Subgroups — C₂×#127×Dic₆₇ #31014346

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×D₄×Dic₆₇	C₂	1	6	yes	maximal, complemented
2	#127×Dic₆₇	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863299	C₂	1	6	yes	maximal, complemented
4	C₂×#16863266	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₆₇	C₂²	1	4	yes	—
8	C₂×D₄×Dic₆₇	—	2	48	no	—
9	C₂×D₄×Dic₆₇	C₂²	1	48	yes	complemented
10	C₂²×C₄⋊Dic₆₇	C₂²	1	6	yes	—
11	C₂²×C₂²⋊Dic₆₇	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₆₇	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	3	yes	—
14	#16863299	C₂²	1	48	yes	complemented
15	#16863266	C₂²	1	8	yes	complemented
16	C₄²×Dic₆₇	C₂²	1	8	yes	complemented
17	C₂³×D₄×C₆₇	C₂²	1	6	yes	—
18	C₂×#127×C₆₇	C₄	1	16	yes	complemented
19	C₂×#127×C₆₇	C₂²	1	8	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×Dic₆₇	—	2	48	no	—
22	C₂³×Dic₆₇	C₂³	1	4	yes	—
23	C₂³×Dic₆₇	C₂³	1	16	yes	—
24	C₂³×Dic₆₇	C₂³	1	3	yes	—
25	C₂³×Dic₆₇	C₂³	1	1	yes	characteristic
26	D₄×Dic₆₇	—	2	192	no	—
27	C₂×C₄⋊Dic₆₇	—	2	24	no	—
28	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
29	C₂×C₂²⋊Dic₆₇	—	2	48	no	—
30	C₂×C₂²⋊Dic₆₇	C₂³	1	48	yes	—
31	C₂×C₄×Dic₆₇	—	2	12	no	—
32	C₂×C₄×Dic₆₇	D₄	1	24	yes	complemented
33	C₂×C₄×Dic₆₇	C₂³	1	12	yes	—
34	C₂×C₄×Dic₆₇	C₂³	1	12	yes	—
35	C₂⁵×C₆₇	C₂³	1	4	yes	—
36	C₂²×D₄×C₆₇	—	2	24	no	—
37	C₂²×D₄×C₆₇	—	2	24	no	—
38	C₂²×D₄×C₆₇	C₂×C₄	1	48	yes	complemented
39	C₂²×D₄×C₆₇	C₂³	1	24	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
41	#127×C₆₇	C₂×C₄	1	64	yes	complemented
42	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
43	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
44	C₂²×Dic₆₇	—	2	12	no	—
45	C₂²×Dic₆₇	—	2	24	no	—
46	C₂²×Dic₆₇	—	2	96	no	—
47	C₂²×Dic₆₇	—	4	64	no	—
48	C₂²×Dic₆₇	C₂×D₄	1	12	yes	—
49	C₂²×Dic₆₇	C₂×D₄	1	12	yes	—
50	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
51	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
52	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
53	C₄⋊Dic₆₇	—	2	48	no	—
54	C₂²⋊Dic₆₇	—	2	96	no	—
55	C₄×Dic₆₇	—	2	24	no	—
56	C₄×Dic₆₇	C₂×D₄	1	48	yes	complemented
57	C₂⁴×C₆₇	—	2	24	no	—
58	C₂⁴×C₆₇	—	2	24	no	—
59	C₂⁴×C₆₇	C₂²×C₄	1	16	yes	—
60	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
61	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₆₇	—	2	192	no	—
63	C₂×D₄×C₆₇	—	2	48	no	—
64	C₂×D₄×C₆₇	—	2	48	no	—
65	C₂×D₄×C₆₇	C₂²×C₄	1	96	yes	complemented
66	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₆₇	#41	1	12	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	16	yes	complemented
71	C₂³×C₆₇	—	2	96	no	—
72	C₂³×C₆₇	—	2	24	no	—
73	C₂³×C₆₇	—	2	24	no	—
74	C₂³×C₆₇	—	4	16	no	—
75	C₂³×C₆₇	—	4	48	no	—
76	C₂³×C₆₇	C₂³×C₄	1	16	yes	—
77	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
78	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
79	C₂³×C₆₇	C₂×#41	1	6	yes	—
80	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
81	D₄×C₆₇	—	2	192	no	—
82	C₂×C₄×C₆₇	C₄×D₄	1	48	yes	complemented
83	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
85	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
86	C₂×Dic₆₇	—	2	24	no	—
87	C₂×Dic₆₇	—	2	12	no	—
88	C₂×Dic₆₇	—	2	24	no	—
89	C₂×Dic₆₇	—	4	16	no	—
90	C₂×Dic₆₇	—	4	64	no	—
91	C₂×Dic₆₇	#127	1	16	yes	direct product factor
92	C₂×Dic₆₇	C₂²×D₄	1	12	yes	—
93	C₂×Dic₆₇	C₂²×D₄	1	12	yes	—
94	C₂²×C₆₇	—	2	48	no	—
95	C₂²×C₆₇	—	4	64	no	—
96	C₂²×C₆₇	—	4	8	no	—
97	C₂²×C₆₇	—	4	24	no	—
98	C₂²×C₆₇	C₂×C₄×D₄	1	12	yes	—
99	C₂²×C₆₇	C₂×#127	1	4	yes	—
100	C₂²×C₆₇	#532	1	12	yes	—
101	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
102	C₂²×C₆₇	C₂³×D₄	1	3	yes	—
103	C₂²×C₆₇	C₂²×#41	1	3	yes	—
104	C₄×C₆₇	C₂×C₄×D₄	1	24	yes	complemented
105	Dic₆₇	—	2	12	no	—
106	Dic₆₇	—	4	8	no	—
107	Dic₆₇	C₂×#127	1	8	yes	direct product factor
108	C₂×C₄×#127	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	4	16	no	—
110	C₂×C₆₇	C₄×#127	1	8	yes	complemented
111	C₂×C₆₇	C₂²×C₄×D₄	1	3	yes	—
112	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#532	1	3	yes	—
114	C₂²×#127	D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
120	C₆₇	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	C₂×D₆₇	1	6	yes	—
122	C₂⁴×C₄	—	67	4	no	—
123	C₂×#127	Dic₆₇	1	16	yes	direct product factor
124	C₂×#127	C₂×D₆₇	1	8	yes	—
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	—	67	3	no	—
130	C₂²×C₄²	—	67	3	no	—
131	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
132	#390	—	67	48	no	—
133	#384	—	67	8	no	—
134	C₄³	—	67	8	no	—
135	C₂⁵	C₂²×D₆₇	1	4	yes	—
136	C₂²×D₄	—	2	24	no	—
137	C₂²×D₄	—	2	24	no	—
138	C₂²×D₄	C₂×Dic₆₇	1	48	yes	complemented
139	C₂²×D₄	C₂²×D₆₇	1	24	yes	—
140	C₂³×C₄	—	67	1	no	—
141	C₂³×C₄	—	67	3	no	—
142	C₂³×C₄	—	67	16	no	—
143	C₂³×C₄	—	67	4	no	—
144	C₂³×C₄	—	134	48	no	—
145	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
146	#127	C₂×Dic₆₇	1	64	yes	direct product factor
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	—	67	24	no	—
153	C₂×C₄²	—	67	12	no	—
154	C₂×C₄²	—	67	12	no	—
155	C₂×C₄²	—	134	12	no	—
156	C₂×C₄²	C₂×Dic₆₇	1	8	yes	complemented
157	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
158	C₂⁴	—	2	24	no	—
159	C₂⁴	—	2	24	no	—
160	C₂⁴	C₂²×Dic₆₇	1	16	yes	—
161	C₂⁴	C₂³×D₆₇	1	8	yes	—
162	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
163	C₂×D₄	—	2	192	no	—
164	C₂×D₄	—	2	48	no	—
165	C₂×D₄	—	2	48	no	—
166	C₂×D₄	C₂²×Dic₆₇	1	96	yes	complemented
167	C₂²×C₄	—	268	64	no	—
168	C₂²×C₄	—	67	12	no	—
169	C₂²×C₄	—	67	2	no	—
170	C₂²×C₄	—	67	12	no	—
171	C₂²×C₄	—	67	6	no	—
172	C₂²×C₄	—	67	8	no	—
173	C₂²×C₄	—	134	96	no	—
174	C₂²×C₄	—	134	24	no	—
175	C₂²×C₄	—	134	12	no	—
176	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
177	C₂²×C₄	D₄×D₆₇	1	12	yes	—
178	C₂²×C₄	#11761627	1	12	yes	—
179	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
180	C₄⋊C₄	—	134	48	no	—
181	C₂²⋊C₄	—	134	96	no	—
182	C₄²	—	67	48	no	—
183	C₄²	—	134	24	no	—
184	C₄²	C₂²×Dic₆₇	1	16	yes	complemented
185	C₂³	—	2	96	no	—
186	C₂³	—	2	24	no	—
187	C₂³	—	2	24	no	—
188	C₂³	—	4	16	no	—
189	C₂³	—	4	48	no	—
190	C₂³	C₂³×Dic₆₇	1	16	yes	—
191	C₂³	C₂³×Dic₆₇	1	2	yes	—
192	C₂³	C₂×D₄×D₆₇	1	6	yes	—
193	C₂³	C₂×#11761627	1	6	yes	—
194	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
195	D₄	—	2	192	no	—
196	C₂×C₄	—	268	16	no	—
197	C₂×C₄	—	268	64	no	—
198	C₂×C₄	—	67	16	no	—
199	C₂×C₄	—	67	12	no	—
200	C₂×C₄	—	67	12	no	—
201	C₂×C₄	—	134	24	no	—
202	C₂×C₄	—	134	24	no	—
203	C₂×C₄	—	134	12	no	—
204	C₂×C₄	D₄×Dic₆₇	1	48	yes	complemented
205	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
206	C₂×C₄	C₂×D₄×D₆₇	1	12	yes	—
207	C₂×C₄	C₂×#11761627	1	12	yes	—
208	C₂²	—	2	48	no	—
209	C₂²	—	4	8	no	—
210	C₂²	—	4	24	no	—
211	C₂²	—	4	64	no	—
212	C₂²	C₂×D₄×Dic₆₇	1	12	yes	—
213	C₂²	#127×D₆₇	1	4	yes	—
214	C₂²	#16863944	1	12	yes	—
215	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
216	C₂²	C₂²×D₄×D₆₇	1	3	yes	—
217	C₂²	C₂²×#11761627	1	3	yes	—
218	C₄	—	268	8	no	—
219	C₄	—	67	8	no	—
220	C₄	—	134	12	no	—
221	C₄	C₂×D₄×Dic₆₇	1	24	yes	complemented
222	C₂	—	4	16	no	—
223	C₂	#127×Dic₆₇	1	8	yes	direct product factor
224	C₂	C₂²×D₄×Dic₆₇	1	3	yes	—
225	C₂	C₂×#127×D₆₇	1	1	yes	characteristic
226	C₂	C₂×#16863944	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₆₇	C₂	semidirect
2. #127×Dic₆₇	C₂	direct (N × H)
3. C₂×#16863299	C₂	semidirect
4. C₂×#16863266	C₂	semidirect
5. C₂×C₄²×Dic₆₇	C₂	semidirect
9. C₂×D₄×Dic₆₇	C₂²	semidirect
14. #16863299	C₂²	semidirect
15. #16863266	C₂²	semidirect
16. C₄²×Dic₆₇	C₂²	semidirect
18. C₂×#127×C₆₇	C₄	semidirect
32. C₂×C₄×Dic₆₇	D₄	semidirect
38. C₂²×D₄×C₆₇	C₂×C₄	semidirect
41. #127×C₆₇	C₂×C₄	semidirect
42. C₂×C₄²×C₆₇	C₂×C₄	semidirect
56. C₄×Dic₆₇	C₂×D₄	semidirect
65. C₂×D₄×C₆₇	C₂²×C₄	semidirect
70. C₄²×C₆₇	C₂²×C₄	semidirect
82. C₂×C₄×C₆₇	C₄×D₄	semidirect
91. C₂×Dic₆₇	#127	direct (N × H)
104. C₄×C₆₇	C₂×C₄×D₄	semidirect
107. Dic₆₇	C₂×#127	direct (N × H)
110. C₂×C₆₇	C₄×#127	semidirect
120. C₆₇	C₂×C₄×#127	semidirect
123. C₂×#127	Dic₆₇	direct (N × H)
138. C₂²×D₄	C₂×Dic₆₇	semidirect
146. #127	C₂×Dic₆₇	direct (N × H)
156. C₂×C₄²	C₂×Dic₆₇	semidirect
166. C₂×D₄	C₂²×Dic₆₇	semidirect
184. C₄²	C₂²×Dic₆₇	semidirect
204. C₂×C₄	D₄×Dic₆₇	semidirect
221. C₄	C₂×D₄×Dic₆₇	semidirect
223. C₂	#127×Dic₆₇	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×D₄×Dic₆₇	2
#127×Dic₆₇	2
C₂×#16863299	2
C₂×#16863266	2
C₂×C₄²×Dic₆₇	2
C₂²×#127×C₆₇	2
C₂×C₄×#127	67

Subgroups — C2×#127×Dic67 #31014346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#127×Dic₆₇ #31014346