Subgroups — C₂×#16863164 #22559944

Order 8576.

432 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863164	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₆₇	C₄	1	2	yes	—
6	C₂×#11761612	—	2	1	no	—
7	C₂×#11761612	—	2	2	no	—
8	C₂×#11761612	C₄	1	4	yes	complemented
9	C₂×#11761612	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₈	—	2	1	no	—
11	C₂×C₄⋊Dic₆₇	—	2	1	no	—
12	C₂×C₄⋊Dic₆₇	—	2	1	no	—
13	C₂×C₄⋊Dic₆₇	—	2	1	no	—
14	C₂×C₄⋊Dic₆₇	—	2	1	no	—
15	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
16	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
17	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	—	2	2	no	—
19	C₂×M₁₆×C₆₇	—	2	1	no	—
20	C₂×M₁₆×C₆₇	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₈×C₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	—	2	2	no	—
24	C₂²×Dic₆₇	—	2	2	no	—
25	C₂²×Dic₆₇	—	4	4	no	—
26	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
27	#11761612	—	2	4	no	—
28	#11761612	—	2	2	no	—
29	#11761612	C₂×C₄	1	4	yes	complemented
30	C₂×C₆₇⋊C₈	—	2	4	no	—
31	C₂×C₆₇⋊C₈	—	2	2	no	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	—	2	2	no	—
34	C₄⋊Dic₆₇	—	2	2	no	—
35	C₄⋊Dic₆₇	—	2	2	no	—
36	C₂⁴×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	D₄	1	2	yes	—
43	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	Q₈	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	M₁₆×C₆₇	—	2	4	no	—
48	M₁₆×C₆₇	—	2	2	no	—
49	M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂³×C₆₇	—	2	2	no	—
53	C₂³×C₆₇	—	2	2	no	—
54	C₂³×C₆₇	C₄²	1	2	yes	—
55	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄²	1	2	yes	—
60	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₆₇	—	2	4	no	—
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	4	4	no	—
80	C₂×Dic₆₇	—	4	4	no	—
81	C₈×C₆₇	—	2	4	no	—
82	C₆₇⋊C₈	—	2	4	no	—
83	C₂²×C₆₇	—	2	4	no	—
84	C₂²×C₆₇	—	2	4	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	—	2	1	no	—
87	C₂²×C₆₇	—	2	1	no	—
88	C₂²×C₆₇	—	2	1	no	—
89	C₂²×C₆₇	#95	1	2	yes	—
90	C₂²×C₆₇	#95	1	2	yes	—
91	C₂²×C₆₇	#100	1	2	yes	—
92	C₂²×C₆₇	#101	1	2	yes	—
93	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₆₇	#95	1	2	yes	—
97	C₄×C₆₇	#95	1	2	yes	—
98	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	Dic₆₇	—	4	4	no	—
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	—	2	1	no	—
105	C₂×C₆₇	—	2	1	no	—
106	C₂×C₆₇	#343	1	4	yes	complemented
107	C₂×C₆₇	C₂×#95	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#100	1	1	yes	characteristic
109	C₂×C₆₇	C₂×#101	1	1	yes	characteristic
110	C₂×#343	—	67	1	no	maximal, Sylow
111	C₆₇	C₂×#343	1	1	yes	characteristic, complemented, Sylow
112	C₂²×M₁₆	—	67	1	no	—
113	C₂²×M₁₆	D₆₇	1	1	yes	characteristic
114	C₂²×C₄⋊C₄	—	67	1	no	—
115	#343	—	67	4	no	—
116	C₂³×C₄	—	67	2	no	—
117	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	67	4	no	—
121	C₂×M₁₆	—	67	2	no	—
122	C₂×M₁₆	—	134	1	no	—
123	C₂×M₁₆	—	134	2	no	—
124	C₂×M₁₆	Dic₆₇	1	4	yes	complemented
125	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	134	1	no	—
128	C₂×C₄⋊C₄	—	67	2	no	—
129	C₂×C₄⋊C₄	—	67	2	no	—
130	C₂×C₄⋊C₄	—	134	1	no	—
131	C₂×C₄⋊C₄	—	134	1	no	—
132	C₂×C₄⋊C₄	—	134	1	no	—
133	C₂×C₄⋊C₄	—	134	1	no	—
134	C₂⁴	C₄×D₆₇	1	1	yes	characteristic
135	C₂²×C₄	—	268	4	no	—
136	C₂²×C₄	—	67	4	no	—
137	C₂²×C₄	—	134	2	no	—
138	C₂²×C₄	—	134	2	no	—
139	C₂²×C₄	Dic₁₃₄	1	2	yes	—
140	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
141	C₂²×C₄	C₄×D₆₇	1	2	yes	—
142	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
143	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
145	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
146	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
149	M₁₆	—	2	4	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	67	4	no	—
152	M₁₆	—	134	4	no	—
153	M₁₆	—	134	2	no	—
154	M₁₆	C₂×Dic₆₇	1	4	yes	complemented
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	134	4	no	—
158	C₂×C₈	—	134	2	no	—
159	C₄⋊C₄	—	134	2	no	—
160	C₄⋊C₄	—	134	2	no	—
161	C₄⋊C₄	—	134	2	no	—
162	C₄⋊C₄	—	134	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₄×Dic₆₇	1	2	yes	—
166	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
167	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂³	Dic₆₇⋊C₄	1	2	yes	—
169	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
170	C₂×C₄	—	268	4	no	—
171	C₂×C₄	—	268	4	no	—
172	C₂×C₄	—	268	4	no	—
173	C₂×C₄	—	134	4	no	—
174	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
175	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
177	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
178	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
179	C₂×C₄	C₂²⋊Dic₆₇	1	2	yes	—
180	C₂×C₄	C₂²⋊Dic₆₇	1	2	yes	—
181	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
182	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
183	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
184	C₂×C₄	C₄⋊Dic₆₇	1	2	yes	—
185	C₂×C₄	C₄⋊Dic₆₇	1	2	yes	—
186	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	134	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₆₇⋊#95	1	2	yes	—
201	C₂²	C₆₇⋊#95	1	2	yes	—
202	C₂²	#14523959	1	2	yes	—
203	C₂²	#14523961	1	2	yes	—
204	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
205	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
206	C₂²	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
207	C₄	—	268	4	no	—
208	C₄	C₆₇⋊#95	1	2	yes	—
209	C₄	C₆₇⋊#95	1	2	yes	—
210	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
211	C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
212	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
213	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#16863164	1	4	yes	direct product factor
218	C₂	C₂×C₆₇⋊#95	1	1	yes	characteristic
219	C₂	C₂×#14523959	1	1	yes	characteristic
220	C₂	C₂×#14523961	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂²×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₄
hypercenter	C₂²×M₁₆	D₆₇
hypocenter	C₆₇	C₂×#343
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16863164	C₂	direct (N × H)
8. C₂×#11761612	C₄	semidirect
20. C₂×M₁₆×C₆₇	C₄	semidirect
29. #11761612	C₂×C₄	semidirect
49. M₁₆×C₆₇	C₂×C₄	semidirect
106. C₂×C₆₇	#343	semidirect
111. C₆₇	C₂×#343	semidirect
124. C₂×M₁₆	Dic₆₇	semidirect
154. M₁₆	C₂×Dic₆₇	semidirect
217. C₂	#16863164	direct (N × H)

Subgroups — C2×#16863164 #22559944

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863164 #22559944