Subgroups — C₂×#22562295 #31010249

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863632	C₂	1	2	yes	maximal, complemented
3	#22562295	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863222	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863198	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₃₆	C₂²	1	2	yes	—
8	C₂×#14524017	—	2	2	no	—
9	C₂×D₁₀₇₂	—	2	4	no	—
10	C₂×D₁₀₇₂	C₂²	1	4	yes	complemented
11	#16863632	—	2	4	no	—
12	#16863632	C₂²	1	8	yes	complemented
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523965	—	2	2	no	—
15	#16863222	C₂²	1	4	yes	complemented
16	#16863198	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₆₇	—	2	4	no	—
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	complemented
22	C₂²×D₂₆₈	—	2	2	no	—
23	C₂²×C₂²⋊D₆₇	—	4	2	no	—
24	C₂×D₅₃₆	—	2	4	no	—
25	C₂×D₅₃₆	—	2	2	no	—
26	C₂×D₅₃₆	—	2	4	no	—
27	C₂×D₅₃₆	C₂³	1	4	yes	—
28	#14524017	—	2	8	no	—
29	#14524017	—	4	4	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×(C₂×D₆₇)⋊C₄	—	4	2	no	—
32	D₁₀₇₂	—	2	8	no	—
33	#14523977	—	2	2	no	—
34	#14523977	C₂³	1	4	yes	—
35	#14523965	—	2	8	no	—
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	—	2	8	no	—
45	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
46	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
48	C₂³×D₆₇	—	4	2	no	—
49	C₂×D₂₆₈	—	2	4	no	—
50	C₂×D₂₆₈	—	4	4	no	—
51	C₂×D₂₆₈	—	4	4	no	—
52	C₂×D₂₆₈	—	4	2	no	—
53	C₂×C₂²⋊D₆₇	—	4	8	no	—
54	C₂×C₂²⋊D₆₇	—	8	8	no	—
55	C₂²×Dic₆₇	—	4	2	no	—
56	D₅₃₆	—	2	4	no	—
57	D₅₃₆	—	2	4	no	—
58	D₅₃₆	—	4	4	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	(C₂×D₆₇)⋊C₄	—	4	8	no	—
62	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	D₈	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	#41	1	2	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₆₇	—	2	2	no	—
79	C₁₆×C₆₇	C₂×D₄	1	4	yes	complemented
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	D₁₆	1	4	yes	complemented
83	C₂³×C₆₇	C₂×D₈	1	2	yes	—
84	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂²×D₆₇	—	4	4	no	—
86	C₂²×D₆₇	—	8	4	no	—
87	C₂²×D₆₇	—	8	4	no	—
88	C₂²×D₆₇	—	8	4	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	#121	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₆₇	—	8	16	no	—
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	8	4	no	—
101	D₂₆₈	—	4	4	no	—
102	D₂₆₈	—	4	4	no	—
103	D₂₆₈	—	8	4	no	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	#121	1	2	yes	—
106	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	8	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	#466	1	2	yes	—
112	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
113	C₂²×C₆₇	C₂×D₁₆	1	4	yes	complemented
114	C₂²×C₆₇	#508	1	2	yes	—
115	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₆₇	—	8	4	no	—
117	C₂×D₆₇	—	8	4	no	—
118	C₂×D₆₇	—	8	4	no	—
119	C₂×D₆₇	—	16	8	no	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	#466	1	2	yes	—
122	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
123	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
124	Dic₆₇	—	8	4	no	—
125	C₂×#2215	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	4	no	—
127	C₂×C₆₇	#2215	1	4	yes	complemented
128	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
129	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
131	D₆₇	—	16	4	no	—
132	C₂²×D₁₆	—	67	1	no	—
133	C₂³×C₁₆	D₆₇	1	1	yes	characteristic, complemented
134	C₂×#466	—	67	2	no	—
135	#2215	—	67	8	no	—
136	C₂×#366	—	67	1	no	—
137	C₂×#357	—	67	2	no	—
138	C₆₇	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
139	C₂²×D₈	—	67	2	no	—
140	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₂×#121	—	134	2	no	—
142	C₂×D₁₆	—	67	4	no	—
143	C₂×D₁₆	—	134	4	no	—
144	C₂²×C₁₆	—	2	4	no	—
145	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
146	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
147	C₂²×C₁₆	C₂×D₆₇	1	4	yes	complemented
148	#466	—	67	8	no	—
149	#466	—	134	4	no	—
150	C₂×#107	—	67	2	no	—
151	C₂×#102	—	134	2	no	—
152	#366	—	67	4	no	—
153	#357	—	67	8	no	—
154	C₂²×D₄	—	268	2	no	—
155	C₂²×D₄	—	134	2	no	—
156	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
157	C₂×D₈	—	67	4	no	—
158	C₂×D₈	—	134	4	no	—
159	C₂×D₈	—	134	2	no	—
160	C₂×D₈	—	134	4	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
164	C₂²×C₈	D₂₆₈	1	2	yes	—
165	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
166	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
167	#121	—	268	4	no	—
168	#121	—	134	8	no	—
169	C₂×C₄⋊C₄	—	134	2	no	—
170	C₂×C₂²⋊C₄	—	268	2	no	—
171	D₁₆	—	134	8	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	8	no	—
174	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
176	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
177	#107	—	67	4	no	—
178	#107	—	134	2	no	—
179	#102	—	134	8	no	—
180	C₂⁴	—	268	2	no	—
181	C₂⁴	D₅₃₆	1	1	yes	characteristic
182	C₂×D₄	—	268	8	no	—
183	C₂×D₄	—	268	4	no	—
184	C₂×D₄	—	268	2	no	—
185	C₂×D₄	—	268	4	no	—
186	C₂×D₄	—	536	8	no	—
187	C₂×D₄	—	134	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	268	2	no	—
191	C₂²×C₄	D₅₃₆	1	2	yes	—
192	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
194	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
195	D₈	—	268	4	no	—
196	D₈	—	134	4	no	—
197	D₈	—	134	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
203	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
204	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
205	C₂×C₈	#11761628	1	2	yes	—
206	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
207	C₄⋊C₄	—	268	2	no	—
208	C₄⋊C₄	—	134	4	no	—
209	C₂²⋊C₄	—	268	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	C₂×C₂²⋊D₆₇	1	4	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	268	4	no	—
215	C₂³	—	536	4	no	—
216	C₂³	—	536	4	no	—
217	C₂³	—	536	4	no	—
218	C₂³	D₁₀₇₂	1	4	yes	complemented
219	C₂³	C₂×D₅₃₆	1	2	yes	—
220	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
221	D₄	—	268	4	no	—
222	D₄	—	268	4	no	—
223	D₄	—	536	16	no	—
224	D₄	—	536	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	268	4	no	—
230	C₂×C₄	—	536	4	no	—
231	C₂×C₄	#14524017	1	2	yes	—
232	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
233	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
234	C₂×C₄	#14524065	1	2	yes	—
235	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#14524017	1	2	yes	—
238	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
239	C₈	C₂×#11761628	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	536	4	no	—
244	C₂²	—	536	4	no	—
245	C₂²	—	536	4	no	—
246	C₂²	—	1072	8	no	—
247	C₂²	#16863632	1	2	yes	—
248	C₂²	C₂×D₁₀₇₂	1	2	yes	—
249	C₂²	C₂×D₁₀₇₂	1	4	yes	complemented
250	C₂²	#16863828	1	2	yes	—
251	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	536	4	no	—
254	C₄	#16863632	1	2	yes	—
255	C₄	C₂×#14524017	1	1	yes	characteristic
256	C₄	C₂×#14524065	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	1072	4	no	—
259	C₂	#22562295	1	4	yes	direct product factor
260	C₂	C₂×#16863632	1	1	yes	characteristic
261	C₂	C₂²×D₁₀₇₂	1	1	yes	characteristic
262	C₂	C₂×#16863828	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂³×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2215
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₀₇₂	C₂	semidirect
2. C₂×#16863632	C₂	semidirect
3. #22562295	C₂	direct (N × H)
4. C₂×#16863222	C₂	semidirect
5. C₂×#16863198	C₂	semidirect
6. C₂³×C₁₆×C₆₇	C₂	semidirect
10. C₂×D₁₀₇₂	C₂²	semidirect
12. #16863632	C₂²	semidirect
15. #16863222	C₂²	semidirect
16. #16863198	C₂²	semidirect
21. C₂²×C₁₆×C₆₇	C₂²	semidirect
45. C₂×C₁₆×C₆₇	D₄	semidirect
79. C₁₆×C₆₇	C₂×D₄	semidirect
82. C₂³×C₆₇	D₁₆	semidirect
113. C₂²×C₆₇	C₂×D₁₆	semidirect
127. C₂×C₆₇	#2215	semidirect
133. C₂³×C₁₆	D₆₇	semidirect
138. C₆₇	C₂×#2215	semidirect
147. C₂²×C₁₆	C₂×D₆₇	semidirect
174. C₂×C₁₆	C₂²⋊D₆₇	semidirect
211. C₁₆	C₂×C₂²⋊D₆₇	semidirect
218. C₂³	D₁₀₇₂	semidirect
249. C₂²	C₂×D₁₀₇₂	semidirect
259. C₂	#22562295	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×D₁₀₇₂	2
C₂×#16863632	2
#22562295	2
C₂×#16863222	2
C₂×#16863198	2
C₂³×C₁₆×C₆₇	2
C₂×#2215	67

Subgroups — C2×#22562295 #31010249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562295 #31010249