Subgroups — C₄×M₁₆×Dic₆₇ #30853161

Order 17152.

1734 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₆₇	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#14523953	C₂	1	2	yes	maximal, complemented
5	#97×Dic₆₇	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₆₇	C₄	1	4	yes	—
10	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
11	C₄×#11761612	C₄	1	8	yes	complemented
12	M₁₆×Dic₆₇	C₄	1	32	yes	direct product factor
13	C₂×#14523954	C₂²	1	2	yes	—
14	C₂×#14523953	C₂²	1	8	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	8	yes	—
17	C₄²×Dic₆₇	C₄	1	4	yes	—
18	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
19	C₄²×Dic₆₇	C₂²	1	2	yes	—
20	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₆₇	C₄	1	8	yes	complemented
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
27	C₂×#11761612	C₂×C₄	1	16	yes	—
28	C₂×#11761612	C₂×C₄	1	8	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₄×Dic₆₇	—	2	16	no	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	8	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	16	yes	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
37	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
38	#14523954	C₂×C₄	1	8	yes	complemented
39	#14523953	C₂×C₄	1	32	yes	complemented
40	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	complemented
41	C₈×Dic₆₇	C₂×C₄	1	32	yes	complemented
42	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	C₂×C₄	1	16	yes	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#97×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
54	C₂²×Dic₆₇	—	2	16	no	—
55	C₂²×Dic₆₇	C₄²	1	16	yes	—
56	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
58	#11761612	C₄²	1	32	yes	complemented
59	C₂×C₆₇⋊C₈	C₂²×C₄	1	16	yes	—
60	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
61	C₄×Dic₆₇	—	2	8	no	—
62	C₄×Dic₆₇	—	2	16	no	—
63	C₄×Dic₆₇	C₄²	1	16	yes	—
64	C₄×Dic₆₇	M₁₆	1	16	yes	direct product factor
65	C₄×Dic₆₇	C₂²×C₄	1	8	yes	—
66	C₄×Dic₆₇	C₂²×C₄	1	8	yes	—
67	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	8	no	—
69	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₆₇	C₄²	1	32	yes	complemented
78	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
80	C₄²×C₆₇	C₄²	1	8	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂³×C₆₇	—	2	4	no	—
86	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
87	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
88	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	16	no	—
91	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
93	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
96	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₆₇	—	2	8	no	—
102	C₂×Dic₆₇	—	2	32	no	—
103	C₂×Dic₆₇	C₂×C₄²	1	8	yes	—
104	C₂×Dic₆₇	C₂×C₄²	1	8	yes	—
105	C₂×Dic₆₇	C₂×M₁₆	1	16	yes	—
106	C₈×C₆₇	C₂×C₄²	1	16	yes	complemented
107	C₆₇⋊C₈	C₂×C₄²	1	16	yes	complemented
108	C₂²×C₆₇	—	2	8	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	C₄³	1	4	yes	—
111	C₂²×C₆₇	C₂²×C₄²	1	2	yes	—
112	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
113	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
114	C₄×C₆₇	—	2	4	no	—
115	C₄×C₆₇	C₄³	1	4	yes	—
116	C₄×C₆₇	C₄×M₁₆	1	8	yes	complemented
117	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
118	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
119	Dic₆₇	—	2	8	no	—
120	Dic₆₇	C₄×M₁₆	1	16	yes	direct product factor
121	C₄²×M₁₆	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	4	no	—
123	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
124	C₂×C₆₇	C₂×C₄×M₁₆	1	4	yes	—
125	C₂×C₆₇	C₂×C₄×M₁₆	1	2	yes	—
126	C₂×C₄×M₁₆	—	67	1	no	—
127	C₂×C₄×M₁₆	—	67	4	no	—
128	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
129	C₂×C₄³	—	67	1	no	—
130	C₄×#97	—	67	4	no	—
131	C₄×#97	—	67	2	no	—
132	C₄²×C₈	—	67	2	no	—
133	C₆₇	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	67	2	no	—
135	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
136	C₂²×C₄²	—	67	2	no	—
137	C₂²×C₄²	—	67	4	no	—
138	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
139	C₄×M₁₆	—	67	32	no	—
140	C₄×M₁₆	—	67	8	no	—
141	C₄×M₁₆	Dic₆₇	1	8	yes	direct product factor
142	C₂×#97	—	67	8	no	—
143	C₂×#97	—	67	2	no	—
144	C₂×#97	C₂×D₆₇	1	2	yes	—
145	C₂×C₄×C₈	—	67	8	no	—
146	C₂×C₄×C₈	—	67	2	no	—
147	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
148	C₄³	—	67	4	no	—
149	C₄³	—	67	2	no	—
150	C₄³	—	67	1	no	—
151	C₄³	—	67	1	no	—
152	C₂³×C₄	—	67	4	no	—
153	C₂³×C₄	C₄×D₆₇	1	2	yes	—
154	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
155	C₂×M₁₆	—	67	8	no	—
156	C₂×M₁₆	—	67	16	no	—
157	C₂×M₁₆	C₄×D₆₇	1	16	yes	—
158	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
159	C₂²×C₈	—	67	4	no	—
160	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
161	C₂×C₄²	—	67	4	no	—
162	C₂×C₄²	—	67	4	no	—
163	C₂×C₄²	—	67	16	no	—
164	C₂×C₄²	—	67	2	no	—
165	C₂×C₄²	—	67	2	no	—
166	C₂×C₄²	—	67	8	no	—
167	C₂×C₄²	—	67	4	no	—
168	C₂×C₄²	—	134	16	no	—
169	C₂×C₄²	C₄×D₆₇	1	2	yes	—
170	C₂×C₄²	C₄×D₆₇	1	4	yes	—
171	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
172	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	#97	—	67	32	no	—
175	#97	—	67	8	no	—
176	#97	C₂×Dic₆₇	1	8	yes	complemented
177	C₄×C₈	—	67	32	no	—
178	C₄×C₈	—	67	8	no	—
179	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
180	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	67	4	no	—
183	C₂²×C₄	—	67	16	no	—
184	C₂²×C₄	—	67	4	no	—
185	C₂²×C₄	—	134	16	no	—
186	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
187	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
188	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₆₇	1	8	yes	—
191	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
193	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
194	M₁₆	—	67	32	no	—
195	M₁₆	C₄×Dic₆₇	1	32	yes	direct product factor
196	C₂×C₈	—	67	8	no	—
197	C₂×C₈	—	67	16	no	—
198	C₂×C₈	C₂×C₄×D₆₇	1	16	yes	—
199	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
200	C₄²	—	67	16	no	—
201	C₄²	—	67	16	no	—
202	C₄²	—	67	8	no	—
203	C₄²	—	67	8	no	—
204	C₄²	—	134	16	no	—
205	C₄²	—	134	8	no	—
206	C₄²	C₄×Dic₆₇	1	8	yes	—
207	C₄²	C₂×C₄×D₆₇	1	2	yes	—
208	C₄²	C₂×C₄×D₆₇	1	2	yes	—
209	C₄²	C₂²×Dic₆₇	1	2	yes	—
210	C₄²	C₂²×Dic₆₇	1	2	yes	—
211	C₂³	—	2	4	no	—
212	C₂³	C₄²×D₆₇	1	4	yes	—
213	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
214	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	67	8	no	—
218	C₂×C₄	—	67	16	no	—
219	C₂×C₄	—	67	8	no	—
220	C₂×C₄	—	134	32	no	—
221	C₂×C₄	—	134	8	no	—
222	C₂×C₄	C₄²×D₆₇	1	4	yes	—
223	C₂×C₄	C₄²×D₆₇	1	8	yes	—
224	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
225	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
226	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
227	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
228	C₂×C₄	M₁₆×D₆₇	1	8	yes	—
229	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
231	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
232	C₈	—	67	16	no	—
233	C₈	C₂×C₄×Dic₆₇	1	16	yes	complemented
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₄²×Dic₆₇	1	4	yes	—
237	C₂²	C₂×C₄²×D₆₇	1	2	yes	—
238	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
239	C₂²	C₂×M₁₆×D₆₇	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	67	16	no	—
242	C₄	—	134	8	no	—
243	C₄	C₄²×Dic₆₇	1	4	yes	—
244	C₄	M₁₆×Dic₆₇	1	8	yes	direct product factor
245	C₄	C₂×C₄²×D₆₇	1	2	yes	—
246	C₄	C₂²×C₄×Dic₆₇	1	2	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₂×C₄²×Dic₆₇	1	1	yes	characteristic
249	C₂	C₄×M₁₆×D₆₇	1	4	yes	—
250	C₂	C₂×M₁₆×Dic₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₄²×M₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₄×#14523953	C₂	semidirect
5. #97×Dic₆₇	C₂	semidirect
6. C₄×C₈×Dic₆₇	C₂	semidirect
11. C₄×#11761612	C₄	semidirect
12. M₁₆×Dic₆₇	C₄	direct (N × H)
23. C₄×M₁₆×C₆₇	C₄	semidirect
38. #14523954	C₂×C₄	semidirect
39. #14523953	C₂×C₄	semidirect
40. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
41. C₈×Dic₆₇	C₂×C₄	semidirect
52. #97×C₆₇	C₂×C₄	semidirect
53. C₄×C₈×C₆₇	C₂×C₄	semidirect
58. #11761612	C₄²	semidirect
64. C₄×Dic₆₇	M₁₆	direct (N × H)
77. M₁₆×C₆₇	C₄²	semidirect
106. C₈×C₆₇	C₂×C₄²	semidirect
107. C₆₇⋊C₈	C₂×C₄²	semidirect
116. C₄×C₆₇	C₄×M₁₆	semidirect
120. Dic₆₇	C₄×M₁₆	direct (N × H)
133. C₆₇	C₄²×M₁₆	semidirect
141. C₄×M₁₆	Dic₆₇	direct (N × H)
176. #97	C₂×Dic₆₇	semidirect
179. C₄×C₈	C₂×Dic₆₇	semidirect
195. M₁₆	C₄×Dic₆₇	direct (N × H)
233. C₈	C₂×C₄×Dic₆₇	semidirect
244. C₄	M₁₆×Dic₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11761612	2
C₂×M₁₆×Dic₆₇	2
C₂×C₄²×Dic₆₇	2
C₄×#14523953	2
#97×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₄²×M₁₆	67

Subgroups — C4×M16×Dic67 #30853161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×Dic₆₇ #30853161