Subgroups — C₆₇×#33401 #30712652

Order 17152.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 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	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2153	C₂	1	6	yes	maximal, complemented
4	C₂×#364×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	—	2	8	no	—
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×C₄×C₈×C₆₇	C₂²	1	3	yes	—
14	#364×C₆₇	C₂²	1	4	yes	—
15	#364×C₆₇	C₂²	1	12	yes	—
16	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
17	#131×C₆₇	C₈	1	8	yes	—
18	#131×C₆₇	D₄	1	4	yes	—
19	#131×C₆₇	Q₈	1	4	yes	—
20	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
21	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
24	C₄×Q₈×C₆₇	—	2	1	no	—
25	C₄×D₄×C₆₇	—	2	3	no	—
26	#117×C₆₇	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
28	M₃₂×C₆₇	—	2	12	no	—
29	M₃₂×C₆₇	C₂³	1	8	yes	—
30	C₂×C₁₆×C₆₇	—	2	12	no	—
31	#97×C₆₇	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂³	1	3	yes	—
36	#41×C₆₇	C₄⋊C₄	1	4	yes	—
37	#41×C₆₇	C₂×C₈	1	4	yes	—
38	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
39	C₂×D₄×C₆₇	M₁₆	1	3	yes	—
40	C₂²×C₄×C₆₇	—	2	3	no	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
42	M₁₆×C₆₇	C₄⋊C₄	1	12	yes	—
43	M₁₆×C₆₇	C₂×C₈	1	12	yes	—
44	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₆₇	—	2	3	no	—
52	C₂²⋊C₄×C₆₇	—	2	3	no	—
53	C₄²×C₆₇	—	2	3	no	—
54	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
56	C₁₆×C₆₇	—	2	8	no	—
57	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
58	Q₈×C₆₇	C₄⋊C₈	1	4	yes	—
59	D₄×C₆₇	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	2	3	no	—
62	C₂×C₄×C₆₇	—	4	6	no	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₈×C₆₇	C₂²×C₈	1	2	yes	—
69	C₈×C₆₇	C₂²×C₈	1	6	yes	—
70	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
71	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₆₇	—	4	3	no	—
73	C₂²×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	4	3	no	—
77	C₄×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
78	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
80	#33401	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	—	2	6	no	—
83	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
84	#3407	C₂×C₆₇	1	2	yes	—
85	C₄×#131	C₂×C₆₇	1	1	yes	characteristic
86	#2153	C₂×C₆₇	1	6	yes	complemented
87	C₂×#364	C₂×C₆₇	1	6	yes	complemented
88	C₆₇	#33401	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
90	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
91	C₄×#41	C₄×C₆₇	1	1	yes	characteristic
92	#504	—	2	8	no	—
93	C₂×M₃₂	C₂²×C₆₇	1	12	yes	—
94	#405	C₄×C₆₇	1	3	yes	—
95	#405	C₂²×C₆₇	1	3	yes	—
96	C₄×M₁₆	C₄×C₆₇	1	3	yes	—
97	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
98	#364	C₂²×C₆₇	1	4	yes	—
99	#364	C₂²×C₆₇	1	12	yes	—
100	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
101	#131	C₈×C₆₇	1	8	yes	—
102	#131	D₄×C₆₇	1	4	yes	—
103	#131	Q₈×C₆₇	1	4	yes	—
104	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
105	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
106	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
107	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
108	C₄×Q₈	—	2	1	no	—
109	C₄×D₄	—	2	3	no	—
110	#117	C₂×C₄×C₆₇	1	3	yes	—
111	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
112	M₃₂	—	2	12	no	—
113	M₃₂	C₂³×C₆₇	1	8	yes	—
114	C₂×C₁₆	—	2	12	no	—
115	#97	C₂×C₄×C₆₇	1	6	yes	—
116	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
117	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
118	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
119	C₄×C₈	C₂³×C₆₇	1	3	yes	—
120	#41	C₄⋊C₄×C₆₇	1	4	yes	—
121	#41	C₂×C₈×C₆₇	1	4	yes	—
122	C₂×Q₈	M₁₆×C₆₇	1	1	yes	characteristic
123	C₂×D₄	M₁₆×C₆₇	1	3	yes	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
126	M₁₆	C₄⋊C₄×C₆₇	1	12	yes	—
127	M₁₆	C₂×C₈×C₆₇	1	12	yes	—
128	C₂×C₈	C₂×C₈×C₆₇	1	12	yes	—
129	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
130	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
132	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
133	C₂×C₈	C₂×Q₈×C₆₇	1	6	yes	—
134	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
135	C₄⋊C₄	—	2	3	no	—
136	C₂²⋊C₄	—	2	3	no	—
137	C₄²	—	2	3	no	—
138	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
139	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
140	C₁₆	—	2	8	no	—
141	C₂³	C₂×M₁₆×C₆₇	1	3	yes	—
142	Q₈	C₄⋊C₈×C₆₇	1	4	yes	—
143	D₄	C₄⋊C₈×C₆₇	1	12	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
148	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
149	C₂×C₄	C₂×M₁₆×C₆₇	1	3	yes	—
150	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
151	C₈	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
152	C₈	C₂²×C₈×C₆₇	1	2	yes	—
153	C₈	C₂²×C₈×C₆₇	1	6	yes	—
154	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
155	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊C₈×C₆₇	1	6	yes	—
158	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊C₈×C₆₇	1	6	yes	—
162	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
163	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2153	C₂	semidirect
4. C₂×#364×C₆₇	C₂	semidirect
80. #33401	C₆₇	direct (N × H)
86. #2153	C₂×C₆₇	semidirect
87. C₂×#364	C₂×C₆₇	semidirect
88. C₆₇	#33401	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3407	2
C₄×#131×C₆₇	2
C₆₇×#2153	2
C₂×#364×C₆₇	2
#33401	67

Subgroups — C67×#33401 #30712652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33401 #30712652