Subgroups — C₂×#22562040 #31008994

Order 17152.

742 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523982	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22562040	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₇	C₂²	1	2	yes	—
8	C₂×C₄×D₂₆₈	C₄	1	1	yes	characteristic
9	C₂×#14523982	—	2	8	no	—
10	C₂×#14523982	C₂²	1	8	yes	complemented
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×D₂₆₈	C₄	1	4	yes	—
14	C₈×D₂₆₈	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₆₇)⋊C₈	C₄	1	2	yes	—
17	(C₂×D₆₇)⋊C₁₆	C₂²	1	8	yes	complemented
18	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
22	C₂²×D₂₆₈	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₆₇	—	2	4	no	—
26	C₂×C₈×D₆₇	—	2	4	no	—
27	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₆₇	C₂³	1	4	yes	—
29	C₄×D₂₆₈	C₈	1	4	yes	—
30	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
32	C₂×(C₂×D₆₇)⋊C₄	C₈	1	2	yes	—
33	#14523982	—	2	16	no	—
34	C₂×C₆₇⋊C₁₆	—	2	2	no	—
35	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	complemented
36	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
37	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
39	(C₂×D₆₇)⋊C₈	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₆₇	—	2	4	no	—
44	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂³×D₆₇	C₂×C₈	1	2	yes	—
48	C₂×D₂₆₈	—	2	4	no	—
49	C₂×D₂₆₈	C₂×C₈	1	4	yes	—
50	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₆₇	—	2	4	no	—
52	C₂×C₄×D₆₇	—	2	4	no	—
53	C₂×C₄×D₆₇	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₆₇	C₂²×C₄	1	4	yes	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
57	C₈×D₆₇	—	2	8	no	—
58	C₈×D₆₇	—	2	4	no	—
59	C₈×D₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
61	(C₂×D₆₇)⋊C₄	C₂×C₈	1	8	yes	—
62	C₆₇⋊C₁₆	—	2	2	no	—
63	C₆₇⋊C₁₆	C₂×D₄	1	4	yes	complemented
64	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×C₈	1	2	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
75	C₁₆×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₆₇	—	2	4	no	—
78	C₂²×D₆₇	—	2	4	no	—
79	C₂²×D₆₇	—	2	4	no	—
80	C₂²×D₆₇	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	M₃₂	1	4	yes	complemented
83	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₆₇	#131	1	2	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₆₇	—	2	4	no	—
90	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
91	D₂₆₈	—	2	8	no	—
92	C₄×D₆₇	—	2	4	no	—
93	C₄×D₆₇	—	2	4	no	—
94	C₄×D₆₇	—	2	8	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₈×C₆₇	C₄×D₄	1	2	yes	—
97	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
99	C₆₇⋊C₈	—	2	4	no	—
100	C₂²×C₆₇	C₈×D₄	1	2	yes	—
101	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
102	C₂²×C₆₇	#504	1	2	yes	—
103	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
104	C₂×D₆₇	—	2	4	no	—
105	C₂×D₆₇	—	2	4	no	—
106	C₂×D₆₇	—	2	4	no	—
107	C₂×D₆₇	—	4	8	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	C₈×D₄	1	2	yes	—
110	C₄×C₆₇	C₂×M₃₂	1	4	yes	complemented
111	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
113	Dic₆₇	—	2	4	no	—
114	C₂×#2169	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	#2169	1	4	yes	complemented
116	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
119	D₆₇	—	4	4	no	—
120	C₂²×M₃₂	—	67	2	no	—
121	C₂×C₈×D₄	—	67	1	no	—
122	#2169	—	67	8	no	—
123	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
124	C₂×C₂²⋊C₁₆	—	67	2	no	—
125	C₂×C₄×C₁₆	—	67	1	no	—
126	C₆₇	C₂×#2169	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	—	67	2	no	—
128	C₂×C₄×D₄	—	67	1	no	—
129	C₂×M₃₂	—	67	8	no	—
130	C₂×M₃₂	—	134	8	no	—
131	C₂²×C₁₆	—	67	1	no	—
132	C₂²×C₁₆	—	67	1	no	—
133	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
134	C₈×D₄	—	67	4	no	—
135	C₈×D₄	—	67	4	no	—
136	C₂×C₄⋊C₈	—	67	1	no	—
137	C₂×C₂²⋊C₈	—	67	2	no	—
138	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	complemented
140	C₂²⋊C₁₆	—	67	8	no	—
141	C₄×C₁₆	—	67	4	no	—
142	C₂²×D₄	—	67	1	no	—
143	C₂³×C₄	—	67	2	no	—
144	C₂²×C₈	—	67	2	no	—
145	C₂²×C₈	—	67	4	no	—
146	C₂²×C₈	—	67	4	no	—
147	C₂²×C₈	—	134	4	no	—
148	C₂²×C₈	—	134	4	no	—
149	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
151	C₄×D₄	—	67	4	no	—
152	C₄×D₄	—	67	4	no	—
153	C₂×C₄⋊C₄	—	67	1	no	—
154	C₂×C₂²⋊C₄	—	67	2	no	—
155	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
156	M₃₂	—	134	16	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	—	67	4	no	—
159	C₂×C₁₆	—	67	2	no	—
160	C₂×C₁₆	—	67	2	no	—
161	C₂×C₁₆	—	134	2	no	—
162	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
163	C₄⋊C₈	—	67	4	no	—
164	C₂²⋊C₈	—	67	8	no	—
165	C₄×C₈	C₄×D₆₇	1	2	yes	—
166	C₄×C₈	C₂²×D₆₇	1	2	yes	—
167	C₂⁴	—	67	2	no	—
168	C₂×D₄	—	67	4	no	—
169	C₂×D₄	—	134	4	no	—
170	C₂²×C₄	—	67	2	no	—
171	C₂²×C₄	—	67	4	no	—
172	C₂²×C₄	—	67	4	no	—
173	C₂²×C₄	—	134	4	no	—
174	C₂²×C₄	—	134	4	no	—
175	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	67	4	no	—
180	C₂×C₈	—	134	4	no	—
181	C₂×C₈	—	134	8	no	—
182	C₂×C₈	—	134	4	no	—
183	C₂×C₈	—	134	4	no	—
184	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
186	C₂×C₈	D₄×D₆₇	1	2	yes	—
187	C₂×C₈	#11761629	1	2	yes	—
188	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
189	C₄⋊C₄	—	67	4	no	—
190	C₂²⋊C₄	—	67	8	no	—
191	C₄²	C₈×D₆₇	1	2	yes	—
192	C₄²	C₂×C₄×D₆₇	1	2	yes	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	67	4	no	—
195	C₁₆	—	134	2	no	—
196	C₂³	—	67	4	no	—
197	C₂³	—	134	4	no	—
198	C₂³	—	134	4	no	—
199	C₂³	—	134	4	no	—
200	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
201	D₄	—	134	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	67	4	no	—
204	C₂×C₄	—	134	4	no	—
205	C₂×C₄	—	134	4	no	—
206	C₂×C₄	—	134	4	no	—
207	C₂×C₄	—	134	8	no	—
208	C₂×C₄	#14523982	1	4	yes	complemented
209	C₂×C₄	#14524005	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
213	C₂×C₄	#14524051	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	134	4	no	—
217	C₈	#14524005	1	2	yes	—
218	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
219	C₈	C₂×#11761629	1	1	yes	characteristic
220	C₂²	—	268	8	no	—
221	C₂²	—	134	4	no	—
222	C₂²	—	134	4	no	—
223	C₂²	—	134	4	no	—
224	C₂²	#16863461	1	2	yes	—
225	C₂²	C₂×#14523982	1	2	yes	—
226	C₂²	#16863814	1	2	yes	—
227	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	134	4	no	—
230	C₄	#16863461	1	2	yes	—
231	C₄	C₂×#14523982	1	4	yes	complemented
232	C₄	C₂×#14524005	1	1	yes	characteristic
233	C₄	C₂×#14524051	1	1	yes	characteristic
234	C₂	—	268	4	no	—
235	C₂	#22562040	1	4	yes	direct product factor
236	C₂	C₂×#16863461	1	1	yes	characteristic
237	C₂	C₂²×#14523982	1	1	yes	characteristic
238	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×C₄⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂²×D₂₆₈	C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#2169
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523982	C₂	semidirect
3. #22562040	C₂	direct (N × H)
4. C₂×(C₂×D₆₇)⋊C₁₆	C₂	semidirect
5. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
10. C₂×#14523982	C₂²	semidirect
17. (C₂×D₆₇)⋊C₁₆	C₂²	semidirect
18. C₄×C₆₇⋊C₁₆	C₂²	semidirect
21. C₄⋊C₁₆×C₆₇	C₂²	semidirect
35. C₂×C₆₇⋊C₁₆	D₄	semidirect
63. C₆₇⋊C₁₆	C₂×D₄	semidirect
82. C₂×C₄×C₆₇	M₃₂	semidirect
110. C₄×C₆₇	C₂×M₃₂	semidirect
115. C₂×C₆₇	#2169	semidirect
123. C₂×C₄⋊C₁₆	D₆₇	semidirect
126. C₆₇	C₂×#2169	semidirect
139. C₄⋊C₁₆	C₂×D₆₇	semidirect
208. C₂×C₄	#14523982	semidirect
231. C₄	C₂×#14523982	semidirect
235. C₂	#22562040	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×#14523982	2
C₂×C₈×D₂₆₈	2
#22562040	2
C₂×(C₂×D₆₇)⋊C₁₆	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#2169	67

Subgroups — C2×#22562040 #31008994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562040 #31008994