Subgroups — C₂²×M₁₆×Dic₆₇ #31719217

Order 17152.

3950 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×Dic₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14523953	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₆₇	C₄	1	2	yes	—
8	C₂²×#11761612	C₄	1	16	yes	complemented
9	C₂²×#11761612	C₂²	1	12	yes	—
10	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₆₇	C₄	1	12	yes	—
12	C₂²×C₄×Dic₆₇	C₄	1	2	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₆₇	C₂²	1	12	yes	—
16	M₁₆×Dic₆₇	C₂²	1	64	yes	direct product factor
17	C₂×#14523953	C₂²	1	24	yes	complemented
18	C₂×C₈×Dic₆₇	C₂²	1	24	yes	complemented
19	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₄	1	16	yes	complemented
21	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
23	C₂³×Dic₆₇	—	2	16	no	—
24	C₂³×Dic₆₇	C₂×C₄	1	24	yes	—
25	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
26	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
27	C₂×#11761612	C₂×C₄	1	96	yes	complemented
28	C₂×#11761612	C₂³	1	16	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	16	yes	complemented
30	C₂²×C₆₇⋊C₈	C₂³	1	12	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	48	yes	—
33	C₂×C₄×Dic₆₇	C₂×C₄	1	24	yes	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
36	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
37	C₂×C₄×Dic₆₇	C₂³	1	16	yes	—
38	#14523953	C₂³	1	32	yes	complemented
39	C₈×Dic₆₇	C₂³	1	32	yes	complemented
40	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₆₇	C₂×C₄	1	96	yes	complemented
48	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
50	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
51	C₂²×Dic₆₇	—	2	8	no	—
52	C₂²×Dic₆₇	—	2	96	no	—
53	C₂²×Dic₆₇	M₁₆	1	16	yes	direct product factor
54	C₂²×Dic₆₇	C₂²×C₄	1	12	yes	—
55	C₂²×Dic₆₇	C₂²×C₄	1	32	yes	—
56	C₂²×Dic₆₇	C₂²×C₄	1	12	yes	—
57	#11761612	C₂²×C₄	1	64	yes	complemented
58	C₂×C₆₇⋊C₈	C₂²×C₄	1	48	yes	complemented
59	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
60	C₄×Dic₆₇	C₂²×C₄	1	12	yes	—
61	C₄×Dic₆₇	C₂²×C₄	1	32	yes	—
62	C₄×Dic₆₇	C₂²×C₄	1	12	yes	—
63	C₄×Dic₆₇	C₂⁴	1	4	yes	—
64	C₄×Dic₆₇	C₂⁴	1	4	yes	—
65	C₂⁴×C₆₇	—	2	8	no	—
66	C₂⁴×C₆₇	C₄²	1	8	yes	—
67	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
68	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₄²	1	48	yes	—
70	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
80	M₁₆×C₆₇	C₂²×C₄	1	64	yes	complemented
81	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	complemented
82	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
83	C₂³×C₆₇	—	2	24	no	—
84	C₂³×C₆₇	—	2	32	no	—
85	C₂³×C₆₇	C₂×C₄²	1	24	yes	—
86	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
87	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
88	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
91	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
92	C₂×C₄×C₆₇	C₂×C₄²	1	24	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
95	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
99	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	24	no	—
101	C₂×Dic₆₇	—	2	64	no	—
102	C₂×Dic₆₇	C₂×M₁₆	1	48	yes	direct product factor
103	C₂×Dic₆₇	C₂³×C₄	1	8	yes	—
104	C₂×Dic₆₇	C₂³×C₄	1	8	yes	—
105	C₈×C₆₇	C₂³×C₄	1	16	yes	complemented
106	C₆₇⋊C₈	C₂³×C₄	1	16	yes	complemented
107	C₂²×C₆₇	—	2	48	no	—
108	C₂²×C₆₇	—	2	8	no	—
109	C₂²×C₆₇	C₄×M₁₆	1	16	yes	complemented
110	C₂²×C₆₇	C₂²×C₄²	1	8	yes	—
111	C₂²×C₆₇	C₂²×C₄²	1	6	yes	—
112	C₂²×C₆₇	C₂²×M₁₆	1	12	yes	—
113	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂²×C₄²	1	6	yes	—
115	C₄×C₆₇	C₂²×C₄²	1	8	yes	—
116	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
118	Dic₆₇	—	2	8	no	—
119	Dic₆₇	C₂²×M₁₆	1	16	yes	direct product factor
120	C₂²×C₄×M₁₆	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	8	no	—
122	C₂×C₆₇	C₂×C₄×M₁₆	1	12	yes	complemented
123	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
124	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
125	C₂³×M₁₆	—	67	1	no	—
126	C₂³×M₁₆	D₆₇	1	1	yes	characteristic
127	C₂³×C₄²	—	67	1	no	—
128	C₂×C₄×M₁₆	—	67	24	no	—
129	C₂²×#97	—	67	2	no	—
130	C₂²×C₄×C₈	—	67	2	no	—
131	C₆₇	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
132	C₂⁴×C₄	—	67	2	no	—
133	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
134	C₂²×M₁₆	—	67	16	no	—
135	C₂²×M₁₆	—	67	12	no	—
136	C₂²×M₁₆	Dic₆₇	1	16	yes	direct product factor
137	C₂²×M₁₆	C₂×D₆₇	1	12	yes	—
138	C₂³×C₈	—	67	2	no	—
139	C₂³×C₈	C₂×D₆₇	1	2	yes	—
140	C₂²×C₄²	—	67	2	no	—
141	C₂²×C₄²	—	67	12	no	—
142	C₂²×C₄²	—	67	12	no	—
143	C₂²×C₄²	—	67	1	no	—
144	C₂²×C₄²	—	67	1	no	—
145	C₄×M₁₆	—	67	64	no	—
146	C₂×#97	—	67	24	no	—
147	C₂×C₄×C₈	—	67	24	no	—
148	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
149	C₂³×C₄	—	67	2	no	—
150	C₂³×C₄	—	67	24	no	—
151	C₂³×C₄	—	67	2	no	—
152	C₂³×C₄	—	134	16	no	—
153	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
154	C₂³×C₄	C₄×D₆₇	1	6	yes	—
155	C₂³×C₄	C₄×D₆₇	1	8	yes	—
156	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
157	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
158	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
159	C₂×M₁₆	—	67	96	no	—
160	C₂×M₁₆	—	67	16	no	—
161	C₂×M₁₆	C₂×Dic₆₇	1	96	yes	direct product factor
162	C₂×M₁₆	C₂²×D₆₇	1	16	yes	—
163	C₂²×C₈	—	67	16	no	—
164	C₂²×C₈	—	67	12	no	—
165	C₂²×C₈	C₂×Dic₆₇	1	16	yes	complemented
166	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
167	C₂×C₄²	—	67	6	no	—
168	C₂×C₄²	—	67	24	no	—
169	C₂×C₄²	—	67	48	no	—
170	C₂×C₄²	—	67	6	no	—
171	C₂×C₄²	—	67	6	no	—
172	C₂×C₄²	—	67	6	no	—
173	C₂×C₄²	—	67	16	no	—
174	#97	—	67	32	no	—
175	C₄×C₈	—	67	32	no	—
176	C₂⁴	—	2	8	no	—
177	C₂⁴	C₄×Dic₆₇	1	8	yes	—
178	C₂⁴	C₂×C₄×D₆₇	1	6	yes	—
179	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	—	67	16	no	—
181	C₂²×C₄	—	67	12	no	—
182	C₂²×C₄	—	67	32	no	—
183	C₂²×C₄	—	67	12	no	—
184	C₂²×C₄	—	134	96	no	—
185	C₂²×C₄	—	134	8	no	—
186	C₂²×C₄	C₄×Dic₆₇	1	48	yes	—
187	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
188	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
189	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
190	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
191	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
192	C₂²×C₄	C₂×C₄×D₆₇	1	24	yes	—
193	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
195	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
196	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
197	M₁₆	—	67	64	no	—
198	M₁₆	C₂²×Dic₆₇	1	64	yes	direct product factor
199	C₂×C₈	—	67	48	no	—
200	C₂×C₈	—	67	8	no	—
201	C₂×C₈	C₂²×Dic₆₇	1	48	yes	complemented
202	C₂×C₈	C₂³×D₆₇	1	8	yes	—
203	C₄²	—	67	12	no	—
204	C₄²	—	67	32	no	—
205	C₄²	—	67	12	no	—
206	C₄²	—	67	4	no	—
207	C₄²	—	67	4	no	—
208	C₂³	—	2	32	no	—
209	C₂³	—	2	24	no	—
210	C₂³	C₂×C₄×Dic₆₇	1	24	yes	—
211	C₂³	C₂×C₄×Dic₆₇	1	4	yes	—
212	C₂³	M₁₆×D₆₇	1	8	yes	—
213	C₂³	C₂²×C₄×D₆₇	1	4	yes	—
214	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
215	C₂×C₄	—	67	48	no	—
216	C₂×C₄	—	67	8	no	—
217	C₂×C₄	—	67	8	no	—
218	C₂×C₄	—	134	64	no	—
219	C₂×C₄	—	134	24	no	—
220	C₂×C₄	C₂×C₄×Dic₆₇	1	48	yes	—
221	C₂×C₄	C₂×C₄×Dic₆₇	1	24	yes	—
222	C₂×C₄	C₂×C₄×Dic₆₇	1	12	yes	—
223	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
224	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
225	C₂×C₄	C₂²×C₄×D₆₇	1	6	yes	—
226	C₂×C₄	C₂²×C₄×D₆₇	1	8	yes	—
227	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
228	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
229	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
230	C₈	—	67	16	no	—
231	C₈	C₂³×Dic₆₇	1	16	yes	complemented
232	C₂²	—	2	8	no	—
233	C₂²	—	2	48	no	—
234	C₂²	M₁₆×Dic₆₇	1	16	yes	direct product factor
235	C₂²	C₂²×C₄×Dic₆₇	1	8	yes	—
236	C₂²	C₂²×C₄×Dic₆₇	1	6	yes	—
237	C₂²	C₂×M₁₆×D₆₇	1	12	yes	—
238	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
239	C₄	—	67	16	no	—
240	C₄	—	134	8	no	—
241	C₄	C₂²×C₄×Dic₆₇	1	6	yes	—
242	C₄	C₂²×C₄×Dic₆₇	1	8	yes	—
243	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
244	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
245	C₂	—	2	8	no	—
246	C₂	C₂×M₁₆×Dic₆₇	1	12	yes	direct product factor
247	C₂	C₂³×C₄×Dic₆₇	1	1	yes	characteristic
248	C₂	C₂²×M₁₆×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×M₁₆×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×M₁₆	D₆₇
hypocenter	C₆₇	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×M₁₆×Dic₆₇	C₂	direct (N × H)
4. C₂²×#14523953	C₂	semidirect
5. C₂²×C₈×Dic₆₇	C₂	semidirect
8. C₂²×#11761612	C₄	semidirect
16. M₁₆×Dic₆₇	C₂²	direct (N × H)
17. C₂×#14523953	C₂²	semidirect
18. C₂×C₈×Dic₆₇	C₂²	semidirect
20. C₂²×M₁₆×C₆₇	C₄	semidirect
27. C₂×#11761612	C₂×C₄	semidirect
29. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
38. #14523953	C₂³	semidirect
39. C₈×Dic₆₇	C₂³	semidirect
47. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
49. C₂²×C₈×C₆₇	C₂×C₄	semidirect
53. C₂²×Dic₆₇	M₁₆	direct (N × H)
57. #11761612	C₂²×C₄	semidirect
58. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
80. M₁₆×C₆₇	C₂²×C₄	semidirect
81. C₂×C₈×C₆₇	C₂²×C₄	semidirect
102. C₂×Dic₆₇	C₂×M₁₆	direct (N × H)
105. C₈×C₆₇	C₂³×C₄	semidirect
106. C₆₇⋊C₈	C₂³×C₄	semidirect
109. C₂²×C₆₇	C₄×M₁₆	semidirect
119. Dic₆₇	C₂²×M₁₆	direct (N × H)
122. C₂×C₆₇	C₂×C₄×M₁₆	semidirect
131. C₆₇	C₂²×C₄×M₁₆	semidirect
136. C₂²×M₁₆	Dic₆₇	direct (N × H)
161. C₂×M₁₆	C₂×Dic₆₇	direct (N × H)
165. C₂²×C₈	C₂×Dic₆₇	semidirect
198. M₁₆	C₂²×Dic₆₇	direct (N × H)
201. C₂×C₈	C₂²×Dic₆₇	semidirect
231. C₈	C₂³×Dic₆₇	semidirect
234. C₂²	M₁₆×Dic₆₇	direct (N × H)
246. C₂	C₂×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
C₂³×#11761612	2
C₂³×C₄×Dic₆₇	2
C₂×M₁₆×Dic₆₇	2
C₂²×#14523953	2
C₂²×C₈×Dic₆₇	2
C₂³×M₁₆×C₆₇	2
C₂²×C₄×M₁₆	67

Subgroups — C22×M16×Dic67 #31719217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×Dic₆₇ #31719217