Subgroups — C₂×C₆₇×#1745 #30699613

Order 17152.

1310 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1745	C₂	1	8	yes	maximal, direct product factor
4	C₂×#336×C₆₇	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#97×C₆₇	C₄	1	16	yes	complemented
10	C₂×#97×C₆₇	C₂²	1	8	yes	—
11	C₄³×C₆₇	C₄	1	4	yes	—
12	C₄³×C₆₇	C₂²	1	4	yes	—
13	#336×C₆₇	C₂²	1	16	yes	—
14	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	—	2	16	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	—
19	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
20	C₂×C₄²×C₆₇	—	2	16	no	—
21	C₂×C₄²×C₆₇	—	2	8	no	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	16	yes	—
23	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
28	#97×C₆₇	C₂×C₄	1	32	yes	complemented
29	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	—	2	8	no	—
31	C₂²×C₄×C₆₇	—	2	8	no	—
32	C₂²×C₄×C₆₇	—	2	8	no	—
33	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
34	C₂²×C₄×C₆₇	M₁₆	1	16	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₆₇	#41	1	8	yes	—
40	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	32	no	—
42	C₂×C₈×C₆₇	—	2	16	no	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	—
44	C₄²×C₆₇	—	2	8	no	—
45	C₄²×C₆₇	—	2	32	no	—
46	C₄²×C₆₇	—	2	8	no	—
47	C₄²×C₆₇	C₄²	1	8	yes	—
48	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
49	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
50	C₂³×C₆₇	#117	1	4	yes	—
51	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
52	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	8	no	—
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	2	16	no	—
57	C₂×C₄×C₆₇	—	2	32	no	—
58	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₆₇	#117	1	16	yes	—
61	C₂×C₄×C₆₇	#117	1	4	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	16	yes	—
63	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
65	C₈×C₆₇	—	2	16	no	—
66	C₂²×C₆₇	#376	1	4	yes	—
67	C₂²×C₆₇	C₄×M₁₆	1	8	yes	—
68	C₂²×C₆₇	#432	1	16	yes	—
69	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
70	C₂²×C₆₇	C₂×#117	1	2	yes	—
71	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
72	C₄×C₆₇	—	2	8	no	—
73	C₄×C₆₇	—	2	16	no	—
74	C₄×C₆₇	#376	1	4	yes	—
75	C₄×C₆₇	C₂×#117	1	4	yes	—
76	C₂×#1745	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₆₇	#1745	1	8	yes	direct product factor
78	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
79	C₂×C₆₇	C₂×C₄×M₁₆	1	2	yes	—
80	C₂×C₆₇	C₂×#432	1	4	yes	—
81	C₂²×#97	C₂×C₆₇	1	2	yes	—
82	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
83	#1745	C₂×C₆₇	1	8	yes	direct product factor
84	C₂×#336	C₂×C₆₇	1	4	yes	—
85	C₆₇	C₂×#1745	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
87	C₂²×C₄²	C₄×C₆₇	1	4	yes	—
88	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
89	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
90	C₂×#97	C₄×C₆₇	1	16	yes	complemented
91	C₂×#97	C₂²×C₆₇	1	8	yes	—
92	C₄³	C₄×C₆₇	1	4	yes	—
93	C₄³	C₂²×C₆₇	1	4	yes	—
94	#336	C₂²×C₆₇	1	16	yes	—
95	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
96	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
97	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
98	C₂²×C₈	—	2	16	no	—
99	C₂²×C₈	C₂×C₄×C₆₇	1	16	yes	—
100	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
101	C₂×C₄²	—	2	16	no	—
102	C₂×C₄²	—	2	8	no	—
103	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₆₇	1	16	yes	—
105	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
107	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
108	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
109	#97	C₂×C₄×C₆₇	1	32	yes	complemented
110	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	C₄²×C₆₇	1	8	yes	—
115	C₂²×C₄	M₁₆×C₆₇	1	16	yes	—
116	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
117	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
118	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
120	C₂²×C₄	#41×C₆₇	1	8	yes	—
121	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
122	C₂×C₈	—	2	32	no	—
123	C₂×C₈	—	2	16	no	—
124	C₂×C₈	C₂²×C₄×C₆₇	1	16	yes	—
125	C₄²	—	2	8	no	—
126	C₄²	—	2	32	no	—
127	C₄²	—	2	8	no	—
128	C₄²	C₄²×C₆₇	1	8	yes	—
129	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
130	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
131	C₂³	#117×C₆₇	1	4	yes	—
132	C₂³	C₂×M₁₆×C₆₇	1	8	yes	—
133	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	32	no	—
139	C₂×C₄	C₂×C₄²×C₆₇	1	8	yes	—
140	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
141	C₂×C₄	#117×C₆₇	1	16	yes	—
142	C₂×C₄	#117×C₆₇	1	4	yes	—
143	C₂×C₄	C₂×M₁₆×C₆₇	1	16	yes	—
144	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
145	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
146	C₈	—	2	16	no	—
147	C₂²	#376×C₆₇	1	4	yes	—
148	C₂²	C₄×M₁₆×C₆₇	1	8	yes	—
149	C₂²	#432×C₆₇	1	16	yes	—
150	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
151	C₂²	C₂×#117×C₆₇	1	2	yes	—
152	C₂²	C₂²×M₁₆×C₆₇	1	4	yes	—
153	C₄	—	2	8	no	—
154	C₄	—	2	16	no	—
155	C₄	#376×C₆₇	1	4	yes	—
156	C₄	C₂×#117×C₆₇	1	4	yes	—
157	C₂	C₆₇×#1745	1	8	yes	direct product factor
158	C₂	C₂×#376×C₆₇	1	1	yes	characteristic
159	C₂	C₂×C₄×M₁₆×C₆₇	1	2	yes	—
160	C₂	C₂×#432×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1745	C₂	direct (N × H)
9. C₂×#97×C₆₇	C₄	semidirect
28. #97×C₆₇	C₂×C₄	semidirect
76. C₂×#1745	C₆₇	direct (N × H)
77. C₂×C₆₇	#1745	direct (N × H)
83. #1745	C₂×C₆₇	direct (N × H)
85. C₆₇	C₂×#1745	direct (N × H)
90. C₂×#97	C₄×C₆₇	semidirect
109. #97	C₂×C₄×C₆₇	semidirect
157. C₂	C₆₇×#1745	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×#97×C₆₇	2
C₂×C₄³×C₆₇	2
C₆₇×#1745	2
C₂×#336×C₆₇	2
C₂×#1745	67

Subgroups — C2×C67×#1745 #30699613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1745 #30699613