Subgroups — C₂²×#365×C₆₇ #30712653

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#365×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂²×M₁₆×C₆₇	—	2	1	no	—
6	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
8	C₂×M₃₂×C₆₇	C₄	1	6	yes	complemented
9	C₂×M₃₂×C₆₇	C₂²	1	6	yes	—
10	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
11	C₂×#108×C₆₇	—	2	3	no	—
12	C₂×#108×C₆₇	C₂²	1	6	yes	—
13	C₂×#106×C₆₇	—	2	3	no	—
14	C₂×#106×C₆₇	C₂²	1	6	yes	—
15	#365×C₆₇	C₂²	1	16	yes	direct product factor
16	C₂³×C₄×C₆₇	—	4	1	no	—
17	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	—	2	6	no	—
19	C₂×M₁₆×C₆₇	—	4	3	no	—
20	C₂²×C₈×C₆₇	—	4	1	no	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₆₇	D₄	1	3	yes	—
24	C₂²×C₈×C₆₇	Q₈	1	3	yes	—
25	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
27	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
28	C₂×C₄⋊C₄×C₆₇	—	4	3	no	—
29	M₃₂×C₆₇	C₂×C₄	1	12	yes	complemented
30	M₃₂×C₆₇	C₂³	1	4	yes	—
31	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	complemented
32	#108×C₆₇	—	2	6	no	—
33	#108×C₆₇	C₂³	1	4	yes	—
34	#106×C₆₇	—	2	6	no	—
35	#106×C₆₇	C₂³	1	4	yes	—
36	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	—	4	6	no	—
38	C₂²×C₄×C₆₇	—	8	4	no	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	3	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₆₇	SD₁₆	1	3	yes	—
43	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
45	M₁₆×C₆₇	—	2	4	no	—
46	M₁₆×C₆₇	—	4	6	no	—
47	C₂×C₈×C₆₇	—	4	6	no	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
49	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
50	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
54	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
55	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
56	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	6	no	—
59	C₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
60	C₂³×C₆₇	—	2	4	no	—
61	C₂³×C₆₇	#106	1	3	yes	—
62	C₂³×C₆₇	#106	1	1	yes	characteristic
63	C₂³×C₆₇	C₂×SD₁₆	1	3	yes	—
64	C₂×C₄×C₆₇	—	4	4	no	—
65	C₂×C₄×C₆₇	—	8	12	no	—
66	C₂×C₄×C₆₇	#106	1	6	yes	—
67	C₂×C₄×C₆₇	#106	1	3	yes	—
68	C₂×C₄×C₆₇	#106	1	3	yes	—
69	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
72	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
74	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₈×C₆₇	—	4	4	no	—
76	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
77	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
78	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₆₇	—	2	6	no	—
81	C₂²×C₆₇	—	2	6	no	—
82	C₂²×C₆₇	#365	1	4	yes	direct product factor
83	C₂²×C₆₇	C₂×#106	1	3	yes	—
84	C₂²×C₆₇	C₂×#106	1	3	yes	—
85	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
86	C₄×C₆₇	—	8	4	no	—
87	C₄×C₆₇	C₂×#106	1	3	yes	—
88	C₄×C₆₇	C₂×#106	1	3	yes	—
89	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
91	C₂²×#365	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₆₇	—	2	1	no	—
93	C₂×C₆₇	—	2	3	no	—
94	C₂×C₆₇	C₂×#365	1	6	yes	direct product factor
95	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
96	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic
97	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
98	C₂²×#106	C₂×C₆₇	1	1	yes	characteristic
99	C₂×#365	C₂×C₆₇	1	12	yes	direct product factor
100	C₆₇	C₂²×#365	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×M₁₆	—	2	1	no	—
102	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
103	C₂²×C₄⋊C₄	—	2	1	no	—
104	C₂×M₃₂	C₄×C₆₇	1	6	yes	complemented
105	C₂×M₃₂	C₂²×C₆₇	1	6	yes	—
106	C₂²×C₁₆	C₄×C₆₇	1	2	yes	complemented
107	C₂×#108	—	2	3	no	—
108	C₂×#108	C₂²×C₆₇	1	6	yes	—
109	C₂×#106	—	2	3	no	—
110	C₂×#106	C₂²×C₆₇	1	6	yes	—
111	#365	C₂²×C₆₇	1	16	yes	direct product factor
112	C₂³×C₄	—	4	1	no	—
113	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
114	C₂×M₁₆	—	2	6	no	—
115	C₂×M₁₆	—	4	3	no	—
116	C₂²×C₈	—	4	1	no	—
117	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
119	C₂²×C₈	D₄×C₆₇	1	3	yes	—
120	C₂²×C₈	Q₈×C₆₇	1	3	yes	—
121	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
122	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄⋊C₄	—	4	3	no	—
125	M₃₂	C₂×C₄×C₆₇	1	12	yes	complemented
126	M₃₂	C₂³×C₆₇	1	4	yes	—
127	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	complemented
128	#108	—	2	6	no	—
129	#108	C₂³×C₆₇	1	4	yes	—
130	#106	—	2	6	no	—
131	#106	C₂³×C₆₇	1	4	yes	—
132	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
133	C₂²×C₄	—	4	6	no	—
134	C₂²×C₄	—	8	4	no	—
135	C₂²×C₄	C₄⋊C₄×C₆₇	1	3	yes	—
136	C₂²×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
137	C₂²×C₄	SD₁₆×C₆₇	1	3	yes	—
138	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
139	C₂²×C₄	SD₁₆×C₆₇	1	3	yes	—
140	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
141	M₁₆	—	2	4	no	—
142	M₁₆	—	4	6	no	—
143	C₂×C₈	—	4	6	no	—
144	C₂×C₈	C₄⋊C₄×C₆₇	1	3	yes	—
145	C₂×C₈	C₄⋊C₄×C₆₇	1	6	yes	—
146	C₂×C₈	C₄⋊C₄×C₆₇	1	3	yes	—
147	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
148	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
149	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
150	C₂×C₈	C₂×Q₈×C₆₇	1	3	yes	—
151	C₂×C₈	C₂×Q₈×C₆₇	1	3	yes	—
152	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	6	no	—
155	C₁₆	C₂²×C₄×C₆₇	1	8	yes	complemented
156	C₂³	—	2	4	no	—
157	C₂³	#106×C₆₇	1	3	yes	—
158	C₂³	#106×C₆₇	1	1	yes	characteristic
159	C₂³	C₂×SD₁₆×C₆₇	1	3	yes	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	12	no	—
162	C₂×C₄	#106×C₆₇	1	6	yes	—
163	C₂×C₄	#106×C₆₇	1	3	yes	—
164	C₂×C₄	#106×C₆₇	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
167	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
168	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
169	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
170	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
171	C₈	—	4	4	no	—
172	C₈	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
173	C₈	C₂×C₄⋊C₄×C₆₇	1	3	yes	—
174	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
175	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	#365×C₆₇	1	4	yes	direct product factor
179	C₂²	C₂×#106×C₆₇	1	3	yes	—
180	C₂²	C₂×#106×C₆₇	1	3	yes	—
181	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
182	C₄	—	8	4	no	—
183	C₄	C₂×#106×C₆₇	1	3	yes	—
184	C₄	C₂×#106×C₆₇	1	3	yes	—
185	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
186	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#365×C₆₇	1	6	yes	direct product factor
190	C₂	C₂²×#106×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#106
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#365×C₆₇	C₂	direct (N × H)
8. C₂×M₃₂×C₆₇	C₄	semidirect
10. C₂²×C₁₆×C₆₇	C₄	semidirect
15. #365×C₆₇	C₂²	direct (N × H)
29. M₃₂×C₆₇	C₂×C₄	semidirect
31. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
59. C₁₆×C₆₇	C₂²×C₄	semidirect
82. C₂²×C₆₇	#365	direct (N × H)
91. C₂²×#365	C₆₇	direct (N × H)
94. C₂×C₆₇	C₂×#365	direct (N × H)
99. C₂×#365	C₂×C₆₇	direct (N × H)
100. C₆₇	C₂²×#365	direct (N × H)
104. C₂×M₃₂	C₄×C₆₇	semidirect
106. C₂²×C₁₆	C₄×C₆₇	semidirect
111. #365	C₂²×C₆₇	direct (N × H)
125. M₃₂	C₂×C₄×C₆₇	semidirect
127. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
155. C₁₆	C₂²×C₄×C₆₇	semidirect
178. C₂²	#365×C₆₇	direct (N × H)
189. C₂	C₂×#365×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
C₂²×M₃₂×C₆₇	2
C₂²×#108×C₆₇	2
C₂²×#106×C₆₇	2
C₂×#365×C₆₇	2
C₂²×#365	67

Subgroups — C22×#365×C67 #30712653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#365×C₆₇ #30712653