Subgroups — C₂²×#16864233 #31769725

Order 17152.

60366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864233	C₂	1	96	yes	maximal, direct product factor
2	C₂²×#14524091	C₂	1	5	yes	maximal, complemented
3	C₂²×#14524089	C₂	1	10	yes	maximal, complemented
4	C₂³×#11761629	C₂	1	5	yes	maximal, complemented
5	C₂²×#41×D₆₇	C₂	1	10	yes	maximal, complemented
6	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#16864233	C₂²	1	1024	yes	direct product factor
8	C₂×#14524091	C₂²	1	240	yes	complemented
9	C₂×#14524089	C₂²	1	480	yes	complemented
10	C₂²×#11761629	C₂²	1	10	yes	complemented
11	C₂²×#11761629	C₂²	1	20	yes	complemented
12	C₂²×#11761629	C₂²	1	240	yes	complemented
13	C₂²×#11761628	C₂²	1	30	yes	complemented
14	C₂²×#11761627	C₂²	1	10	yes	complemented
15	C₂×#41×D₆₇	C₂²	1	480	yes	complemented
16	C₂²×Q₈×D₆₇	C₂²	1	10	yes	complemented
17	C₂³×D₂₆₈	C₂²	1	15	yes	complemented
18	C₂²×D₄×D₆₇	C₂²	1	30	yes	complemented
19	C₂³×C₄×D₆₇	C₂²	1	15	yes	complemented
20	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	48	yes	complemented
21	C₂²×#41×C₆₇	C₂²	1	10	yes	complemented
22	C₂³×Q₈×C₆₇	C₂²	1	5	yes	complemented
23	C₂⁴×D₆₇	C₂³	1	15	yes	—
24	#14524091	C₂³	1	1280	yes	complemented
25	#14524089	C₂³	1	2560	yes	complemented
26	C₂×#11761629	C₂³	1	240	yes	complemented
27	C₂×#11761629	C₂³	1	480	yes	complemented
28	C₂×#11761629	C₂³	1	1280	yes	complemented
29	C₂×#11761628	C₂³	1	720	yes	complemented
30	C₂×#11761627	C₂³	1	240	yes	complemented
31	#41×D₆₇	C₂³	1	2560	yes	complemented
32	C₂²×Dic₁₃₄	C₂³	1	10	yes	—
33	C₂×Q₈×D₆₇	C₂³	1	240	yes	complemented
34	C₂²×D₂₆₈	C₂³	1	30	yes	—
35	C₂²×D₂₆₈	C₂³	1	360	yes	complemented
36	C₂²×C₂²⋊D₆₇	C₂³	1	20	yes	—
37	C₂×D₄×D₆₇	C₂³	1	720	yes	complemented
38	C₂³×Dic₆₇	C₂³	1	5	yes	—
39	C₂²×C₄×D₆₇	C₂³	1	10	yes	—
40	C₂²×C₄×D₆₇	C₂³	1	30	yes	—
41	C₂²×C₄×D₆₇	C₂³	1	360	yes	complemented
42	2₋¹⁺⁴×C₆₇	C₂³	1	256	yes	complemented
43	C₂×#41×C₆₇	C₂³	1	240	yes	complemented
44	C₂²×Q₈×C₆₇	C₂³	1	10	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	120	yes	complemented
46	C₂²×D₄×C₆₇	C₂³	1	10	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	15	yes	—
48	C₂³×D₆₇	—	4	60	no	—
49	C₂³×D₆₇	C₂⁴	1	10	yes	—
50	C₂³×D₆₇	C₂⁴	1	180	yes	—
51	#11761629	C₂⁴	1	640	yes	complemented
52	#11761629	C₂⁴	1	1280	yes	complemented
53	#11761628	C₂⁴	1	1920	yes	complemented
54	#11761627	C₂⁴	1	640	yes	complemented
55	C₂×Dic₁₃₄	C₂⁴	1	120	yes	—
56	Q₈×D₆₇	C₂⁴	1	640	yes	complemented
57	C₂×D₂₆₈	C₂⁴	1	360	yes	—
58	C₂×D₂₆₈	C₂⁴	1	960	yes	complemented
59	C₂×C₂²⋊D₆₇	C₂⁴	1	240	yes	—
60	D₄×D₆₇	C₂⁴	1	1920	yes	complemented
61	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂⁴	1	5	yes	—
63	C₂²×Dic₆₇	C₂⁴	1	60	yes	—
64	C₂×C₄×D₆₇	C₂⁴	1	120	yes	—
65	C₂×C₄×D₆₇	C₂⁴	1	360	yes	—
66	C₂×C₄×D₆₇	C₂⁴	1	960	yes	complemented
67	C₂⁴×C₆₇	C₂⁴	1	5	yes	—
68	#41×C₆₇	C₂⁴	1	640	yes	complemented
69	C₂×Q₈×C₆₇	C₂⁴	1	120	yes	—
70	C₂×Q₈×C₆₇	C₂⁴	1	320	yes	complemented
71	C₂×D₄×C₆₇	C₂⁴	1	120	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	10	yes	—
73	C₂²×C₄×C₆₇	C₂⁴	1	180	yes	—
74	C₂³×C₆₇	—	2	20	no	—
75	C₂³×C₆₇	C₂⁵	1	30	yes	—
76	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
77	C₂²×D₆₇	—	2	40	no	—
78	C₂²×D₆₇	—	4	360	no	—
79	C₂²×D₆₇	C₂⁵	1	60	yes	—
80	C₂²×D₆₇	C₂⁵	1	240	yes	—
81	Q₈×C₆₇	C₂⁵	1	160	yes	—
82	D₄×C₆₇	C₂⁵	1	160	yes	—
83	C₂×C₄×C₆₇	C₂⁵	1	60	yes	—
84	C₂×C₄×C₆₇	C₂⁵	1	240	yes	—
85	C₂²⋊D₆₇	C₂⁵	1	320	yes	—
86	C₂×Dic₆₇	C₂⁵	1	6	yes	—
87	C₂×Dic₆₇	C₂⁵	1	30	yes	—
88	C₂×Dic₆₇	C₂⁵	1	80	yes	—
89	D₂₆₈	C₂⁵	1	480	yes	—
90	C₄×D₆₇	C₂⁵	1	160	yes	—
91	C₄×D₆₇	C₂⁵	1	480	yes	—
92	Dic₁₃₄	C₂⁵	1	160	yes	—
93	C₂²×C₆₇	—	2	60	no	—
94	C₂²×C₆₇	#585	1	4	yes	complemented
95	C₂²×C₆₇	C₂⁶	1	20	yes	—
96	C₂²×C₆₇	C₂⁶	1	3	yes	—
97	C₂×D₆₇	—	2	120	no	—
98	C₂×D₆₇	—	4	240	no	—
99	C₂×D₆₇	C₂⁶	1	40	yes	—
100	C₄×C₆₇	C₂⁶	1	40	yes	—
101	Dic₆₇	C₂⁶	1	20	yes	—
102	Dic₆₇	C₂⁶	1	4	yes	—
103	C₂²×#585	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	—	2	20	no	—
105	C₂×C₆₇	C₂×#585	1	6	yes	complemented
106	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
107	D₆₇	—	2	40	no	—
108	C₂×#585	—	67	96	no	—
109	C₂²×2₋¹⁺⁴	—	67	5	no	—
110	C₂²×2₋¹⁺⁴	D₆₇	1	1	yes	characteristic, complemented
111	C₂²×H(5,𝔽₂)	—	67	10	no	—
112	C₂³×#41	—	67	10	no	—
113	C₂³×#41	—	67	5	no	—
114	C₆₇	C₂²×#585	1	1	yes	characteristic, complemented, Sylow
115	#585	—	67	1024	no	—
116	C₂×2₋¹⁺⁴	—	67	240	no	—
117	C₂×2₋¹⁺⁴	C₂×D₆₇	1	48	yes	complemented
118	C₂×H(5,𝔽₂)	—	67	480	no	—
119	C₂²×#41	—	67	480	no	—
120	C₂²×#41	—	67	10	no	—
121	C₂²×#41	—	67	20	no	—
122	C₂²×#41	—	67	240	no	—
123	C₂²×#41	—	67	10	no	—
124	C₂²×#41	—	67	30	no	—
125	C₂²×#41	C₂×D₆₇	1	10	yes	complemented
126	C₂³×Q₈	—	67	10	no	—
127	C₂³×Q₈	C₂×D₆₇	1	5	yes	complemented
128	C₂³×D₄	—	67	30	no	—
129	C₂³×D₄	—	67	15	no	—
130	C₂⁴×C₄	—	67	15	no	—
131	C₂⁵	—	67	15	no	—
132	2₋¹⁺⁴	—	67	1280	no	—
133	2₋¹⁺⁴	C₂²×D₆₇	1	256	yes	complemented
134	H(5,𝔽₂)	—	67	2560	no	—
135	C₂×#41	—	67	2560	no	—
136	C₂×#41	—	67	240	no	—
137	C₂×#41	—	67	480	no	—
138	C₂×#41	—	67	1280	no	—
139	C₂×#41	—	67	240	no	—
140	C₂×#41	—	67	720	no	—
141	C₂×#41	C₂²×D₆₇	1	240	yes	complemented
142	C₂²×Q₈	—	67	240	no	—
143	C₂²×Q₈	—	67	10	no	—
144	C₂²×Q₈	C₂²×D₆₇	1	10	yes	—
145	C₂²×Q₈	C₂²×D₆₇	1	120	yes	complemented
146	C₂²×D₄	—	67	20	no	—
147	C₂²×D₄	—	67	720	no	—
148	C₂²×D₄	—	67	30	no	—
149	C₂²×D₄	—	67	360	no	—
150	C₂²×D₄	C₂²×D₆₇	1	10	yes	—
151	C₂³×C₄	—	67	5	no	—
152	C₂³×C₄	—	67	10	no	—
153	C₂³×C₄	—	67	30	no	—
154	C₂³×C₄	—	67	360	no	—
155	C₂³×C₄	C₂²×D₆₇	1	15	yes	—
156	C₂⁴	—	268	60	no	—
157	C₂⁴	—	67	180	no	—
158	C₂⁴	—	67	10	no	—
159	C₂⁴	C₂³×D₆₇	1	5	yes	—
160	#41	—	67	640	no	—
161	#41	—	67	1280	no	—
162	#41	—	67	640	no	—
163	#41	—	67	1920	no	—
164	#41	C₂³×D₆₇	1	640	yes	complemented
165	C₂×Q₈	—	67	640	no	—
166	C₂×Q₈	—	67	120	no	—
167	C₂×Q₈	C₂³×D₆₇	1	120	yes	—
168	C₂×Q₈	C₂³×D₆₇	1	320	yes	complemented
169	C₂×D₄	—	67	240	no	—
170	C₂×D₄	—	67	1920	no	—
171	C₂×D₄	—	67	360	no	—
172	C₂×D₄	—	67	960	no	—
173	C₂×D₄	C₂³×D₆₇	1	120	yes	—
174	C₂²×C₄	—	67	60	no	—
175	C₂²×C₄	—	67	1	no	—
176	C₂²×C₄	—	67	5	no	—
177	C₂²×C₄	—	67	120	no	—
178	C₂²×C₄	—	67	360	no	—
179	C₂²×C₄	—	67	960	no	—
180	C₂²×C₄	C₂³×D₆₇	1	180	yes	—
181	C₂²×C₄	C₂³×D₆₇	1	10	yes	—
182	C₂³	—	2	20	no	—
183	C₂³	—	268	360	no	—
184	C₂³	—	67	240	no	—
185	C₂³	—	67	60	no	—
186	C₂³	—	134	40	no	—
187	C₂³	C₂⁴×D₆₇	1	30	yes	—
188	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
189	Q₈	—	67	160	no	—
190	Q₈	C₂⁴×D₆₇	1	160	yes	—
191	D₄	—	67	320	no	—
192	D₄	—	67	480	no	—
193	D₄	C₂⁴×D₆₇	1	160	yes	—
194	C₂×C₄	—	67	6	no	—
195	C₂×C₄	—	67	30	no	—
196	C₂×C₄	—	67	80	no	—
197	C₂×C₄	—	67	160	no	—
198	C₂×C₄	—	67	480	no	—
199	C₂×C₄	C₂⁴×D₆₇	1	60	yes	—
200	C₂×C₄	C₂⁴×D₆₇	1	240	yes	—
201	C₂²	—	2	60	no	—
202	C₂²	—	268	240	no	—
203	C₂²	—	67	40	no	—
204	C₂²	—	134	120	no	—
205	C₂²	#16864233	1	4	yes	direct product factor
206	C₂²	C₂⁵×D₆₇	1	20	yes	—
207	C₂²	C₂⁵×D₆₇	1	3	yes	—
208	C₄	—	67	4	no	—
209	C₄	—	67	20	no	—
210	C₄	C₂⁵×D₆₇	1	40	yes	—
211	C₂	—	2	20	no	—
212	C₂	—	134	40	no	—
213	C₂	C₂×#16864233	1	6	yes	direct product factor
214	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂²×2₋¹⁺⁴×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₆₇
hypocenter	C₆₇	C₂²×#585
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#16864233	C₂	direct (N × H)
2. C₂²×#14524091	C₂	semidirect
3. C₂²×#14524089	C₂	semidirect
4. C₂³×#11761629	C₂	semidirect
5. C₂²×#41×D₆₇	C₂	semidirect
6. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
7. #16864233	C₂²	direct (N × H)
8. C₂×#14524091	C₂²	semidirect
9. C₂×#14524089	C₂²	semidirect
10. C₂²×#11761629	C₂²	semidirect
11. C₂²×#11761629	C₂²	semidirect
12. C₂²×#11761629	C₂²	semidirect
13. C₂²×#11761628	C₂²	semidirect
14. C₂²×#11761627	C₂²	semidirect
15. C₂×#41×D₆₇	C₂²	semidirect
16. C₂²×Q₈×D₆₇	C₂²	semidirect
17. C₂³×D₂₆₈	C₂²	semidirect
18. C₂²×D₄×D₆₇	C₂²	semidirect
19. C₂³×C₄×D₆₇	C₂²	semidirect
20. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
21. C₂²×#41×C₆₇	C₂²	semidirect
22. C₂³×Q₈×C₆₇	C₂²	semidirect
24. #14524091	C₂³	semidirect
25. #14524089	C₂³	semidirect
26. C₂×#11761629	C₂³	semidirect
27. C₂×#11761629	C₂³	semidirect
28. C₂×#11761629	C₂³	semidirect
29. C₂×#11761628	C₂³	semidirect
30. C₂×#11761627	C₂³	semidirect
31. #41×D₆₇	C₂³	semidirect
33. C₂×Q₈×D₆₇	C₂³	semidirect
35. C₂²×D₂₆₈	C₂³	semidirect
37. C₂×D₄×D₆₇	C₂³	semidirect
41. C₂²×C₄×D₆₇	C₂³	semidirect
42. 2₋¹⁺⁴×C₆₇	C₂³	semidirect
43. C₂×#41×C₆₇	C₂³	semidirect
45. C₂²×Q₈×C₆₇	C₂³	semidirect
51. #11761629	C₂⁴	semidirect
52. #11761629	C₂⁴	semidirect
53. #11761628	C₂⁴	semidirect
54. #11761627	C₂⁴	semidirect
56. Q₈×D₆₇	C₂⁴	semidirect
58. C₂×D₂₆₈	C₂⁴	semidirect
60. D₄×D₆₇	C₂⁴	semidirect
66. C₂×C₄×D₆₇	C₂⁴	semidirect
68. #41×C₆₇	C₂⁴	semidirect
70. C₂×Q₈×C₆₇	C₂⁴	semidirect
94. C₂²×C₆₇	#585	semidirect
105. C₂×C₆₇	C₂×#585	semidirect
110. C₂²×2₋¹⁺⁴	D₆₇	semidirect
114. C₆₇	C₂²×#585	semidirect
117. C₂×2₋¹⁺⁴	C₂×D₆₇	semidirect
125. C₂²×#41	C₂×D₆₇	semidirect
127. C₂³×Q₈	C₂×D₆₇	semidirect
133. 2₋¹⁺⁴	C₂²×D₆₇	semidirect
141. C₂×#41	C₂²×D₆₇	semidirect
145. C₂²×Q₈	C₂²×D₆₇	semidirect
164. #41	C₂³×D₆₇	semidirect
168. C₂×Q₈	C₂³×D₆₇	semidirect
205. C₂²	#16864233	direct (N × H)
213. C₂	C₂×#16864233	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×#16864233	2
C₂²×#14524091	2
C₂²×#14524089	2
C₂³×#11761629	2
C₂²×#41×D₆₇	2
C₂²×2₋¹⁺⁴×C₆₇	2
C₂²×#585	67

Subgroups — C22×#16864233 #31769725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16864233 #31769725