Subgroups — C₂²×#41×D₁₆₇ #36338934

Order 21376.

6520 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14581951	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14581950	C₂	1	3	yes	maximal, complemented
3	C₂²×#14581949	C₂	1	3	yes	maximal, complemented
4	C₂×#41×D₁₆₇	C₂	1	48	yes	maximal, direct product factor
5	C₂²×Q₈×D₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×D₁₆₇	C₂	1	3	yes	maximal, complemented
7	C₂³×C₄×D₁₆₇	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₆₇	C₂²	1	3	yes	—
10	C₂×#14581951	C₂²	1	24	yes	complemented
11	C₂×#14581950	C₂²	1	72	yes	complemented
12	C₂×#14581949	C₂²	1	72	yes	complemented
13	#41×D₁₆₇	C₂²	1	256	yes	direct product factor
14	C₂²×Dic₃₃₄	C₂²	1	3	yes	complemented
15	C₂×Q₈×D₁₆₇	C₂²	1	24	yes	complemented
16	C₂²×D₆₆₈	C₂²	1	3	yes	complemented
17	C₂²×C₂²⋊D₁₆₇	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₆₇	C₂²	1	72	yes	complemented
19	C₂³×Dic₁₆₇	C₂²	1	3	yes	complemented
20	C₂²×C₄×D₁₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₁₆₇	C₂²	1	3	yes	—
22	C₂²×C₄×D₁₆₇	C₂²	1	3	yes	complemented
23	C₂²×C₄×D₁₆₇	C₂²	1	3	yes	complemented
24	C₂²×C₄×D₁₆₇	C₂²	1	72	yes	complemented
25	C₂×#41×C₁₆₇	C₂²	1	24	yes	complemented
26	C₂²×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₄×C₁₆₇	C₂²	1	3	yes	complemented
28	C₂³×C₄×C₁₆₇	C₂²	1	3	yes	complemented
29	C₂³×D₁₆₇	—	2	24	no	—
30	C₂³×D₁₆₇	C₂³	1	36	yes	—
31	C₂³×D₁₆₇	C₂³	1	3	yes	—
32	C₂³×D₁₆₇	C₂³	1	1	yes	characteristic
33	#14581951	C₂³	1	64	yes	complemented
34	#14581950	C₂³	1	192	yes	complemented
35	#14581949	C₂³	1	192	yes	complemented
36	C₂×Dic₃₃₄	C₂³	1	36	yes	complemented
37	Q₈×D₁₆₇	C₂³	1	64	yes	complemented
38	C₂×D₆₆₈	C₂³	1	36	yes	complemented
39	C₂×C₂²⋊D₁₆₇	C₂³	1	72	yes	complemented
40	D₄×D₁₆₇	C₂³	1	192	yes	complemented
41	C₂²×Dic₁₆₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₆₇	C₂³	1	3	yes	—
43	C₂²×Dic₁₆₇	C₂³	1	36	yes	complemented
44	C₂×C₄×D₁₆₇	C₂³	1	12	yes	—
45	C₂×C₄×D₁₆₇	C₂³	1	36	yes	—
46	C₂×C₄×D₁₆₇	C₂³	1	36	yes	complemented
47	C₂×C₄×D₁₆₇	C₂³	1	36	yes	complemented
48	C₂×C₄×D₁₆₇	C₂³	1	192	yes	complemented
49	C₂⁴×C₁₆₇	C₂³	1	3	yes	—
50	#41×C₁₆₇	C₂³	1	64	yes	complemented
51	C₂×Q₈×C₁₆₇	C₂³	1	12	yes	complemented
52	C₂×D₄×C₁₆₇	C₂³	1	36	yes	complemented
53	C₂²×C₄×C₁₆₇	C₂³	1	3	yes	—
54	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₆₇	C₂³	1	36	yes	complemented
56	C₂²×D₁₆₇	—	2	12	no	—
57	C₂²×D₁₆₇	—	2	144	no	—
58	C₂²×D₁₆₇	#41	1	8	yes	direct product factor
59	C₂²×D₁₆₇	C₂⁴	1	48	yes	—
60	C₂²×D₁₆₇	C₂⁴	1	18	yes	—
61	C₂²×D₁₆₇	C₂⁴	1	6	yes	—
62	Dic₃₃₄	C₂⁴	1	48	yes	complemented
63	D₆₆₈	C₂⁴	1	48	yes	complemented
64	C₂²⋊D₁₆₇	C₂⁴	1	96	yes	complemented
65	C₂×Dic₁₆₇	C₂⁴	1	6	yes	—
66	C₂×Dic₁₆₇	C₂⁴	1	18	yes	—
67	C₂×Dic₁₆₇	C₂⁴	1	48	yes	complemented
68	C₄×D₁₆₇	C₂⁴	1	16	yes	—
69	C₄×D₁₆₇	C₂⁴	1	48	yes	—
70	C₄×D₁₆₇	C₂⁴	1	48	yes	complemented
71	C₄×D₁₆₇	C₂⁴	1	48	yes	complemented
72	C₂³×C₁₆₇	—	2	12	no	—
73	C₂³×C₁₆₇	C₂⁴	1	18	yes	—
74	C₂³×C₁₆₇	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₆₇	C₂⁴	1	16	yes	complemented
76	D₄×C₁₆₇	C₂⁴	1	48	yes	complemented
77	C₂×C₄×C₁₆₇	C₂⁴	1	6	yes	—
78	C₂×C₄×C₁₆₇	C₂⁴	1	18	yes	—
79	C₂×C₄×C₁₆₇	C₂⁴	1	48	yes	complemented
80	C₂²×C₁₆₇	—	2	36	no	—
81	C₂²×C₁₆₇	C₂×#41	1	4	yes	complemented
82	C₂²×C₁₆₇	C₂⁵	1	12	yes	—
83	C₂²×C₁₆₇	C₂⁵	1	3	yes	—
84	C₂×D₁₆₇	—	2	36	no	—
85	C₂×D₁₆₇	—	2	96	no	—
86	C₂×D₁₆₇	C₂×#41	1	24	yes	direct product factor
87	C₂×D₁₆₇	C₂⁵	1	12	yes	—
88	C₂×D₁₆₇	C₂⁵	1	4	yes	—
89	C₄×C₁₆₇	C₂⁵	1	4	yes	—
90	C₄×C₁₆₇	C₂⁵	1	12	yes	—
91	Dic₁₆₇	C₂⁵	1	4	yes	—
92	Dic₁₆₇	C₂⁵	1	12	yes	—
93	C₂×C₁₆₇	—	2	12	no	—
94	C₂×C₁₆₇	C₂²×#41	1	6	yes	complemented
95	C₂×C₁₆₇	C₂⁶	1	1	yes	characteristic
96	D₁₆₇	—	2	12	no	—
97	D₁₆₇	C₂²×#41	1	8	yes	direct product factor
98	C₁₆₇	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
99	C₂³×#41	—	167	1	no	maximal, Sylow
100	C₂²×#41	—	167	48	no	—
101	C₂²×#41	—	167	1	no	—
102	C₂²×#41	—	167	3	no	—
103	C₂²×#41	—	167	3	no	—
104	C₂²×#41	D₁₆₇	1	1	yes	characteristic, direct product factor
105	C₂³×Q₈	—	167	1	no	—
106	C₂³×D₄	—	167	3	no	—
107	C₂⁴×C₄	—	167	3	no	—
108	C₂⁵	—	167	3	no	—
109	C₂×#41	—	167	256	no	—
110	C₂×#41	—	167	24	no	—
111	C₂×#41	—	167	72	no	—
112	C₂×#41	—	167	72	no	—
113	C₂×#41	C₂×D₁₆₇	1	24	yes	direct product factor
114	C₂²×Q₈	—	167	24	no	—
115	C₂²×Q₈	—	167	3	no	—
116	C₂²×Q₈	C₂×D₁₆₇	1	1	yes	characteristic, complemented
117	C₂²×D₄	—	167	6	no	—
118	C₂²×D₄	—	167	72	no	—
119	C₂²×D₄	—	167	3	no	—
120	C₂²×D₄	C₂×D₁₆₇	1	3	yes	complemented
121	C₂³×C₄	—	167	3	no	—
122	C₂³×C₄	—	167	3	no	—
123	C₂³×C₄	—	167	3	no	—
124	C₂³×C₄	—	167	1	no	—
125	C₂³×C₄	—	167	3	no	—
126	C₂³×C₄	—	167	72	no	—
127	C₂³×C₄	C₂×D₁₆₇	1	3	yes	complemented
128	C₂⁴	—	334	24	no	—
129	C₂⁴	—	167	36	no	—
130	C₂⁴	—	167	3	no	—
131	C₂⁴	—	167	1	no	—
132	C₂⁴	C₂²×D₁₆₇	1	3	yes	—
133	#41	—	167	64	no	—
134	#41	—	167	192	no	—
135	#41	—	167	192	no	—
136	#41	C₂²×D₁₆₇	1	64	yes	direct product factor
137	C₂×Q₈	—	167	64	no	—
138	C₂×Q₈	—	167	36	no	—
139	C₂×Q₈	C₂²×D₁₆₇	1	12	yes	complemented
140	C₂×D₄	—	167	72	no	—
141	C₂×D₄	—	167	192	no	—
142	C₂×D₄	—	167	36	no	—
143	C₂×D₄	C₂²×D₁₆₇	1	36	yes	complemented
144	C₂²×C₄	—	167	1	no	—
145	C₂²×C₄	—	167	3	no	—
146	C₂²×C₄	—	167	36	no	—
147	C₂²×C₄	—	167	12	no	—
148	C₂²×C₄	—	167	36	no	—
149	C₂²×C₄	—	167	36	no	—
150	C₂²×C₄	—	167	36	no	—
151	C₂²×C₄	—	167	192	no	—
152	C₂²×C₄	C₂²×D₁₆₇	1	3	yes	—
153	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₆₇	1	36	yes	complemented
155	C₂³	—	2	12	no	—
156	C₂³	—	334	144	no	—
157	C₂³	—	334	12	no	—
158	C₂³	—	167	48	no	—
159	C₂³	—	167	18	no	—
160	C₂³	—	167	8	no	—
161	C₂³	—	167	6	no	—
162	C₂³	C₂³×D₁₆₇	1	18	yes	—
163	C₂³	C₂³×D₁₆₇	1	1	yes	characteristic
164	Q₈	—	167	48	no	—
165	Q₈	C₂³×D₁₆₇	1	16	yes	complemented
166	D₄	—	167	96	no	—
167	D₄	—	167	48	no	—
168	D₄	C₂³×D₁₆₇	1	48	yes	complemented
169	C₂×C₄	—	167	6	no	—
170	C₂×C₄	—	167	18	no	—
171	C₂×C₄	—	167	48	no	—
172	C₂×C₄	—	167	16	no	—
173	C₂×C₄	—	167	48	no	—
174	C₂×C₄	—	167	48	no	—
175	C₂×C₄	—	167	48	no	—
176	C₂×C₄	C₂³×D₁₆₇	1	6	yes	—
177	C₂×C₄	C₂³×D₁₆₇	1	18	yes	—
178	C₂×C₄	C₂³×D₁₆₇	1	48	yes	complemented
179	C₂²	—	2	36	no	—
180	C₂²	—	334	96	no	—
181	C₂²	—	334	36	no	—
182	C₂²	—	167	12	no	—
183	C₂²	—	167	24	no	—
184	C₂²	—	167	4	no	—
185	C₂²	#41×D₁₆₇	1	4	yes	direct product factor
186	C₂²	C₂⁴×D₁₆₇	1	12	yes	—
187	C₂²	C₂⁴×D₁₆₇	1	3	yes	—
188	C₄	—	167	4	no	—
189	C₄	—	167	12	no	—
190	C₄	C₂⁴×D₁₆₇	1	4	yes	—
191	C₄	C₂⁴×D₁₆₇	1	12	yes	—
192	C₂	—	2	12	no	—
193	C₂	—	334	12	no	—
194	C₂	—	167	8	no	—
195	C₂	C₂×#41×D₁₆₇	1	6	yes	direct product factor
196	C₂	C₂⁵×D₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₆₇
commutator, cosocle	C₂×C₁₆₇	C₂⁶
Frattini	C₂	C₂⁵×D₁₆₇
Fitting	C₂²×#41×C₁₆₇	C₂
socle	C₂³×C₁₆₇	C₂⁴
hypercenter	C₂²×#41	D₁₆₇
hypocenter	C₁₆₇	C₂³×#41
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#14581951	C₂	semidirect
2. C₂²×#14581950	C₂	semidirect
3. C₂²×#14581949	C₂	semidirect
4. C₂×#41×D₁₆₇	C₂	direct (N × H)
5. C₂²×Q₈×D₁₆₇	C₂	semidirect
6. C₂²×D₄×D₁₆₇	C₂	semidirect
7. C₂³×C₄×D₁₆₇	C₂	semidirect
8. C₂²×#41×C₁₆₇	C₂	semidirect
10. C₂×#14581951	C₂²	semidirect
11. C₂×#14581950	C₂²	semidirect
12. C₂×#14581949	C₂²	semidirect
13. #41×D₁₆₇	C₂²	direct (N × H)
14. C₂²×Dic₃₃₄	C₂²	semidirect
15. C₂×Q₈×D₁₆₇	C₂²	semidirect
16. C₂²×D₆₆₈	C₂²	semidirect
17. C₂²×C₂²⋊D₁₆₇	C₂²	semidirect
18. C₂×D₄×D₁₆₇	C₂²	semidirect
19. C₂³×Dic₁₆₇	C₂²	semidirect
22. C₂²×C₄×D₁₆₇	C₂²	semidirect
23. C₂²×C₄×D₁₆₇	C₂²	semidirect
24. C₂²×C₄×D₁₆₇	C₂²	semidirect
25. C₂×#41×C₁₆₇	C₂²	semidirect
26. C₂²×Q₈×C₁₆₇	C₂²	semidirect
27. C₂²×D₄×C₁₆₇	C₂²	semidirect
28. C₂³×C₄×C₁₆₇	C₂²	semidirect
33. #14581951	C₂³	semidirect
34. #14581950	C₂³	semidirect
35. #14581949	C₂³	semidirect
36. C₂×Dic₃₃₄	C₂³	semidirect
37. Q₈×D₁₆₇	C₂³	semidirect
38. C₂×D₆₆₈	C₂³	semidirect
39. C₂×C₂²⋊D₁₆₇	C₂³	semidirect
40. D₄×D₁₆₇	C₂³	semidirect
43. C₂²×Dic₁₆₇	C₂³	semidirect
46. C₂×C₄×D₁₆₇	C₂³	semidirect
47. C₂×C₄×D₁₆₇	C₂³	semidirect
48. C₂×C₄×D₁₆₇	C₂³	semidirect
50. #41×C₁₆₇	C₂³	semidirect
51. C₂×Q₈×C₁₆₇	C₂³	semidirect
52. C₂×D₄×C₁₆₇	C₂³	semidirect
55. C₂²×C₄×C₁₆₇	C₂³	semidirect
58. C₂²×D₁₆₇	#41	direct (N × H)
62. Dic₃₃₄	C₂⁴	semidirect
63. D₆₆₈	C₂⁴	semidirect
64. C₂²⋊D₁₆₇	C₂⁴	semidirect
67. C₂×Dic₁₆₇	C₂⁴	semidirect
70. C₄×D₁₆₇	C₂⁴	semidirect
71. C₄×D₁₆₇	C₂⁴	semidirect
75. Q₈×C₁₆₇	C₂⁴	semidirect
76. D₄×C₁₆₇	C₂⁴	semidirect
79. C₂×C₄×C₁₆₇	C₂⁴	semidirect
81. C₂²×C₁₆₇	C₂×#41	semidirect
86. C₂×D₁₆₇	C₂×#41	direct (N × H)
94. C₂×C₁₆₇	C₂²×#41	semidirect
97. D₁₆₇	C₂²×#41	direct (N × H)
98. C₁₆₇	C₂³×#41	semidirect
104. C₂²×#41	D₁₆₇	direct (N × H)
113. C₂×#41	C₂×D₁₆₇	direct (N × H)
116. C₂²×Q₈	C₂×D₁₆₇	semidirect
120. C₂²×D₄	C₂×D₁₆₇	semidirect
127. C₂³×C₄	C₂×D₁₆₇	semidirect
136. #41	C₂²×D₁₆₇	direct (N × H)
139. C₂×Q₈	C₂²×D₁₆₇	semidirect
143. C₂×D₄	C₂²×D₁₆₇	semidirect
154. C₂²×C₄	C₂²×D₁₆₇	semidirect
165. Q₈	C₂³×D₁₆₇	semidirect
168. D₄	C₂³×D₁₆₇	semidirect
178. C₂×C₄	C₂³×D₁₆₇	semidirect
185. C₂²	#41×D₁₆₇	direct (N × H)
195. C₂	C₂×#41×D₁₆₇	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×#14581951	2
C₂²×#14581950	2
C₂²×#14581949	2
C₂×#41×D₁₆₇	2
C₂²×Q₈×D₁₆₇	2
C₂²×D₄×D₁₆₇	2
C₂³×C₄×D₁₆₇	2
C₂²×#41×C₁₆₇	2
C₂³×#41	167

Subgroups — C22×#41×D167 #36338934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×D₁₆₇ #36338934