Subgroups — C₆₇×#33329 #30712580

Order 17152.

716 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	C₆₇×#3407	C₂	1	2	yes	maximal
2	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2110	C₂	1	6	yes	maximal, complemented
4	C₂×#347×C₆₇	C₂	1	6	yes	maximal, complemented
5	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
8	#504×C₆₇	C₄	1	16	yes	complemented
9	C₂×M₃₂×C₆₇	C₂²	1	12	yes	—
10	#405×C₆₇	C₄	1	3	yes	—
11	#405×C₆₇	C₂²	1	3	yes	—
12	C₄×M₁₆×C₆₇	C₄	1	3	yes	—
13	C₂×#97×C₆₇	C₂²	1	3	yes	—
14	#347×C₆₇	C₂²	1	4	yes	—
15	#347×C₆₇	C₂²	1	12	yes	—
16	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
17	#131×C₆₇	—	2	4	no	—
18	#131×C₆₇	C₂×C₄	1	8	yes	—
19	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
20	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
23	C₄×Q₈×C₆₇	—	2	1	no	—
24	C₄×D₄×C₆₇	—	2	3	no	—
25	#117×C₆₇	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
27	M₃₂×C₆₇	C₂×C₄	1	24	yes	complemented
28	M₃₂×C₆₇	C₂³	1	8	yes	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	24	yes	complemented
30	#97×C₆₇	C₂×C₄	1	3	yes	—
31	#97×C₆₇	C₂×C₄	1	3	yes	—
32	#97×C₆₇	C₂³	1	1	yes	characteristic
33	#97×C₆₇	C₂³	1	3	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
35	#41×C₆₇	C₄²	1	4	yes	—
36	#41×C₆₇	M₁₆	1	4	yes	—
37	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
38	C₂×D₄×C₆₇	M₁₆	1	3	yes	—
39	C₂²×C₄×C₆₇	—	2	3	no	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
41	M₁₆×C₆₇	—	2	6	no	—
42	M₁₆×C₆₇	C₄²	1	12	yes	—
43	C₂×C₈×C₆₇	—	2	6	no	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
48	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	—	2	3	no	—
50	C₂²⋊C₄×C₆₇	—	2	3	no	—
51	C₄²×C₆₇	—	2	3	no	—
52	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
54	C₁₆×C₆₇	C₂²×C₄	1	16	yes	—
55	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
56	Q₈×C₆₇	#97	1	4	yes	—
57	D₄×C₆₇	#97	1	12	yes	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	3	no	—
60	C₂×C₄×C₆₇	—	8	3	no	—
61	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₆₇	—	2	1	no	—
66	C₈×C₆₇	—	2	3	no	—
67	C₈×C₆₇	C₂×C₄²	1	6	yes	—
68	C₈×C₆₇	C₂³×C₄	1	2	yes	—
69	C₂²×C₆₇	—	4	3	no	—
70	C₂²×C₆₇	C₂×#97	1	6	yes	—
71	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
72	C₄×C₆₇	—	4	3	no	—
73	C₄×C₆₇	—	4	1	no	—
74	C₄×C₆₇	C₂×#97	1	6	yes	—
75	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
76	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
77	#33329	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	—	2	6	no	—
80	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
81	#3407	C₂×C₆₇	1	2	yes	—
82	#2875	C₂×C₆₇	1	1	yes	characteristic
83	#2110	C₂×C₆₇	1	6	yes	complemented
84	C₂×#347	C₂×C₆₇	1	6	yes	complemented
85	C₆₇	#33329	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
87	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
88	C₄×#41	C₄×C₆₇	1	1	yes	characteristic
89	#504	C₄×C₆₇	1	16	yes	complemented
90	C₂×M₃₂	C₂²×C₆₇	1	12	yes	—
91	#405	C₄×C₆₇	1	3	yes	—
92	#405	C₂²×C₆₇	1	3	yes	—
93	C₄×M₁₆	C₄×C₆₇	1	3	yes	—
94	C₂×#97	C₂²×C₆₇	1	3	yes	—
95	#347	C₂²×C₆₇	1	4	yes	—
96	#347	C₂²×C₆₇	1	12	yes	—
97	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
98	#131	—	2	4	no	—
99	#131	C₂×C₄×C₆₇	1	8	yes	—
100	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
101	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
102	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
103	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×D₄	—	2	3	no	—
106	#117	C₂×C₄×C₆₇	1	3	yes	—
107	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
108	M₃₂	C₂×C₄×C₆₇	1	24	yes	complemented
109	M₃₂	C₂³×C₆₇	1	8	yes	—
110	C₂×C₁₆	C₂×C₄×C₆₇	1	24	yes	complemented
111	#97	C₂×C₄×C₆₇	1	3	yes	—
112	#97	C₂×C₄×C₆₇	1	3	yes	—
113	#97	C₂³×C₆₇	1	1	yes	characteristic
114	#97	C₂³×C₆₇	1	3	yes	—
115	C₄×C₈	C₂×C₄×C₆₇	1	6	yes	—
116	#41	C₄²×C₆₇	1	4	yes	—
117	#41	M₁₆×C₆₇	1	4	yes	—
118	C₂×Q₈	M₁₆×C₆₇	1	1	yes	characteristic
119	C₂×D₄	M₁₆×C₆₇	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
122	M₁₆	—	2	6	no	—
123	M₁₆	C₄²×C₆₇	1	12	yes	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
126	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
128	C₂×C₈	C₂²×C₄×C₆₇	1	12	yes	—
129	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
130	C₄⋊C₄	—	2	3	no	—
131	C₂²⋊C₄	—	2	3	no	—
132	C₄²	—	2	3	no	—
133	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
134	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
135	C₁₆	C₂²×C₄×C₆₇	1	16	yes	—
136	C₂³	C₂×M₁₆×C₆₇	1	3	yes	—
137	Q₈	#97×C₆₇	1	4	yes	—
138	D₄	#97×C₆₇	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×C₄²×C₆₇	1	6	yes	—
143	C₂×C₄	C₂×M₁₆×C₆₇	1	6	yes	—
144	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
145	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄²×C₆₇	1	6	yes	—
149	C₈	C₂³×C₄×C₆₇	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#97×C₆₇	1	6	yes	—
152	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×#97×C₆₇	1	6	yes	—
156	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
157	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#97×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×#97
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2110	C₂	semidirect
4. C₂×#347×C₆₇	C₂	semidirect
8. #504×C₆₇	C₄	semidirect
27. M₃₂×C₆₇	C₂×C₄	semidirect
29. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
77. #33329	C₆₇	direct (N × H)
83. #2110	C₂×C₆₇	semidirect
84. C₂×#347	C₂×C₆₇	semidirect
85. C₆₇	#33329	direct (N × H)
89. #504	C₄×C₆₇	semidirect
108. M₃₂	C₂×C₄×C₆₇	semidirect
110. C₂×C₁₆	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3407	2
C₆₇×#2875	2
C₆₇×#2110	2
C₂×#347×C₆₇	2
#33329	67

Subgroups — C67×#33329 #30712580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33329 #30712580