Subgroups — C₂×C₄×D₁₆×C₆₇ #30712695

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₄×D₁₆×C₆₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#357×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₆₇	—	2	2	no	—
9	C₂×D₁₆×C₆₇	C₄	1	8	yes	direct product factor
10	C₂×D₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×D₈×C₆₇	—	2	4	no	—
14	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#366×C₆₇	C₂²	1	4	yes	complemented
19	#357×C₆₇	C₂²	1	8	yes	complemented
20	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂³×C₄×C₆₇	—	4	2	no	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₂×D₈×C₆₇	—	2	4	no	—
25	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
26	C₂×D₈×C₆₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₆₇	—	2	8	no	—
30	C₄×D₄×C₆₇	—	4	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	D₁₆×C₆₇	C₂×C₄	1	16	yes	direct product factor
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
37	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
39	#107×C₆₇	—	2	2	no	—
40	#107×C₆₇	C₂³	1	4	yes	—
41	#102×C₆₇	—	2	8	no	—
42	C₄×C₈×C₆₇	D₄	1	2	yes	—
43	C₄×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂⁴×C₆₇	—	4	2	no	—
45	C₂×D₄×C₆₇	—	2	8	no	—
46	C₂×D₄×C₆₇	—	2	4	no	—
47	C₂×D₄×C₆₇	—	4	2	no	—
48	C₂×D₄×C₆₇	—	4	4	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	—	4	8	no	—
51	C₂²×C₄×C₆₇	—	8	8	no	—
52	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	D₈×C₆₇	—	2	8	no	—
56	D₈×C₆₇	—	2	4	no	—
57	D₈×C₆₇	C₂²×C₄	1	8	yes	complemented
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	2	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₂²⋊C₄×C₆₇	—	4	8	no	—
67	C₄²×C₆₇	D₈	1	2	yes	—
68	C₄²×C₆₇	C₂×D₄	1	2	yes	—
69	C₁₆×C₆₇	—	2	2	no	—
70	C₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₆₇	—	4	4	no	—
72	C₂³×C₆₇	—	4	8	no	—
73	C₂³×C₆₇	—	8	4	no	—
74	C₂³×C₆₇	—	8	4	no	—
75	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
76	D₄×C₆₇	—	2	8	no	—
77	D₄×C₆₇	—	4	4	no	—
78	D₄×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	—	8	4	no	—
82	C₂×C₄×C₆₇	—	8	16	no	—
83	C₂×C₄×C₆₇	D₁₆	1	4	yes	direct product factor
84	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₆₇	#135	1	2	yes	—
89	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₆₇	—	2	2	no	—
91	C₈×C₆₇	C₄×D₄	1	2	yes	—
92	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
94	C₂²×C₆₇	—	4	8	no	—
95	C₂²×C₆₇	—	8	4	no	—
96	C₂²×C₆₇	—	8	4	no	—
97	C₂²×C₆₇	—	8	16	no	—
98	C₂²×C₆₇	C₄×D₈	1	2	yes	—
99	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
100	C₂²×C₆₇	#508	1	2	yes	—
101	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₆₇	—	2	2	no	—
103	C₄×C₆₇	—	8	4	no	—
104	C₄×C₆₇	C₄×D₈	1	2	yes	—
105	C₄×C₆₇	C₂×D₁₆	1	4	yes	direct product factor
106	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
108	C₂×C₄×D₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₆₇	—	8	8	no	—
110	C₂×C₆₇	C₄×D₁₆	1	4	yes	direct product factor
111	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
112	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
114	C₂²×D₁₆	C₂×C₆₇	1	1	yes	characteristic
115	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
116	C₄×D₁₆	C₂×C₆₇	1	8	yes	direct product factor
117	C₂×#366	C₂×C₆₇	1	1	yes	characteristic, complemented
118	C₂×#357	C₂×C₆₇	1	2	yes	complemented
119	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
120	C₆₇	C₂×C₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
122	C₂×C₄×D₄	—	2	2	no	—
123	C₂×D₁₆	C₄×C₆₇	1	8	yes	direct product factor
124	C₂×D₁₆	C₂²×C₆₇	1	4	yes	—
125	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
127	C₄×D₈	—	2	4	no	—
128	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
129	C₂×#107	C₂²×C₆₇	1	2	yes	—
130	C₂×#102	—	2	2	no	—
131	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
132	#366	C₂²×C₆₇	1	4	yes	complemented
133	#357	C₂²×C₆₇	1	8	yes	complemented
134	C₄×C₁₆	C₂²×C₆₇	1	4	yes	complemented
135	C₂²×D₄	—	2	2	no	—
136	C₂³×C₄	—	4	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
140	C₂×D₈	C₂³×C₆₇	1	4	yes	—
141	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
143	C₄×D₄	—	2	8	no	—
144	C₄×D₄	—	4	4	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₂²⋊C₄	—	4	2	no	—
147	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
148	D₁₆	C₂×C₄×C₆₇	1	16	yes	direct product factor
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
151	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
152	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
153	#107	—	2	2	no	—
154	#107	C₂³×C₆₇	1	4	yes	—
155	#102	—	2	8	no	—
156	C₄×C₈	D₄×C₆₇	1	2	yes	—
157	C₄×C₈	C₂³×C₆₇	1	2	yes	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	—	8	8	no	—
166	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
167	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
169	D₈	—	2	8	no	—
170	D₈	—	2	4	no	—
171	D₈	C₂²×C₄×C₆₇	1	8	yes	complemented
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
174	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
175	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
176	C₂×C₈	#41×C₆₇	1	2	yes	—
177	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	D₈×C₆₇	1	2	yes	—
182	C₄²	C₂×D₄×C₆₇	1	2	yes	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
185	C₂³	—	4	4	no	—
186	C₂³	—	4	8	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	4	no	—
189	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	D₄	—	4	4	no	—
192	D₄	—	4	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	4	no	—
196	C₂×C₄	—	8	16	no	—
197	C₂×C₄	D₁₆×C₆₇	1	4	yes	direct product factor
198	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
199	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
200	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
201	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
202	C₂×C₄	#135×C₆₇	1	2	yes	—
203	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	C₄×D₄×C₆₇	1	2	yes	—
206	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
207	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
208	C₂²	—	4	8	no	—
209	C₂²	—	8	4	no	—
210	C₂²	—	8	4	no	—
211	C₂²	—	8	16	no	—
212	C₂²	C₄×D₈×C₆₇	1	2	yes	—
213	C₂²	C₂×D₁₆×C₆₇	1	2	yes	—
214	C₂²	#508×C₆₇	1	2	yes	—
215	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	8	4	no	—
218	C₄	C₄×D₈×C₆₇	1	2	yes	—
219	C₄	C₂×D₁₆×C₆₇	1	4	yes	direct product factor
220	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
221	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
222	C₂	—	8	8	no	—
223	C₂	C₄×D₁₆×C₆₇	1	4	yes	direct product factor
224	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
225	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic
226	C₂	C₂×#508×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₁₆×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₆₇	C₂	semidirect
3. C₄×D₁₆×C₆₇	C₂	direct (N × H)
4. C₂×#366×C₆₇	C₂	semidirect
5. C₂×#357×C₆₇	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
9. C₂×D₁₆×C₆₇	C₄	direct (N × H)
14. C₄×D₈×C₆₇	C₂²	semidirect
18. #366×C₆₇	C₂²	semidirect
19. #357×C₆₇	C₂²	semidirect
20. C₄×C₁₆×C₆₇	C₂²	semidirect
25. C₂×D₈×C₆₇	C₂×C₄	semidirect
34. D₁₆×C₆₇	C₂×C₄	direct (N × H)
36. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
57. D₈×C₆₇	C₂²×C₄	semidirect
70. C₁₆×C₆₇	C₂²×C₄	semidirect
83. C₂×C₄×C₆₇	D₁₆	direct (N × H)
105. C₄×C₆₇	C₂×D₁₆	direct (N × H)
108. C₂×C₄×D₁₆	C₆₇	direct (N × H)
110. C₂×C₆₇	C₄×D₁₆	direct (N × H)
115. C₂×C₄×D₈	C₂×C₆₇	semidirect
116. C₄×D₁₆	C₂×C₆₇	direct (N × H)
117. C₂×#366	C₂×C₆₇	semidirect
118. C₂×#357	C₂×C₆₇	semidirect
119. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
120. C₆₇	C₂×C₄×D₁₆	direct (N × H)
123. C₂×D₁₆	C₄×C₆₇	direct (N × H)
128. C₄×D₈	C₂²×C₆₇	semidirect
132. #366	C₂²×C₆₇	semidirect
133. #357	C₂²×C₆₇	semidirect
134. C₄×C₁₆	C₂²×C₆₇	semidirect
139. C₂×D₈	C₂×C₄×C₆₇	semidirect
148. D₁₆	C₂×C₄×C₆₇	direct (N × H)
150. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
171. D₈	C₂²×C₄×C₆₇	semidirect
184. C₁₆	C₂²×C₄×C₆₇	semidirect
197. C₂×C₄	D₁₆×C₆₇	direct (N × H)
219. C₄	C₂×D₁₆×C₆₇	direct (N × H)
223. C₂	C₄×D₁₆×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×D₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₄×D₁₆×C₆₇	2
C₂×#366×C₆₇	2
C₂×#357×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×C₄×D₁₆	67

Subgroups — C2×C4×D16×C67 #30712695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₁₆×C₆₇ #30712695