Subgroups — C₂×#22562013 #31008903

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
2	#22562013	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	C₄	1	1	yes	characteristic
10	C₂×#14524002	C₄	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16863351	C₄	1	4	yes	—
14	#16863351	C₂²	1	4	yes	—
15	C₂×#14523953	C₄	1	1	yes	characteristic
16	C₂×#14523953	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	Dic₆₇⋊C₁₆	C₂²	1	4	yes	complemented
20	Dic₆₇⋊C₁₆	C₂²	1	4	yes	complemented
21	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
22	#16863171	C₂²	1	4	yes	complemented
23	C₁₆×Dic₆₇	C₂²	1	4	yes	complemented
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
28	C₂×#11761612	—	2	4	no	—
29	C₂×#11761612	C₂×C₄	1	4	yes	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	#14524002	C₈	1	4	yes	—
33	#14524002	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₆₇	C₈	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₆₇	C₈	1	1	yes	characteristic
38	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₁₆	—	2	2	no	—
41	C₂×C₆₇⋊C₁₆	—	2	2	no	—
42	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
44	#14523953	C₂×C₄	1	4	yes	—
45	#14523953	C₂×C₄	1	4	yes	—
46	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
47	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
48	C₈×Dic₆₇	C₂³	1	2	yes	—
49	C₈×Dic₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₆₇	—	2	2	no	—
58	C₂×C₁₆×C₆₇	—	2	2	no	—
59	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
65	#11761612	—	2	8	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
70	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
71	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
72	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
73	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
74	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
75	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
76	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
77	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
78	C₆₇⋊C₁₆	—	2	2	no	—
79	C₆₇⋊C₁₆	—	2	2	no	—
80	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₁₆×C₆₇	—	2	2	no	—
97	C₁₆×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	2	2	no	—
114	C₂×Dic₆₇	—	2	2	no	—
115	C₂×Dic₆₇	—	2	2	no	—
116	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
117	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
118	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
119	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	#117	1	2	yes	—
122	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₆₇⋊C₈	—	2	2	no	—
125	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	4	4	no	—
130	C₂²×C₆₇	#431	1	2	yes	—
131	C₂²×C₆₇	#504	1	2	yes	—
132	C₂²×C₆₇	#504	1	2	yes	—
133	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	#431	1	2	yes	—
136	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
138	Dic₆₇	—	2	2	no	—
139	Dic₆₇	—	2	2	no	—
140	Dic₆₇	—	2	2	no	—
141	Dic₆₇	—	2	2	no	—
142	C₂×#2164	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	4	2	no	—
144	C₂×C₆₇	#2164	1	4	yes	complemented
145	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
148	C₂×#405	—	67	1	no	—
149	#2164	—	67	8	no	—
150	C₂×C₄⋊C₁₆	—	67	1	no	—
151	C₂×C₄⋊C₁₆	—	67	1	no	—
152	C₂×C₂²⋊C₁₆	—	67	1	no	—
153	C₂×C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
154	C₂×#346	—	67	1	no	—
155	C₂×C₄×C₁₆	—	67	1	no	—
156	C₆₇	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	67	1	no	—
158	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂×#117	—	67	1	no	—
160	C₂²×C₁₆	—	67	1	no	—
161	C₂²×C₁₆	—	67	1	no	—
162	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
163	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
164	#405	—	67	4	no	—
165	#405	—	67	4	no	—
166	C₂×#97	—	67	1	no	—
167	C₂×#97	—	67	1	no	—
168	C₂×C₄×C₈	—	67	1	no	—
169	C₂×C₄×C₈	—	67	1	no	—
170	C₄⋊C₁₆	—	67	4	no	—
171	C₄⋊C₁₆	—	67	4	no	—
172	C₂²⋊C₁₆	—	67	4	no	—
173	C₂²⋊C₁₆	C₂×D₆₇	1	4	yes	—
174	#346	—	67	4	no	—
175	C₄×C₁₆	—	67	4	no	—
176	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
177	C₂×M₁₆	—	67	4	no	—
178	C₂×M₁₆	—	134	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	67	1	no	—
182	C₂²×C₈	—	67	1	no	—
183	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₆₇	1	2	yes	—
185	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
186	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	#117	—	67	4	no	—
188	#117	—	67	4	no	—
189	C₂×C₄⋊C₄	—	67	1	no	—
190	C₂×C₄⋊C₄	—	67	1	no	—
191	C₂×C₂²⋊C₄	—	67	1	no	—
192	C₂×C₂²⋊C₄	—	67	1	no	—
193	C₂×C₄²	—	67	1	no	—
194	C₂×C₄²	—	67	1	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	67	2	no	—
198	C₂×C₁₆	—	67	2	no	—
199	C₂×C₁₆	—	134	2	no	—
200	C₂×C₁₆	—	134	2	no	—
201	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
202	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
203	#97	—	67	4	no	—
204	#97	—	67	4	no	—
205	C₄×C₈	—	67	2	no	—
206	C₄×C₈	—	67	2	no	—
207	C₄×C₈	—	67	2	no	—
208	C₄×C₈	—	67	2	no	—
209	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	67	1	no	—
213	C₂²×C₄	—	67	1	no	—
214	C₂²×C₄	—	67	1	no	—
215	C₂²×C₄	—	67	1	no	—
216	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₈×D₆₇	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
220	M₁₆	—	134	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	67	4	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	67	2	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₈	#11761627	1	2	yes	—
232	C₂×C₈	#11761627	1	2	yes	—
233	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
234	C₄⋊C₄	—	67	4	no	—
235	C₄⋊C₄	—	67	4	no	—
236	C₂²⋊C₄	—	67	4	no	—
237	C₂²⋊C₄	—	67	4	no	—
238	C₄²	—	67	2	no	—
239	C₄²	—	67	2	no	—
240	C₄²	—	67	2	no	—
241	C₄²	—	67	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	134	2	no	—
245	C₁₆	—	134	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×C₈×D₆₇	1	2	yes	—
250	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	67	2	no	—
256	C₂×C₄	—	67	2	no	—
257	C₂×C₄	—	67	2	no	—
258	C₂×C₄	—	67	2	no	—
259	C₂×C₄	—	134	2	no	—
260	C₂×C₄	—	134	2	no	—
261	C₂×C₄	—	134	2	no	—
262	C₂×C₄	—	134	2	no	—
263	C₂×C₄	#14523999	1	2	yes	—
264	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
265	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
267	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	67	4	no	—
270	C₈	—	134	2	no	—
271	C₈	#14523999	1	2	yes	—
272	C₈	C₂×#11761627	1	1	yes	characteristic
273	C₈	C₂×#11761627	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	#16863446	1	2	yes	—
279	C₂²	#16863812	1	2	yes	—
280	C₂²	#16863814	1	2	yes	—
281	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	134	2	no	—
284	C₄	—	134	2	no	—
285	C₄	—	134	2	no	—
286	C₄	—	134	2	no	—
287	C₄	#16863446	1	2	yes	—
288	C₄	C₂×#14523999	1	1	yes	characteristic
289	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#22562013	1	4	yes	direct product factor
292	C₂	C₂×#16863446	1	1	yes	characteristic
293	C₂	C₂×#16863812	1	1	yes	characteristic
294	C₂	C₂×#16863814	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₂²⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2164
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22562013	C₂	direct (N × H)
3. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
4. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
6. C₂×#16863171	C₂	semidirect
7. C₂×C₁₆×Dic₆₇	C₂	semidirect
19. Dic₆₇⋊C₁₆	C₂²	semidirect
20. Dic₆₇⋊C₁₆	C₂²	semidirect
22. #16863171	C₂²	semidirect
23. C₁₆×Dic₆₇	C₂²	semidirect
144. C₂×C₆₇	#2164	semidirect
156. C₆₇	C₂×#2164	semidirect
291. C₂	#22562013	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×#16863351	2
#22562013	2
C₂×Dic₆₇⋊C₁₆	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#16863171	2
C₂×C₁₆×Dic₆₇	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₂×#2164	67

Subgroups — C2×#22562013 #31008903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562013 #31008903