Subgroups — C₁₆×#11761628 #31009348

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
3	C₁₆×D₂₆₈	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₂²⋊D₆₇	C₂	1	4	yes	maximal, complemented
5	#22562008	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562006	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₆×D₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11761628	C₄	1	1	yes	characteristic
10	C₂×C₁₆×D₆₇	C₂²	1	4	yes	—
11	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	C₈×D₂₆₈	C₄	1	1	yes	characteristic
14	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₆₇	C₄	1	2	yes	—
16	C₈×C₂²⋊D₆₇	C₂²	1	2	yes	—
17	#16863449	C₄	1	1	yes	characteristic
18	#16863447	C₄	1	2	yes	—
19	C₄×C₈×D₆₇	C₂²	1	2	yes	—
20	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
21	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
23	(C₂×D₆₇)⋊C₁₆	C₂²	1	4	yes	—
24	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
25	C₁₆×Dic₆₇	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#11761628	C₈	1	1	yes	characteristic
32	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₆₇	C₂³	1	2	yes	—
34	C₄×Dic₁₃₄	C₈	1	1	yes	characteristic
35	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₆₈	C₈	1	1	yes	characteristic
37	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₆₇	C₈	1	2	yes	—
39	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
40	#14524002	C₈	1	1	yes	characteristic
41	#14524000	C₈	1	2	yes	—
42	C₄²×D₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
44	C₁₆×D₆₇	—	2	8	no	—
45	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
48	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
51	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
54	C₈×Dic₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₆₇	—	2	4	no	—
59	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#11761628	C₁₆	1	8	yes	direct product factor
66	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
67	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₆₇	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
74	C₈×D₆₇	—	2	4	no	—
75	C₈×D₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
77	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
78	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
79	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
80	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
81	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
82	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
83	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
84	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
85	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
86	C₆₇⋊C₁₆	—	2	4	no	—
87	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₆₇	—	2	2	no	—
101	C₁₆×C₆₇	#41	1	4	yes	complemented
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₆₇	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	complemented
108	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂²⋊D₆₇	C₂×C₁₆	1	8	yes	complemented
115	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
116	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
117	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
118	D₂₆₈	C₂×C₁₆	1	4	yes	complemented
119	C₄×D₆₇	—	2	2	no	—
120	C₄×D₆₇	—	2	2	no	—
121	C₄×D₆₇	—	2	4	no	—
122	C₄×D₆₇	C₂×C₁₆	1	8	yes	complemented
123	Dic₁₃₄	C₂×C₁₆	1	4	yes	complemented
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	#131	1	2	yes	—
127	C₈×C₆₇	C₂×#41	1	2	yes	—
128	C₆₇⋊C₈	—	2	2	no	—
129	C₆₇⋊C₈	—	2	2	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
133	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
134	C₂×D₆₇	—	2	2	no	—
135	C₂×D₆₇	—	2	2	no	—
136	C₂×D₆₇	C₂²×C₁₆	1	4	yes	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	C₂²×C₁₆	1	2	yes	—
140	C₄×C₆₇	C₂²×C₁₆	1	2	yes	—
141	C₄×C₆₇	#504	1	2	yes	—
142	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
144	Dic₆₇	—	2	2	no	—
145	Dic₆₇	—	2	2	no	—
146	Dic₆₇	C₂²×C₁₆	1	4	yes	—
147	C₁₆×#41	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
150	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
152	D₆₇	—	2	4	no	—
153	C₈×#41	—	67	1	no	—
154	Q₈×C₁₆	—	67	2	no	—
155	D₄×C₁₆	—	67	4	no	—
156	D₄×C₁₆	—	67	2	no	—
157	#2162	—	67	1	no	—
158	#2162	—	67	2	no	—
159	C₂×C₄×C₁₆	—	67	2	no	—
160	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic, complemented
161	C₆₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	—	67	1	no	—
163	C₂²×C₁₆	—	67	4	no	—
164	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
165	C₈×Q₈	—	67	1	no	—
166	C₈×Q₈	—	67	1	no	—
167	C₈×D₄	—	67	2	no	—
168	C₈×D₄	—	67	2	no	—
169	C₈×D₄	—	67	1	no	—
170	C₈×D₄	—	67	1	no	—
171	#431	—	67	1	no	—
172	#431	—	67	2	no	—
173	C₂×C₄×C₈	—	67	2	no	—
174	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
175	C₄⋊C₁₆	—	67	4	no	—
176	C₄⋊C₁₆	—	67	2	no	—
177	C₂²⋊C₁₆	—	67	2	no	—
178	C₂²⋊C₁₆	—	67	4	no	—
179	C₄×C₁₆	—	67	4	no	—
180	C₄×C₁₆	—	67	2	no	—
181	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
182	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
183	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#41	—	67	1	no	—
185	C₂²×C₈	—	67	2	no	—
186	C₂²×C₈	—	67	2	no	—
187	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₄×Q₈	—	67	1	no	—
190	C₄×Q₈	—	67	1	no	—
191	C₄×D₄	—	67	2	no	—
192	C₄×D₄	—	67	2	no	—
193	C₄×D₄	—	67	1	no	—
194	C₄×D₄	—	67	1	no	—
195	#117	—	67	1	no	—
196	#117	—	67	2	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₁₆	—	2	4	no	—
200	C₂×C₁₆	—	67	4	no	—
201	C₂×C₁₆	—	134	8	no	—
202	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
203	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
204	C₄⋊C₈	—	67	2	no	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	67	1	no	—
207	C₄⋊C₈	—	67	1	no	—
208	C₂²⋊C₈	—	67	2	no	—
209	C₂²⋊C₈	—	67	2	no	—
210	C₂²⋊C₈	—	67	2	no	—
211	C₄×C₈	—	67	2	no	—
212	C₄×C₈	—	67	2	no	—
213	C₄×C₈	—	67	2	no	—
214	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
215	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	#41	—	67	8	no	—
219	C₂×Q₈	—	67	1	no	—
220	C₂×D₄	—	67	2	no	—
221	C₂×D₄	—	67	1	no	—
222	C₂²×C₄	—	67	2	no	—
223	C₂²×C₄	—	67	2	no	—
224	C₂²×C₄	—	67	2	no	—
225	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	67	2	no	—
231	C₂×C₈	—	67	2	no	—
232	C₂×C₈	—	134	4	no	—
233	C₂×C₈	—	134	4	no	—
234	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
238	C₄⋊C₄	—	67	2	no	—
239	C₄⋊C₄	—	67	2	no	—
240	C₄⋊C₄	—	67	1	no	—
241	C₄⋊C₄	—	67	1	no	—
242	C₂²⋊C₄	—	67	2	no	—
243	C₂²⋊C₄	—	67	2	no	—
244	C₂²⋊C₄	—	67	2	no	—
245	C₄²	—	67	2	no	—
246	C₄²	—	67	2	no	—
247	C₄²	—	67	2	no	—
248	C₄²	C₈×D₆₇	1	1	yes	characteristic
249	C₄²	C₈×D₆₇	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
252	C₁₆	—	2	2	no	—
253	C₁₆	—	134	4	no	—
254	C₁₆	#11761628	1	4	yes	direct product factor
255	C₂³	—	67	2	no	—
256	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
257	Q₈	—	67	4	no	—
258	D₄	—	67	8	no	—
259	D₄	—	67	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	67	8	no	—
265	C₂×C₄	—	67	2	no	—
266	C₂×C₄	—	67	2	no	—
267	C₂×C₄	—	134	4	no	—
268	C₂×C₄	—	134	2	no	—
269	C₂×C₄	—	134	2	no	—
270	C₂×C₄	C₁₆×D₆₇	1	4	yes	complemented
271	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	134	2	no	—
280	C₈	—	134	2	no	—
281	C₈	#14524049	1	2	yes	—
282	C₈	C₂×#11761628	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	67	4	no	—
286	C₂²	—	134	2	no	—
287	C₂²	—	134	2	no	—
288	C₂²	C₂×C₁₆×D₆₇	1	2	yes	—
289	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	67	4	no	—
293	C₄	—	134	2	no	—
294	C₄	—	134	2	no	—
295	C₄	C₂×C₁₆×D₆₇	1	2	yes	—
296	C₄	C₂×C₁₆×D₆₇	1	2	yes	—
297	C₄	#16863812	1	2	yes	—
298	C₄	C₄×#11761628	1	1	yes	characteristic
299	C₄	C₂×#14524049	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	134	4	no	—
302	C₂	C₈×#11761628	1	1	yes	characteristic
303	C₂	C₂²×C₁₆×D₆₇	1	1	yes	characteristic
304	C₂	C₂×#16863812	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂×#11761628	C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₁₆×#41
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₁₃₄	C₂	semidirect
3. C₁₆×D₂₆₈	C₂	semidirect
4. C₁₆×C₂²⋊D₆₇	C₂	semidirect
5. #22562008	C₂	semidirect
6. #22562006	C₂	semidirect
7. C₄×C₁₆×D₆₇	C₂	semidirect
8. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
65. #11761628	C₁₆	direct (N × H)
101. C₁₆×C₆₇	#41	semidirect
107. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
114. C₂²⋊D₆₇	C₂×C₁₆	semidirect
118. D₂₆₈	C₂×C₁₆	semidirect
122. C₄×D₆₇	C₂×C₁₆	semidirect
123. Dic₁₃₄	C₂×C₁₆	semidirect
160. C₂×C₄×C₁₆	D₆₇	semidirect
161. C₆₇	C₁₆×#41	semidirect
254. C₁₆	#11761628	direct (N × H)
270. C₂×C₄	C₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761628	2
C₁₆×Dic₁₃₄	2
C₁₆×D₂₆₈	2
C₁₆×C₂²⋊D₆₇	2
#22562008	2
#22562006	2
C₄×C₁₆×D₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₁₆×#41	67

Subgroups — C16×#11761628 #31009348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#11761628 #31009348