9	no	—
187	C₂×C₄²	—	71	24	no	—
188	C₂×C₄²	C₂²×D₇₁	1	3	yes	complemented
189	C₂⁴	—	4	12	no	—
190	C₂⁴	—	284	36	no	—
191	C₂⁴	—	284	128	no	—
192	C₂⁴	—	284	96	no	—
193	C₂⁴	—	71	24	no	—
194	C₂⁴	—	71	1	no	—
195	C₂⁴	—	71	24	no	—
196	C₂⁴	—	71	6	no	—
197	C₂⁴	—	142	72	no	—
198	C₂⁴	C₂³×D₇₁	1	6	yes	—
199	C₂⁴	C₂³×D₇₁	1	8	yes	complemented
200	C₂×Q₈	—	4	6	no	—
201	C₂×Q₈	—	284	18	no	—
202	C₂×Q₈	—	284	48	no	—
203	C₂×Q₈	—	71	36	no	—
204	C₂×Q₈	C₂³×D₇₁	1	12	yes	complemented
205	C₂×D₄	—	4	18	no	—
206	C₂×D₄	—	284	18	no	—
207	C₂×D₄	—	284	144	no	—
208	C₂×D₄	—	284	36	no	—
209	C₂×D₄	—	71	72	no	—
210	C₂×D₄	—	71	36	no	—
211	C₂×D₄	C₂³×D₇₁	1	36	yes	—
212	C₂²×C₄	—	71	9	no	—
213	C₂²×C₄	—	71	36	no	—
214	C₂²×C₄	—	142	36	no	—
215	C₂²×C₄	C₂³×D₇₁	1	9	yes	—
216	C₂²⋊C₄	—	71	72	no	—
217	C₂²⋊C₄	—	71	144	no	—
218	C₂²⋊C₄	C₂³×D₇₁	1	72	yes	complemented
219	C₄²	—	71	36	no	—
220	C₄²	C₂³×D₇₁	1	12	yes	complemented
221	C₂³	—	2	36	no	—
222	C₂³	—	4	24	no	—
223	C₂³	—	4	32	no	—
224	C₂³	—	284	72	no	—
225	C₂³	—	284	256	no	—
226	C₂³	—	284	96	no	—
227	C₂³	—	284	96	no	—
228	C₂³	—	71	12	no	—
229	C₂³	—	71	2	no	—
230	C₂³	—	71	12	no	—
231	C₂³	—	142	144	no	—
232	C₂³	—	142	36	no	—
233	C₂³	C₂⁴×D₇₁	1	12	yes	—
234	C₂³	C₂⁴×D₇₁	1	1	yes	characteristic
235	Q₈	—	4	12	no	—
236	Q₈	—	284	36	no	—
237	D₄	—	4	36	no	—
238	D₄	—	284	36	no	—
239	D₄	—	284	72	no	—
240	C₂×C₄	—	2	18	no	—
241	C₂×C₄	—	71	18	no	—
242	C₂×C₄	—	142	72	no	—
243	C₂×C₄	—	142	18	no	—
244	C₂×C₄	C₂⁴×D₇₁	1	18	yes	—
245	C₂²	—	2	36	no	—
246	C₂²	—	4	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	284	96	no	—
249	C₂²	—	284	96	no	—
250	C₂²	—	284	24	no	—
251	C₂²	—	71	12	no	—
252	C₂²	—	71	16	no	—
253	C₂²	—	142	36	no	—
254	C₂²	H(5,𝔽₂)×D₇₁	1	6	yes	—
255	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
256	C₄	—	2	18	no	—
257	C₄	—	142	18	no	—
258	C₂	—	4	12	no	—
259	C₂	—	284	12	no	—
260	C₂	—	71	8	no	—
261	C₂	H(3,𝔽₄)×D₇₁	1	4	yes	direct product factor
262	C₂	C₂×H(5,𝔽₂)×D₇₁	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂×H(3,𝔽₄)×C₇₁	C₂
socle	C₂³×C₇₁	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₇₁
hypocenter	C₇₁	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. C₂×#17986742	C₂	semidirect
2. C₂×#17986741	C₂	semidirect
3. H(3,𝔽₄)×D₇₁	C₂	direct (N × H)
4. C₂×#124×D₇₁	C₂	semidirect
5. C₂×#120×D₇₁	C₂	semidirect
6. C₂×H(3,𝔽₄)×C₇₁	C₂	semidirect
7. C₂⁵×D₇₁	C₂²	semidirect
8. C₂²×Q₈×D₇₁	C₂²	semidirect
10. #17986742	C₂²	semidirect
11. #17986741	C₂²	semidirect
12. C₂×#14556117	C₂²	semidirect
13. C₂×#14556116	C₂²	semidirect
14. C₂×#14556115	C₂²	semidirect
15. C₂×#14556114	C₂²	semidirect
16. #124×D₇₁	C₂²	semidirect
17. C₂×#14556098	C₂²	semidirect
18. C₂×#14556097	C₂²	semidirect
19. C₂×#14556096	C₂²	semidirect
20. #120×D₇₁	C₂²	semidirect
21. C₂×C₂²⋊C₄×D₇₁	C₂²	semidirect
22. C₂×C₄²×D₇₁	C₂²	semidirect
23. H(3,𝔽₄)×C₇₁	C₂²	semidirect
24. C₂×#124×C₇₁	C₂²	semidirect
25. C₂×#120×C₇₁	C₂²	semidirect
28. C₂⁴×D₇₁	C₂³	semidirect
29. C₂⁴×D₇₁	C₂³	semidirect
30. C₂²×Dic₁₄₂	C₂³	semidirect
32. C₂×Q₈×D₇₁	C₂³	semidirect
38. #14556117	C₂³	semidirect
39. #14556116	C₂³	semidirect
40. #14556115	C₂³	semidirect
41. #14556114	C₂³	semidirect
42. #14556098	C₂³	semidirect
43. #14556097	C₂³	semidirect
44. #14556096	C₂³	semidirect
45. C₂×C₂²⋊Dic₇₁	C₂³	semidirect
46. C₂×(C₂×D₇₁)⋊C₄	C₂³	semidirect
47. C₂²⋊C₄×D₇₁	C₂³	semidirect
48. C₂×C₄×Dic₇₁	C₂³	semidirect
49. C₄²×D₇₁	C₂³	semidirect
50. C₂⁵×C₇₁	C₂³	semidirect
51. C₂²×Q₈×C₇₁	C₂³	semidirect
53. #124×C₇₁	C₂³	semidirect
54. #120×C₇₁	C₂³	semidirect
55. C₂×C₂²⋊C₄×C₇₁	C₂³	semidirect
56. C₂×C₄²×C₇₁	C₂³	semidirect
64. C₂³×D₇₁	C₂⁴	semidirect
66. C₂×Dic₁₄₂	C₂⁴	semidirect
76. C₂²⋊Dic₇₁	C₂⁴	semidirect
77. (C₂×D₇₁)⋊C₄	C₂⁴	semidirect
78. C₄×Dic₇₁	C₂⁴	semidirect
81. C₂⁴×C₇₁	C₂⁴	semidirect
83. C₂×Q₈×C₇₁	C₂⁴	semidirect
87. C₂²⋊C₄×C₇₁	C₂⁴	semidirect
88. C₄²×C₇₁	C₂⁴	semidirect
122. C₂×D₇₁	H(3,𝔽₄)	direct (N × H)
128. C₂×C₇₁	C₂×H(3,𝔽₄)	semidirect
131. D₇₁	C₂×H(3,𝔽₄)	direct (N × H)
135. C₂×H(3,𝔽₄)	D₇₁	direct (N × H)
138. C₇₁	C₂²×H(3,𝔽₄)	semidirect
144. H(3,𝔽₄)	C₂×D₇₁	direct (N × H)
150. C₂×#124	C₂×D₇₁	semidirect
155. C₂×#120	C₂×D₇₁	semidirect
162. C₂⁵	C₂²×D₇₁	semidirect
166. C₂²×Q₈	C₂²×D₇₁	semidirect
177. #124	C₂²×D₇₁	semidirect
181. #120	C₂²×D₇₁	semidirect
185. C₂×C₂²⋊C₄	C₂²×D₇₁	semidirect
188. C₂×C₄²	C₂²×D₇₁	semidirect
199. C₂⁴	C₂³×D₇₁	semidirect
204. C₂×Q₈	C₂³×D₇₁	semidirect
218. C₂²⋊C₄	C₂³×D₇₁	semidirect
220. C₄²	C₂³×D₇₁	semidirect
261. C₂	H(3,𝔽₄)×D₇₁	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×#17986742	2
C₂×#17986741	2
H(3,𝔽₄)×D₇₁	2
C₂×#124×D₇₁	2
C₂×#120×D₇₁	2
C₂×H(3,𝔽₄)×C₇₁	2
C₂²×H(3,𝔽₄)	71

Subgroups — C2×H(3,𝔽4)×D71 #32904323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(3,𝔽₄)×D₇₁ #32904323