Subgroups — C₂³×#14524090 #31769720

Order 17152.

59806 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524090	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11761628	C₂	1	3	yes	maximal, complemented
3	C₂³×#11761627	C₂	1	6	yes	maximal, complemented
4	C₂⁴×Dic₁₃₄	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×D₆₇	C₂	1	2	yes	maximal, complemented
6	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524090	C₂²	1	1792	yes	direct product factor
8	C₂²×#11761628	C₂²	1	168	yes	complemented
9	C₂²×#11761627	C₂²	1	336	yes	complemented
10	C₂³×Dic₁₃₄	C₂²	1	6	yes	—
11	C₂³×Dic₁₃₄	C₂²	1	3	yes	—
12	C₂³×Dic₁₃₄	C₂²	1	168	yes	complemented
13	C₂²×Q₈×D₆₇	C₂²	1	112	yes	complemented
14	C₂³×D₂₆₈	C₂²	1	1	yes	characteristic
15	C₂³×C₂²⋊D₆₇	C₂²	1	6	yes	—
16	C₂⁴×Dic₆₇	C₂²	1	6	yes	—
17	C₂³×C₄×D₆₇	C₂²	1	6	yes	—
18	C₂²×#41×C₆₇	C₂²	1	56	yes	complemented
19	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×D₄×C₆₇	C₂²	1	3	yes	—
21	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
22	C₂⁴×D₆₇	C₂³	1	2	yes	—
23	#14524090	C₂³	1	4096	yes	direct product factor
24	C₂×#11761628	C₂³	1	1344	yes	complemented
25	C₂×#11761627	C₂³	1	2688	yes	complemented
26	C₂²×Dic₁₃₄	C₂³	1	168	yes	—
27	C₂²×Dic₁₃₄	C₂³	1	84	yes	—
28	C₂²×Dic₁₃₄	C₂³	1	1344	yes	complemented
29	C₂×Q₈×D₆₇	C₂³	1	896	yes	complemented
30	C₂²×D₂₆₈	C₂³	1	28	yes	—
31	C₂²×C₂²⋊D₆₇	C₂³	1	168	yes	—
32	C₂³×Dic₆₇	C₂³	1	6	yes	—
33	C₂³×Dic₆₇	C₂³	1	168	yes	—
34	C₂²×C₄×D₆₇	C₂³	1	168	yes	—
35	C₂⁵×C₆₇	C₂³	1	3	yes	—
36	C₂×#41×C₆₇	C₂³	1	448	yes	complemented
37	C₂²×Q₈×C₆₇	C₂³	1	28	yes	—
38	C₂²×D₄×C₆₇	C₂³	1	84	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	84	yes	—
42	C₂³×D₆₇	—	2	16	no	—
43	C₂³×D₆₇	C₂⁴	1	28	yes	—
44	#11761628	C₂⁴	1	1536	yes	complemented
45	#11761627	C₂⁴	1	3072	yes	complemented
46	C₂×Dic₁₃₄	C₂⁴	1	672	yes	—
47	C₂×Dic₁₃₄	C₂⁴	1	336	yes	—
48	C₂×Dic₁₃₄	C₂⁴	1	1536	yes	complemented
49	Q₈×D₆₇	C₂⁴	1	1024	yes	complemented
50	C₂×D₂₆₈	C₂⁴	1	112	yes	—
51	C₂×C₂²⋊D₆₇	C₂⁴	1	672	yes	—
52	C₂²×Dic₆₇	C₂⁴	1	84	yes	—
53	C₂²×Dic₆₇	C₂⁴	1	672	yes	—
54	C₂×C₄×D₆₇	C₂⁴	1	672	yes	—
55	C₂⁴×C₆₇	—	2	24	no	—
56	C₂⁴×C₆₇	C₂⁴	1	42	yes	—
57	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
58	#41×C₆₇	C₂⁴	1	512	yes	complemented
59	C₂×Q₈×C₆₇	C₂⁴	1	112	yes	—
60	C₂×D₄×C₆₇	C₂⁴	1	336	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	14	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	42	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	336	yes	—
64	C₂³×C₆₇	—	2	168	no	—
65	C₂³×C₆₇	2₋¹⁺⁴	1	8	yes	complemented
66	C₂³×C₆₇	C₂⁵	1	84	yes	—
67	C₂³×C₆₇	C₂⁵	1	7	yes	—
68	C₂²×D₆₇	—	2	112	no	—
69	C₂²×D₆₇	C₂⁵	1	56	yes	—
70	Q₈×C₆₇	C₂⁵	1	64	yes	—
71	D₄×C₆₇	C₂⁵	1	192	yes	—
72	C₂×C₄×C₆₇	C₂⁵	1	28	yes	—
73	C₂×C₄×C₆₇	C₂⁵	1	84	yes	—
74	C₂×C₄×C₆₇	C₂⁵	1	192	yes	—
75	C₂²⋊D₆₇	C₂⁵	1	384	yes	—
76	C₂×Dic₆₇	C₂⁵	1	168	yes	—
77	C₂×Dic₆₇	C₂⁵	1	384	yes	—
78	D₂₆₈	C₂⁵	1	64	yes	—
79	C₄×D₆₇	C₂⁵	1	384	yes	—
80	Dic₁₃₄	C₂⁵	1	384	yes	—
81	Dic₁₃₄	C₂⁵	1	192	yes	—
82	C₂²×C₆₇	—	2	168	no	—
83	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	28	yes	complemented
84	C₂²×C₆₇	C₂⁶	1	24	yes	—
85	C₂²×C₆₇	C₂⁶	1	7	yes	—
86	C₂×D₆₇	—	2	112	no	—
87	C₂×D₆₇	C₂⁶	1	16	yes	—
88	C₄×C₆₇	C₂⁶	1	8	yes	—
89	C₄×C₆₇	C₂⁶	1	24	yes	—
90	Dic₆₇	C₂⁶	1	48	yes	—
91	C₂³×2₋¹⁺⁴	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	—	2	24	no	—
93	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	14	yes	complemented
94	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
95	D₆₇	—	2	16	no	—
96	C₂²×2₋¹⁺⁴	—	67	112	no	—
97	C₂³×#41	—	67	6	no	—
98	C₂³×#41	—	67	3	no	—
99	C₂³×#41	D₆₇	1	1	yes	characteristic, complemented
100	C₂⁴×Q₈	—	67	2	no	—
101	C₂⁴×Q₈	—	67	3	no	—
102	C₆₇	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
103	C₂×2₋¹⁺⁴	—	67	1792	no	—
104	C₂²×#41	—	67	336	no	—
105	C₂²×#41	—	67	168	no	—
106	C₂²×#41	C₂×D₆₇	1	56	yes	complemented
107	C₂³×Q₈	—	67	112	no	—
108	C₂³×Q₈	—	67	6	no	—
109	C₂³×Q₈	—	67	3	no	—
110	C₂³×Q₈	—	67	168	no	—
111	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
112	C₂³×D₄	—	67	6	no	—
113	C₂³×D₄	—	67	1	no	—
114	C₂³×D₄	C₂×D₆₇	1	3	yes	—
115	C₂⁴×C₄	—	67	6	no	—
116	C₂⁴×C₄	—	67	6	no	—
117	C₂⁴×C₄	C₂×D₆₇	1	3	yes	—
118	C₂⁵	—	67	2	no	—
119	C₂⁵	C₂²×D₆₇	1	3	yes	—
120	2₋¹⁺⁴	—	67	4096	no	—
121	C₂×#41	—	67	2688	no	—
122	C₂×#41	—	67	1344	no	—
123	C₂×#41	C₂²×D₆₇	1	448	yes	complemented
124	C₂²×Q₈	—	67	896	no	—
125	C₂²×Q₈	—	67	168	no	—
126	C₂²×Q₈	—	67	84	no	—
127	C₂²×Q₈	—	67	1344	no	—
128	C₂²×Q₈	C₂²×D₆₇	1	28	yes	—
129	C₂²×D₄	—	67	168	no	—
130	C₂²×D₄	—	67	28	no	—
131	C₂²×D₄	C₂²×D₆₇	1	84	yes	—
132	C₂³×C₄	—	67	168	no	—
133	C₂³×C₄	—	67	6	no	—
134	C₂³×C₄	—	67	168	no	—
135	C₂³×C₄	C₂²×D₆₇	1	84	yes	—
136	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
137	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
138	C₂⁴	—	2	24	no	—
139	C₂⁴	—	67	28	no	—
140	C₂⁴	—	134	16	no	—
141	C₂⁴	C₂³×D₆₇	1	42	yes	—
142	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
143	#41	—	67	3072	no	—
144	#41	—	67	1536	no	—
145	#41	C₂³×D₆₇	1	512	yes	complemented
146	C₂×Q₈	—	67	1024	no	—
147	C₂×Q₈	—	67	672	no	—
148	C₂×Q₈	—	67	336	no	—
149	C₂×Q₈	—	67	1536	no	—
150	C₂×Q₈	C₂³×D₆₇	1	112	yes	—
151	C₂×D₄	—	67	672	no	—
152	C₂×D₄	—	67	112	no	—
153	C₂×D₄	C₂³×D₆₇	1	336	yes	—
154	C₂²×C₄	—	67	84	no	—
155	C₂²×C₄	—	67	672	no	—
156	C₂²×C₄	—	67	672	no	—
157	C₂²×C₄	C₂³×D₆₇	1	42	yes	—
158	C₂²×C₄	C₂³×D₆₇	1	14	yes	—
159	C₂²×C₄	C₂³×D₆₇	1	336	yes	—
160	C₂³	—	2	168	no	—
161	C₂³	—	67	56	no	—
162	C₂³	—	134	112	no	—
163	C₂³	#14524090	1	8	yes	direct product factor
164	C₂³	C₂⁴×D₆₇	1	84	yes	—
165	C₂³	C₂⁴×D₆₇	1	7	yes	—
166	Q₈	—	67	384	no	—
167	Q₈	—	67	192	no	—
168	Q₈	C₂⁴×D₆₇	1	64	yes	—
169	D₄	—	67	384	no	—
170	D₄	—	67	64	no	—
171	D₄	C₂⁴×D₆₇	1	192	yes	—
172	C₂×C₄	—	67	168	no	—
173	C₂×C₄	—	67	384	no	—
174	C₂×C₄	—	67	384	no	—
175	C₂×C₄	C₂⁴×D₆₇	1	84	yes	—
176	C₂×C₄	C₂⁴×D₆₇	1	28	yes	—
177	C₂×C₄	C₂⁴×D₆₇	1	192	yes	—
178	C₂²	—	2	168	no	—
179	C₂²	—	67	16	no	—
180	C₂²	—	134	112	no	—
181	C₂²	C₂×#14524090	1	28	yes	direct product factor
182	C₂²	C₂⁵×D₆₇	1	24	yes	—
183	C₂²	C₂⁵×D₆₇	1	7	yes	—
184	C₄	—	67	48	no	—
185	C₄	C₂⁵×D₆₇	1	24	yes	—
186	C₄	C₂⁵×D₆₇	1	8	yes	—
187	C₂	—	2	24	no	—
188	C₂	—	134	16	no	—
189	C₂	C₂²×#14524090	1	14	yes	direct product factor
190	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂³×#41×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂³×#41	D₆₇
hypocenter	C₆₇	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524090	C₂	direct (N × H)
2. C₂³×#11761628	C₂	semidirect
3. C₂³×#11761627	C₂	semidirect
4. C₂⁴×Dic₁₃₄	C₂	semidirect
5. C₂³×Q₈×D₆₇	C₂	semidirect
6. C₂³×#41×C₆₇	C₂	semidirect
7. C₂×#14524090	C₂²	direct (N × H)
8. C₂²×#11761628	C₂²	semidirect
9. C₂²×#11761627	C₂²	semidirect
12. C₂³×Dic₁₃₄	C₂²	semidirect
13. C₂²×Q₈×D₆₇	C₂²	semidirect
18. C₂²×#41×C₆₇	C₂²	semidirect
23. #14524090	C₂³	direct (N × H)
24. C₂×#11761628	C₂³	semidirect
25. C₂×#11761627	C₂³	semidirect
28. C₂²×Dic₁₃₄	C₂³	semidirect
29. C₂×Q₈×D₆₇	C₂³	semidirect
36. C₂×#41×C₆₇	C₂³	semidirect
44. #11761628	C₂⁴	semidirect
45. #11761627	C₂⁴	semidirect
48. C₂×Dic₁₃₄	C₂⁴	semidirect
49. Q₈×D₆₇	C₂⁴	semidirect
58. #41×C₆₇	C₂⁴	semidirect
65. C₂³×C₆₇	2₋¹⁺⁴	semidirect
83. C₂²×C₆₇	C₂×2₋¹⁺⁴	semidirect
93. C₂×C₆₇	C₂²×2₋¹⁺⁴	semidirect
99. C₂³×#41	D₆₇	semidirect
102. C₆₇	C₂³×2₋¹⁺⁴	semidirect
106. C₂²×#41	C₂×D₆₇	semidirect
123. C₂×#41	C₂²×D₆₇	semidirect
145. #41	C₂³×D₆₇	semidirect
163. C₂³	#14524090	direct (N × H)
181. C₂²	C₂×#14524090	direct (N × H)
189. C₂	C₂²×#14524090	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×#14524090	2
C₂³×#11761628	2
C₂³×#11761627	2
C₂⁴×Dic₁₃₄	2
C₂³×Q₈×D₆₇	2
C₂³×#41×C₆₇	2
C₂³×2₋¹⁺⁴	67

Subgroups — C23×#14524090 #31769720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14524090 #31769720