Subgroups — C₂×#366×D₆₇ #31008850

Order 17152.

782 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×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×D₆₇	C₂	1	2	yes	maximal
3	C₂×#16863222	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863221	C₂	1	2	yes	maximal, complemented
5	#366×D₆₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×D₆₇	—	2	2	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₆₇	C₄	1	8	yes	complemented
11	C₂×C₁₆×D₆₇	C₂²	1	4	yes	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523976	—	2	2	no	—
14	#107×D₆₇	—	2	4	no	—
15	#107×D₆₇	C₂²	1	8	yes	—
16	#16863222	C₂²	1	4	yes	complemented
17	#16863221	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	#366×C₆₇	C₂²	1	4	yes	complemented
21	C₂²×C₄×D₆₇	—	4	2	no	—
22	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₈×D₆₇	D₄	1	2	yes	—
26	C₂×C₈×D₆₇	Q₈	1	4	yes	—
27	C₂×C₈×D₆₇	C₂³	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×Dic₆₇⋊C₄	—	4	2	no	—
30	C₄⋊C₄×D₆₇	—	2	8	no	—
31	C₄⋊C₄×D₆₇	—	4	4	no	—
32	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
33	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
34	C₁₆×D₆₇	C₂×C₄	1	16	yes	complemented
35	#14523977	—	2	2	no	—
36	#14523977	C₂³	1	4	yes	—
37	#14523976	—	2	8	no	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
41	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
42	#107×C₆₇	—	2	2	no	—
43	#107×C₆₇	C₂³	1	4	yes	—
44	C₂³×D₆₇	D₈	1	1	yes	characteristic
45	C₂²×Dic₆₇	—	4	2	no	—
46	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
47	C₂×C₄×D₆₇	—	4	8	no	—
48	C₂×C₄×D₆₇	—	8	8	no	—
49	C₂×C₄×D₆₇	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×D₆₇	D₈	1	2	yes	—
51	C₂×C₄×D₆₇	Q₁₆	1	4	yes	—
52	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
53	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
54	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
55	C₈×D₆₇	C₄⋊C₄	1	8	yes	—
56	C₈×D₆₇	C₂²×C₄	1	4	yes	—
57	C₈×D₆₇	C₂×Q₈	1	4	yes	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	Dic₆₇⋊C₄	—	4	8	no	—
61	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	complemented
62	C₂²×C₄×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
72	C₂²×D₆₇	#107	1	4	yes	—
73	C₂²×D₆₇	D₁₆	1	4	yes	—
74	C₂²×D₆₇	Q₃₂	1	4	yes	—
75	C₂²×D₆₇	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	8	4	no	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₆₇	—	4	4	no	—
83	C₂×Dic₆₇	—	8	4	no	—
84	C₂×Dic₆₇	#107	1	4	yes	—
85	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
86	C₄×D₆₇	—	8	16	no	—
87	C₄×D₆₇	#107	1	8	yes	—
88	C₄×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₄×D₆₇	C₂×Q₁₆	1	4	yes	—
90	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
94	C₂²×C₆₇	C₂×#107	1	2	yes	—
95	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
96	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
97	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
98	C₂×D₆₇	#366	1	16	yes	direct product factor
99	C₂×D₆₇	C₂×#107	1	4	yes	—
100	C₂×D₆₇	C₂×D₁₆	1	4	yes	—
101	C₂×D₆₇	C₂×Q₃₂	1	4	yes	—
102	C₄×C₆₇	—	8	4	no	—
103	C₄×C₆₇	C₂×#107	1	2	yes	—
104	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
106	Dic₆₇	—	8	4	no	—
107	Dic₆₇	C₂×#107	1	4	yes	—
108	C₂²×#366	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	C₂×#366	1	4	yes	complemented
110	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
111	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
112	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
113	D₆₇	C₂×#366	1	8	yes	direct product factor
114	C₂³×C₁₆	—	67	1	no	—
115	C₂²×#107	—	67	2	no	—
116	C₂×#366	—	67	1	no	—
117	C₂×#366	—	67	2	no	—
118	C₂×#366	—	67	8	no	—
119	C₂×#366	D₆₇	1	1	yes	characteristic, direct product factor
120	C₆₇	C₂²×#366	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₈	—	67	1	no	—
122	C₂²×C₄⋊C₄	—	134	2	no	—
123	C₂²×C₁₆	—	67	1	no	—
124	C₂²×C₁₆	—	67	8	no	—
125	C₂²×C₁₆	—	67	4	no	—
126	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
127	C₂×#107	—	67	2	no	—
128	C₂×#107	—	67	8	no	—
129	C₂×#107	—	134	4	no	—
130	C₂×#107	—	134	2	no	—
131	C₂×#107	C₂×D₆₇	1	2	yes	—
132	#366	—	67	4	no	—
133	#366	—	67	8	no	—
134	#366	C₂×D₆₇	1	4	yes	direct product factor
135	C₂³×C₄	—	268	2	no	—
136	C₂³×C₄	—	67	1	no	—
137	C₂²×C₈	—	67	1	no	—
138	C₂²×C₈	—	67	4	no	—
139	C₂²×C₈	—	67	2	no	—
140	C₂²×C₈	—	67	4	no	—
141	C₂²×C₈	—	67	2	no	—
142	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	268	4	no	—
145	C₂×C₄⋊C₄	—	268	2	no	—
146	C₂×C₄⋊C₄	—	134	8	no	—
147	C₂×C₄⋊C₄	—	134	2	no	—
148	C₂×C₁₆	—	67	4	no	—
149	C₂×C₁₆	—	67	2	no	—
150	C₂×C₁₆	—	67	16	no	—
151	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
153	#107	—	2	2	no	—
154	#107	—	67	4	no	—
155	#107	—	134	8	no	—
156	#107	—	134	2	no	—
157	#107	C₂²×D₆₇	1	4	yes	—
158	C₂⁴	—	67	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	268	8	no	—
161	C₂²×C₄	—	268	2	no	—
162	C₂²×C₄	—	536	8	no	—
163	C₂²×C₄	—	67	1	no	—
164	C₂²×C₄	—	67	4	no	—
165	C₂²×C₄	—	67	2	no	—
166	C₂²×C₄	—	67	4	no	—
167	C₂²×C₄	—	67	2	no	—
168	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
169	C₂×C₈	—	67	4	no	—
170	C₂×C₈	—	67	2	no	—
171	C₂×C₈	—	67	8	no	—
172	C₂×C₈	—	67	4	no	—
173	C₂×C₈	—	67	4	no	—
174	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
175	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₆₇	1	2	yes	—
177	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	268	8	no	—
181	C₄⋊C₄	—	268	2	no	—
182	C₄⋊C₄	—	134	4	no	—
183	C₁₆	—	67	4	no	—
184	C₁₆	C₂×C₄×D₆₇	1	4	yes	complemented
185	C₂³	—	67	4	no	—
186	C₂³	—	67	4	no	—
187	C₂³	—	67	2	no	—
188	C₂³	—	67	4	no	—
189	C₂³	D₈×D₆₇	1	1	yes	characteristic
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	—	268	4	no	—
193	C₂×C₄	—	536	16	no	—
194	C₂×C₄	—	536	4	no	—
195	C₂×C₄	—	67	2	no	—
196	C₂×C₄	—	67	4	no	—
197	C₂×C₄	—	67	4	no	—
198	C₂×C₄	—	67	4	no	—
199	C₂×C₄	—	67	8	no	—
200	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
201	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
202	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
203	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
204	C₈	—	67	4	no	—
205	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
206	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
207	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
208	C₂²	—	67	4	no	—
209	C₂²	—	67	4	no	—
210	C₂²	—	67	4	no	—
211	C₂²	—	67	16	no	—
212	C₂²	#107×D₆₇	1	2	yes	—
213	C₂²	D₁₆×D₆₇	1	2	yes	—
214	C₂²	Q₃₂×D₆₇	1	2	yes	—
215	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
216	C₄	—	8	4	no	—
217	C₄	—	536	4	no	—
218	C₄	—	67	4	no	—
219	C₄	#107×D₆₇	1	2	yes	—
220	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
221	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
222	C₂	—	67	8	no	—
223	C₂	#366×D₆₇	1	4	yes	direct product factor
224	C₂	C₂×#107×D₆₇	1	1	yes	characteristic
225	C₂	C₂×D₁₆×D₆₇	1	1	yes	characteristic
226	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂³×D₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	C₂²×#366
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. C₂×#16863222	C₂	semidirect
4. C₂×#16863221	C₂	semidirect
5. #366×D₆₇	C₂	direct (N × H)
6. C₂×#366×C₆₇	C₂	semidirect
10. C₂×C₁₆×D₆₇	C₄	semidirect
16. #16863222	C₂²	semidirect
17. #16863221	C₂²	semidirect
20. #366×C₆₇	C₂²	semidirect
32. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
34. C₁₆×D₆₇	C₂×C₄	semidirect
40. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
61. C₆₇⋊C₁₆	C₂²×C₄	semidirect
70. C₁₆×C₆₇	C₂²×C₄	semidirect
98. C₂×D₆₇	#366	direct (N × H)
109. C₂×C₆₇	C₂×#366	semidirect
113. D₆₇	C₂×#366	direct (N × H)
119. C₂×#366	D₆₇	direct (N × H)
120. C₆₇	C₂²×#366	semidirect
134. #366	C₂×D₆₇	direct (N × H)
151. C₂×C₁₆	C₄×D₆₇	semidirect
184. C₁₆	C₂×C₄×D₆₇	semidirect
223. C₂	#366×D₆₇	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×C₁₆×D₆₇	2
C₂×#107×D₆₇	2
C₂×#16863222	2
C₂×#16863221	2
#366×D₆₇	2
C₂×#366×C₆₇	2
C₂²×#366	67

Subgroups — C2×#366×D67 #31008850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#366×D₆₇ #31008850