Subgroups — C₂×C₄×#14524040 #31014345

Order 17152.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524040	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₂₆₈	C₂	1	6	yes	maximal, complemented
3	C₄×#14524040	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16863298	C₂	1	6	yes	maximal, complemented
5	C₂×#16863266	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₆₈	C₂²	1	6	yes	—
8	C₂³×C₄×D₆₇	C₂²	1	4	yes	—
9	C₂×#14524040	C₄	1	16	yes	direct product factor
10	C₂×#14524040	C₂²	1	8	yes	—
11	C₂×C₄×D₂₆₈	—	2	48	no	—
12	C₂×C₄×D₂₆₈	C₂²	1	48	yes	complemented
13	C₂²×C₄⋊Dic₆₇	C₂²	1	6	yes	—
14	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	12	yes	—
15	#16863298	C₂²	1	48	yes	complemented
16	#16863266	C₂²	1	8	yes	complemented
17	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
20	C₄³×C₆₇	C₂²	1	8	yes	complemented
21	C₂⁴×D₆₇	C₂³	1	4	yes	—
22	C₂²×D₂₆₈	—	2	24	no	—
23	C₂²×D₂₆₈	—	2	24	no	—
24	C₂²×D₂₆₈	C₂×C₄	1	48	yes	complemented
25	C₂²×D₂₆₈	C₂³	1	24	yes	—
26	C₂³×Dic₆₇	C₂³	1	4	yes	—
27	C₂²×C₄×D₆₇	—	2	48	no	—
28	C₂²×C₄×D₆₇	C₂³	1	16	yes	—
29	#14524040	C₂×C₄	1	64	yes	direct product factor
30	C₄×D₂₆₈	—	2	192	no	—
31	C₂×C₄⋊Dic₆₇	—	2	24	no	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
33	C₂×(C₂×D₆₇)⋊C₄	—	2	48	no	—
34	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	48	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	—	2	12	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
40	C₂×C₄²×C₆₇	D₄	1	24	yes	complemented
41	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
44	C₂³×D₆₇	—	2	24	no	—
45	C₂³×D₆₇	—	2	24	no	—
46	C₂³×D₆₇	C₂²×C₄	1	16	yes	—
47	C₂³×D₆₇	C₂⁴	1	8	yes	—
48	C₂×D₂₆₈	—	2	192	no	—
49	C₂×D₂₆₈	—	2	48	no	—
50	C₂×D₂₆₈	—	2	48	no	—
51	C₂×D₂₆₈	C₂²×C₄	1	96	yes	complemented
52	C₂²×Dic₆₇	—	2	24	no	—
53	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
54	C₂×C₄×D₆₇	—	2	96	no	—
55	C₂×C₄×D₆₇	—	4	64	no	—
56	C₄⋊Dic₆₇	—	2	48	no	—
57	(C₂×D₆₇)⋊C₄	—	2	96	no	—
58	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	—	2	12	no	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₆₇	#41	1	12	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
68	C₄²×C₆₇	—	2	24	no	—
69	C₄²×C₆₇	C₂²×C₄	1	16	yes	complemented
70	C₄²×C₆₇	C₂×D₄	1	48	yes	complemented
71	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
72	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
73	C₂³×C₆₇	C₂×#41	1	6	yes	—
74	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
75	C₂²×D₆₇	—	2	96	no	—
76	C₂²×D₆₇	—	2	24	no	—
77	C₂²×D₆₇	—	2	24	no	—
78	C₂²×D₆₇	—	4	16	no	—
79	C₂²×D₆₇	—	4	48	no	—
80	C₂²×D₆₇	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₆₇	—	2	24	no	—
82	C₂×C₄×C₆₇	—	2	12	no	—
83	C₂×C₄×C₆₇	C₄×D₄	1	48	yes	complemented
84	C₂×C₄×C₆₇	#127	1	16	yes	complemented
85	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
90	C₂×Dic₆₇	—	2	24	no	—
91	C₂×Dic₆₇	—	4	16	no	—
92	D₂₆₈	—	2	192	no	—
93	C₄×D₆₇	—	4	64	no	—
94	C₂²×C₆₇	C₂×C₄×D₄	1	12	yes	—
95	C₂²×C₆₇	C₂×#127	1	4	yes	—
96	C₂²×C₆₇	#532	1	12	yes	—
97	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
98	C₂²×C₆₇	C₂³×D₄	1	3	yes	—
99	C₂²×C₆₇	C₂²×#41	1	3	yes	—
100	C₂×D₆₇	—	2	48	no	—
101	C₂×D₆₇	—	4	64	no	—
102	C₂×D₆₇	—	4	8	no	—
103	C₂×D₆₇	—	4	24	no	—
104	C₄×C₆₇	—	2	12	no	—
105	C₄×C₆₇	C₂×C₄×D₄	1	24	yes	complemented
106	C₄×C₆₇	C₂×#127	1	8	yes	complemented
107	Dic₆₇	—	4	8	no	—
108	C₂×C₄×#127	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	C₄×#127	1	8	yes	complemented
110	C₂×C₆₇	C₂²×C₄×D₄	1	3	yes	—
111	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#532	1	3	yes	—
113	D₆₇	—	4	16	no	—
114	C₂²×#127	—	67	1	no	—
115	C₂²×C₄×D₄	—	67	6	no	—
116	C₄×#127	—	67	16	no	—
117	C₂×#390	—	67	6	no	—
118	C₂×#384	—	67	1	no	—
119	C₂×C₄³	D₆₇	1	1	yes	characteristic, complemented
120	C₆₇	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	—	67	6	no	—
122	C₂⁴×C₄	—	67	4	no	—
123	C₂×#127	—	67	16	no	—
124	C₂×#127	—	67	8	no	—
125	C₂×C₄×D₄	—	67	48	no	—
126	C₂×C₄×D₄	—	134	48	no	—
127	C₂²×C₄⋊C₄	—	67	6	no	—
128	C₂²×C₂²⋊C₄	—	67	12	no	—
129	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
130	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
131	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
132	#390	—	67	48	no	—
133	#384	—	67	8	no	—
134	C₄³	C₂×D₆₇	1	8	yes	complemented
135	C₂⁵	—	67	4	no	—
136	C₂²×D₄	—	67	48	no	—
137	C₂²×D₄	—	67	24	no	—
138	C₂²×D₄	—	134	24	no	—
139	C₂²×D₄	—	134	24	no	—
140	C₂³×C₄	—	67	4	no	—
141	C₂³×C₄	—	67	16	no	—
142	C₂³×C₄	—	134	48	no	—
143	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
144	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
145	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
146	#127	—	67	64	no	—
147	C₄×D₄	—	134	192	no	—
148	C₂×C₄⋊C₄	—	67	24	no	—
149	C₂×C₄⋊C₄	—	134	24	no	—
150	C₂×C₂²⋊C₄	—	67	48	no	—
151	C₂×C₂²⋊C₄	—	134	48	no	—
152	C₂×C₄²	—	2	12	no	—
153	C₂×C₄²	C₄×D₆₇	1	8	yes	complemented
154	C₂×C₄²	D₂₆₈	1	24	yes	complemented
155	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
156	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
157	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
158	C₂⁴	—	67	16	no	—
159	C₂⁴	—	67	8	no	—
160	C₂⁴	—	134	24	no	—
161	C₂⁴	—	134	24	no	—
162	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
163	C₂×D₄	—	67	96	no	—
164	C₂×D₄	—	134	192	no	—
165	C₂×D₄	—	134	48	no	—
166	C₂×D₄	—	134	48	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	268	64	no	—
169	C₂²×C₄	—	67	8	no	—
170	C₂²×C₄	—	134	24	no	—
171	C₂²×C₄	—	134	96	no	—
172	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
173	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
174	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
175	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
176	C₂²×C₄	#11761628	1	12	yes	—
177	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
178	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
179	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
180	C₄⋊C₄	—	134	48	no	—
181	C₂²⋊C₄	—	134	96	no	—
182	C₄²	—	2	24	no	—
183	C₄²	C₂×C₄×D₆₇	1	16	yes	complemented
184	C₄²	C₂×D₂₆₈	1	48	yes	complemented
185	C₂³	—	268	16	no	—
186	C₂³	—	268	48	no	—
187	C₂³	—	67	16	no	—
188	C₂³	—	134	96	no	—
189	C₂³	—	134	24	no	—
190	C₂³	—	134	24	no	—
191	C₂³	C₂²×C₄×D₆₇	1	2	yes	—
192	C₂³	C₂²×D₂₆₈	1	6	yes	—
193	C₂³	C₂×#11761628	1	6	yes	—
194	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
195	D₄	—	134	192	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	268	16	no	—
199	C₂×C₄	—	268	64	no	—
200	C₂×C₄	—	134	24	no	—
201	C₂×C₄	C₄×D₂₆₈	1	48	yes	complemented
202	C₂×C₄	#14524040	1	16	yes	direct product factor
203	C₂×C₄	C₂²×C₄×D₆₇	1	12	yes	—
204	C₂×C₄	C₂²×D₂₆₈	1	12	yes	—
205	C₂×C₄	C₂²×D₂₆₈	1	12	yes	—
206	C₂×C₄	C₂²×D₂₆₈	1	12	yes	—
207	C₂×C₄	C₂×#11761628	1	12	yes	—
208	C₂²	—	268	8	no	—
209	C₂²	—	268	24	no	—
210	C₂²	—	268	64	no	—
211	C₂²	—	134	48	no	—
212	C₂²	C₂×C₄×D₂₆₈	1	12	yes	—
213	C₂²	C₂×#14524040	1	4	yes	—
214	C₂²	#16863946	1	12	yes	—
215	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
216	C₂²	C₂³×D₂₆₈	1	3	yes	—
217	C₂²	C₂²×#11761628	1	3	yes	—
218	C₄	—	2	12	no	—
219	C₄	—	268	8	no	—
220	C₄	C₂×C₄×D₂₆₈	1	24	yes	complemented
221	C₄	C₂×#14524040	1	8	yes	direct product factor
222	C₂	—	268	16	no	—
223	C₂	C₄×#14524040	1	8	yes	direct product factor
224	C₂	C₂²×C₄×D₂₆₈	1	3	yes	—
225	C₂	C₂²×#14524040	1	1	yes	characteristic
226	C₂	C₂×#16863946	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₇
Fitting	C₂×C₄³×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂²×#14524040	C₂
cosocle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	C₂×C₄×#127
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₆₈	C₂	semidirect
3. C₄×#14524040	C₂	direct (N × H)
4. C₂×#16863298	C₂	semidirect
5. C₂×#16863266	C₂	semidirect
6. C₂×C₄³×C₆₇	C₂	semidirect
9. C₂×#14524040	C₄	direct (N × H)
12. C₂×C₄×D₂₆₈	C₂²	semidirect
15. #16863298	C₂²	semidirect
16. #16863266	C₂²	semidirect
20. C₄³×C₆₇	C₂²	semidirect
24. C₂²×D₂₆₈	C₂×C₄	semidirect
29. #14524040	C₂×C₄	direct (N × H)
39. C₂×C₄²×C₆₇	C₂×C₄	semidirect
40. C₂×C₄²×C₆₇	D₄	semidirect
51. C₂×D₂₆₈	C₂²×C₄	semidirect
69. C₄²×C₆₇	C₂²×C₄	semidirect
70. C₄²×C₆₇	C₂×D₄	semidirect
83. C₂×C₄×C₆₇	C₄×D₄	semidirect
84. C₂×C₄×C₆₇	#127	semidirect
105. C₄×C₆₇	C₂×C₄×D₄	semidirect
106. C₄×C₆₇	C₂×#127	semidirect
109. C₂×C₆₇	C₄×#127	semidirect
119. C₂×C₄³	D₆₇	semidirect
120. C₆₇	C₂×C₄×#127	semidirect
134. C₄³	C₂×D₆₇	semidirect
153. C₂×C₄²	C₄×D₆₇	semidirect
154. C₂×C₄²	D₂₆₈	semidirect
183. C₄²	C₂×C₄×D₆₇	semidirect
184. C₄²	C₂×D₂₆₈	semidirect
201. C₂×C₄	C₄×D₂₆₈	semidirect
202. C₂×C₄	#14524040	direct (N × H)
220. C₄	C₂×C₄×D₂₆₈	semidirect
221. C₄	C₂×#14524040	direct (N × H)
223. C₂	C₄×#14524040	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×#14524040	2
C₂²×C₄×D₂₆₈	2
C₄×#14524040	2
C₂×#16863298	2
C₂×#16863266	2
C₂×C₄³×C₆₇	2
C₂×C₄×#127	67

Subgroups — C2×C4×#14524040 #31014345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14524040 #31014345