Subgroups — #2253×D₆₇ #31011309

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₆₇	C₂	1	2	yes	maximal
2	#22562488	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562487	C₂	1	2	yes	maximal, complemented
4	#22562486	C₂	1	2	yes	maximal, complemented
5	#22562485	C₂	1	2	yes	maximal, complemented
6	#367×D₆₇	C₂	1	2	yes	maximal
7	#366×D₆₇	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#125×D₆₇	—	2	2	no	—
11	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
13	#16863798	C₂²	1	2	yes	—
14	#16863797	C₂²	1	4	yes	—
15	#16863796	—	2	2	no	—
16	#16863795	—	2	2	no	—
17	#107×D₆₇	C₂²	1	4	yes	—
18	#106×D₆₇	—	2	2	no	—
19	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
20	#16863224	C₂²	1	2	yes	—
21	#16863223	C₂²	1	4	yes	—
22	#16863222	C₂²	1	2	yes	—
23	#16863221	C₂²	1	4	yes	—
24	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
27	#499×C₆₇	C₂²	1	2	yes	—
28	#367×C₆₇	C₂²	1	2	yes	—
29	#366×C₆₇	C₂²	1	2	yes	—
30	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
33	#14524035	—	2	2	no	—
34	#14524034	—	2	4	no	—
35	#14524033	—	4	4	no	—
36	C₄⋊C₄×D₆₇	—	2	4	no	—
37	C₄⋊C₄×D₆₇	—	4	4	no	—
38	C₄²×D₆₇	D₄	1	1	yes	characteristic
39	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₁₆×D₆₇	Q₈	1	4	yes	—
42	C₁₆×D₆₇	Q₈	1	4	yes	—
43	#14523977	C₂³	1	4	yes	—
44	#14523976	—	2	4	no	—
45	#14523975	—	2	2	no	—
46	#14523974	—	2	4	no	—
47	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
48	C₈×Dic₆₇	D₄	1	1	yes	characteristic
49	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
50	#125×C₆₇	—	2	2	no	—
51	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	#107×C₆₇	C₂³	1	4	yes	—
54	#106×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₆₇	—	4	4	no	—
57	C₂×C₄×D₆₇	D₈	1	2	yes	—
58	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
61	C₈×D₆₇	Q₁₆	1	4	yes	—
62	C₈×D₆₇	C₂×Q₈	1	2	yes	—
63	C₈×D₆₇	C₂×Q₈	1	2	yes	—
64	C₄⋊Dic₆₇	—	2	4	no	—
65	C₄⋊Dic₆₇	—	4	4	no	—
66	Dic₆₇⋊C₄	—	4	4	no	—
67	Dic₆₇⋊C₄	—	4	4	no	—
68	Dic₆₇⋊C₄	—	4	4	no	—
69	C₄×Dic₆₇	D₈	1	2	yes	—
70	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
71	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
72	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₄⋊C₄×C₆₇	—	4	4	no	—
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
79	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₇	—	4	4	no	—
85	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
86	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
87	C₄×D₆₇	—	2	4	no	—
88	C₄×D₆₇	—	8	8	no	—
89	C₄×D₆₇	C₄⋊Q₈	1	2	yes	—
90	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
91	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
94	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
95	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₆₇	#500	1	2	yes	—
98	C₂×D₆₇	#508	1	4	yes	—
99	C₄×C₆₇	—	2	2	no	—
100	C₄×C₆₇	—	8	4	no	—
101	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₆₇	—	2	2	no	—
104	Dic₆₇	—	8	4	no	—
105	Dic₆₇	#500	1	2	yes	—
106	C₂×#2253	—	67	1	no	maximal, Sylow
107	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#508	1	2	yes	—
109	D₆₇	#2253	1	4	yes	direct product factor
110	C₂×#499	—	67	2	no	—
111	#2253	—	67	2	no	—
112	#2253	—	67	2	no	—
113	#2253	—	67	2	no	—
114	#2253	—	67	1	no	—
115	#2253	D₆₇	1	1	yes	characteristic, direct product factor
116	C₂×#367	—	67	2	no	—
117	C₂×#366	—	67	2	no	—
118	C₂×C₄×C₁₆	—	67	1	no	—
119	C₆₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
120	C₂×#125	—	134	2	no	—
121	C₂²×C₁₆	—	67	1	no	—
122	C₂²×C₁₆	—	67	1	no	—
123	#499	—	67	4	no	—
124	#499	—	67	2	no	—
125	#499	—	134	2	no	—
126	#499	—	134	2	no	—
127	#499	C₂×D₆₇	1	2	yes	—
128	C₂×#107	—	67	4	no	—
129	C₂×#106	—	134	2	no	—
130	C₂×C₄×C₈	—	67	1	no	—
131	#367	—	67	2	no	—
132	#367	—	67	4	no	—
133	#367	C₂×D₆₇	1	2	yes	—
134	#366	—	67	2	no	—
135	#366	—	67	4	no	—
136	#366	C₂×D₆₇	1	2	yes	—
137	C₄×C₁₆	—	67	1	no	—
138	C₄×C₁₆	—	67	1	no	—
139	C₄×C₁₆	—	67	1	no	—
140	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
141	C₂²×C₈	—	67	1	no	—
142	C₂²×C₈	—	67	1	no	—
143	#125	—	2	2	no	—
144	#125	—	268	4	no	—
145	#125	—	134	4	no	—
146	#125	—	134	2	no	—
147	C₂×C₄⋊C₄	—	268	4	no	—
148	C₂×C₄⋊C₄	—	134	4	no	—
149	C₂×C₄²	—	67	1	no	—
150	C₂×C₁₆	—	67	1	no	—
151	C₂×C₁₆	—	67	1	no	—
152	C₂×C₁₆	—	67	4	no	—
153	C₂×C₁₆	—	67	4	no	—
154	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
156	#107	—	67	4	no	—
157	#107	—	134	4	no	—
158	#107	C₂²×D₆₇	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	134	4	no	—
161	#106	—	134	2	no	—
162	C₄×C₈	—	67	1	no	—
163	C₄×C₈	—	67	1	no	—
164	C₄×C₈	—	67	1	no	—
165	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₂²×C₄	—	268	4	no	—
167	C₂²×C₄	—	67	1	no	—
168	C₂²×C₄	—	67	2	no	—
169	C₂×C₈	—	67	1	no	—
170	C₂×C₈	—	67	1	no	—
171	C₂×C₈	—	67	4	no	—
172	C₂×C₈	—	67	2	no	—
173	C₂×C₈	—	67	2	no	—
174	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	268	4	no	—
179	C₄⋊C₄	—	268	4	no	—
180	C₄⋊C₄	—	268	4	no	—
181	C₄⋊C₄	—	268	4	no	—
182	C₄⋊C₄	—	134	4	no	—
183	C₄²	—	67	2	no	—
184	C₄²	—	67	1	no	—
185	C₄²	D₄×D₆₇	1	1	yes	characteristic
186	C₁₆	—	67	2	no	—
187	C₁₆	—	67	2	no	—
188	C₁₆	Q₈×D₆₇	1	2	yes	—
189	C₁₆	Q₈×D₆₇	1	2	yes	—
190	C₂³	—	67	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	268	4	no	—
193	C₂×C₄	—	536	8	no	—
194	C₂×C₄	—	67	1	no	—
195	C₂×C₄	—	67	2	no	—
196	C₂×C₄	—	67	2	no	—
197	C₂×C₄	—	67	2	no	—
198	C₂×C₄	—	134	4	no	—
199	C₂×C₄	D₈×D₆₇	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
201	C₈	—	67	2	no	—
202	C₈	—	67	2	no	—
203	C₈	Q₁₆×D₆₇	1	2	yes	—
204	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
206	C₂²	—	67	4	no	—
207	C₂²	—	67	2	no	—
208	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	536	4	no	—
212	C₄	—	67	2	no	—
213	C₄	—	134	2	no	—
214	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
216	C₂	—	67	4	no	—
217	C₂	#500×D₆₇	1	1	yes	characteristic
218	C₂	#508×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2253	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2253	D₆₇
hypocenter	C₆₇	C₂×#2253
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #22562488	C₂	semidirect
3. #22562487	C₂	semidirect
4. #22562486	C₂	semidirect
5. #22562485	C₂	semidirect
9. C₆₇×#2253	C₂	semidirect
109. D₆₇	#2253	direct (N × H)
115. #2253	D₆₇	direct (N × H)
119. C₆₇	C₂×#2253	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#499×D₆₇	2
#22562488	2
#22562487	2
#22562486	2
#22562485	2
#367×D₆₇	2
#366×D₆₇	2
C₄×C₁₆×D₆₇	2
C₆₇×#2253	2
C₂×#2253	67

Subgroups — #2253×D67 #31011309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2253×D₆₇ #31011309