Subgroups — C₄²×#41×C₆₇ #30713244

Order 17152.

4086 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₄²×D₄×C₆₇	C₂	1	12	yes	maximal, complemented
4	C₄×#117×C₆₇	C₂	1	9	yes	maximal, complemented
5	C₂×C₄³×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₆₇	C₄	1	48	yes	direct product factor
8	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
9	C₂×C₄×D₄×C₆₇	C₂²	1	18	yes	—
10	C₂×#117×C₆₇	C₂²	1	9	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	12	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	9	yes	—
13	C₄×C₄⋊C₄×C₆₇	C₂²	1	36	yes	—
14	C₄×C₂²⋊C₄×C₆₇	C₂²	1	36	yes	—
15	#376×C₆₇	C₂²	1	6	yes	—
16	C₄³×C₆₇	C₂²	1	9	yes	—
17	C₄³×C₆₇	C₂²	1	12	yes	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#41×C₆₇	C₂×C₄	1	24	yes	—
20	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	18	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
24	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	complemented
25	C₄×D₄×C₆₇	C₂×C₄	1	144	yes	complemented
26	#117×C₆₇	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄×C₆₇	C₂³	1	18	yes	—
29	C₂×C₄²×C₆₇	—	2	48	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	72	yes	complemented
31	C₂×C₄²×C₆₇	C₂³	1	36	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	18	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
36	#95×C₆₇	C₂³	1	24	yes	—
37	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
38	#41×C₆₇	C₄²	1	64	yes	direct product factor
39	C₂×Q₈×C₆₇	C₂²×C₄	1	12	yes	—
40	C₂×D₄×C₆₇	C₂²×C₄	1	36	yes	—
41	C₂²×C₄×C₆₇	—	2	72	no	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	72	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	36	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	18	yes	—
45	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
47	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
48	C₄⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
49	C₂²⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
50	C₄²×C₆₇	—	2	72	no	—
51	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
52	C₄²×C₆₇	C₂²×C₄	1	36	yes	—
53	C₄²×C₆₇	C₂²×C₄	1	72	yes	—
54	C₄²×C₆₇	#41	1	16	yes	direct product factor
55	C₂³×C₆₇	—	2	12	no	—
56	C₂³×C₆₇	C₂³×C₄	1	18	yes	—
57	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
58	Q₈×C₆₇	C₂×C₄²	1	16	yes	complemented
59	D₄×C₆₇	C₂×C₄²	1	48	yes	complemented
60	C₂×C₄×C₆₇	—	2	36	no	—
61	C₂×C₄×C₆₇	—	2	144	no	—
62	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	complemented
63	C₂×C₄×C₆₇	C₂³×C₄	1	36	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	18	yes	—
67	C₂×C₄×C₆₇	C₂×#41	1	24	yes	—
68	C₂²×C₆₇	—	2	36	no	—
69	C₂²×C₆₇	C₂²×C₄²	1	12	yes	—
70	C₂²×C₆₇	C₂⁴×C₄	1	3	yes	—
71	C₂²×C₆₇	C₂²×#41	1	4	yes	—
72	C₄×C₆₇	—	2	36	no	—
73	C₄×C₆₇	C₂²×C₄²	1	4	yes	—
74	C₄×C₆₇	C₂²×C₄²	1	12	yes	—
75	C₄×C₆₇	C₄×#41	1	24	yes	direct product factor
76	C₄²×#41	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₆₇	—	2	12	no	—
78	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
79	C₂×C₆₇	C₂×C₄×#41	1	6	yes	—
80	C₂×C₄×#41	C₂×C₆₇	1	3	yes	—
81	C₄²×Q₈	C₂×C₆₇	1	4	yes	complemented
82	C₄²×D₄	C₂×C₆₇	1	12	yes	complemented
83	C₄×#117	C₂×C₆₇	1	9	yes	complemented
84	C₂×C₄³	C₂×C₆₇	1	3	yes	complemented
85	C₆₇	C₄²×#41	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
87	C₄×#41	C₄×C₆₇	1	48	yes	direct product factor
88	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
89	C₂×C₄×D₄	C₂²×C₆₇	1	18	yes	—
90	C₂×#117	C₂²×C₆₇	1	9	yes	—
91	C₂²×C₄²	C₂²×C₆₇	1	12	yes	—
92	C₂²×C₄²	C₂²×C₆₇	1	9	yes	—
93	C₄×C₄⋊C₄	C₂²×C₆₇	1	36	yes	—
94	C₄×C₂²⋊C₄	C₂²×C₆₇	1	36	yes	—
95	#376	C₂²×C₆₇	1	6	yes	—
96	C₄³	C₂²×C₆₇	1	9	yes	—
97	C₄³	C₂²×C₆₇	1	12	yes	—
98	C₄³	C₂²×C₆₇	1	1	yes	characteristic
99	C₂×#41	C₂×C₄×C₆₇	1	24	yes	—
100	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
101	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
102	C₂³×C₄	C₂³×C₆₇	1	18	yes	—
103	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
104	C₄×Q₈	C₂×C₄×C₆₇	1	48	yes	complemented
105	C₄×D₄	C₂×C₄×C₆₇	1	144	yes	complemented
106	#117	C₂×C₄×C₆₇	1	72	yes	complemented
107	C₂×C₄⋊C₄	C₂³×C₆₇	1	18	yes	—
108	C₂×C₂²⋊C₄	C₂³×C₆₇	1	18	yes	—
109	C₂×C₄²	—	2	48	no	—
110	C₂×C₄²	C₂×C₄×C₆₇	1	72	yes	complemented
111	C₂×C₄²	C₂³×C₆₇	1	36	yes	—
112	C₂×C₄²	C₂³×C₆₇	1	9	yes	—
113	C₂×C₄²	C₂³×C₆₇	1	18	yes	—
114	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
115	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
116	#95	C₂³×C₆₇	1	24	yes	—
117	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
118	#41	C₄²×C₆₇	1	64	yes	direct product factor
119	C₂×Q₈	C₂²×C₄×C₆₇	1	12	yes	—
120	C₂×D₄	C₂²×C₄×C₆₇	1	36	yes	—
121	C₂²×C₄	—	2	72	no	—
122	C₂²×C₄	C₂²×C₄×C₆₇	1	72	yes	—
123	C₂²×C₄	C₂²×C₄×C₆₇	1	36	yes	—
124	C₂²×C₄	C₂⁴×C₆₇	1	18	yes	—
125	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
127	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
128	C₄⋊C₄	C₂²×C₄×C₆₇	1	72	yes	—
129	C₂²⋊C₄	C₂²×C₄×C₆₇	1	72	yes	—
130	C₄²	—	2	72	no	—
131	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
132	C₄²	C₂²×C₄×C₆₇	1	36	yes	—
133	C₄²	C₂²×C₄×C₆₇	1	72	yes	—
134	C₄²	#41×C₆₇	1	16	yes	direct product factor
135	C₂³	—	2	12	no	—
136	C₂³	C₂³×C₄×C₆₇	1	18	yes	—
137	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
138	Q₈	C₂×C₄²×C₆₇	1	16	yes	complemented
139	D₄	C₂×C₄²×C₆₇	1	48	yes	complemented
140	C₂×C₄	—	2	36	no	—
141	C₂×C₄	—	2	144	no	—
142	C₂×C₄	C₂×C₄²×C₆₇	1	48	yes	complemented
143	C₂×C₄	C₂³×C₄×C₆₇	1	36	yes	—
144	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
145	C₂×C₄	C₂³×C₄×C₆₇	1	18	yes	—
146	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
147	C₂×C₄	C₂×#41×C₆₇	1	24	yes	—
148	C₂²	—	2	36	no	—
149	C₂²	C₂²×C₄²×C₆₇	1	12	yes	—
150	C₂²	C₂⁴×C₄×C₆₇	1	3	yes	—
151	C₂²	C₂²×#41×C₆₇	1	4	yes	—
152	C₄	—	2	36	no	—
153	C₄	C₂²×C₄²×C₆₇	1	4	yes	—
154	C₄	C₂²×C₄²×C₆₇	1	12	yes	—
155	C₄	C₄×#41×C₆₇	1	24	yes	direct product factor
156	C₂	—	2	12	no	—
157	C₂	C₂³×C₄²×C₆₇	1	1	yes	characteristic
158	C₂	C₂×C₄×#41×C₆₇	1	6	yes	—

	subgroup	G/H
center	C₄³×C₆₇	C₂²
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈×C₆₇	C₂	semidirect
3. C₄²×D₄×C₆₇	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
5. C₂×C₄³×C₆₇	C₂	semidirect
7. C₄×#41×C₆₇	C₄	direct (N × H)
24. C₄×Q₈×C₆₇	C₂×C₄	semidirect
25. C₄×D₄×C₆₇	C₂×C₄	semidirect
26. #117×C₆₇	C₂×C₄	semidirect
30. C₂×C₄²×C₆₇	C₂×C₄	semidirect
38. #41×C₆₇	C₄²	direct (N × H)
54. C₄²×C₆₇	#41	direct (N × H)
58. Q₈×C₆₇	C₂×C₄²	semidirect
59. D₄×C₆₇	C₂×C₄²	semidirect
62. C₂×C₄×C₆₇	C₂×C₄²	semidirect
75. C₄×C₆₇	C₄×#41	direct (N × H)
76. C₄²×#41	C₆₇	direct (N × H)
81. C₄²×Q₈	C₂×C₆₇	semidirect
82. C₄²×D₄	C₂×C₆₇	semidirect
83. C₄×#117	C₂×C₆₇	semidirect
84. C₂×C₄³	C₂×C₆₇	semidirect
85. C₆₇	C₄²×#41	direct (N × H)
87. C₄×#41	C₄×C₆₇	direct (N × H)
104. C₄×Q₈	C₂×C₄×C₆₇	semidirect
105. C₄×D₄	C₂×C₄×C₆₇	semidirect
106. #117	C₂×C₄×C₆₇	semidirect
110. C₂×C₄²	C₂×C₄×C₆₇	semidirect
118. #41	C₄²×C₆₇	direct (N × H)
134. C₄²	#41×C₆₇	direct (N × H)
138. Q₈	C₂×C₄²×C₆₇	semidirect
139. D₄	C₂×C₄²×C₆₇	semidirect
142. C₂×C₄	C₂×C₄²×C₆₇	semidirect
155. C₄	C₄×#41×C₆₇	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×C₄×#41×C₆₇	2
C₄²×Q₈×C₆₇	2
C₄²×D₄×C₆₇	2
C₄×#117×C₆₇	2
C₂×C₄³×C₆₇	2
C₄²×#41	67

Subgroups — C42×#41×C67 #30713244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#41×C₆₇ #30713244