Subgroups — #1455×D₆₇ #30821535

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₁₆×D₆₇	C₂	1	3	yes	maximal
2	#22558513	C₂	1	3	yes	maximal, complemented
3	#22558512	C₂	1	3	yes	maximal, complemented
4	#22558511	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×D₆₇	C₂	1	3	yes	maximal, complemented
6	C₈²×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₆₇	C₄	1	6	yes	—
9	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
10	C₄×#11761612	C₂²	1	3	yes	—
11	#16863348	C₄	1	6	yes	—
12	M₁₆×Dic₆₇	C₄	1	6	yes	—
13	C₄×#11761611	C₂²	1	3	yes	—
14	C₄×#11761611	C₂²	1	3	yes	—
15	#97×D₆₇	C₄	1	6	yes	—
16	C₄×C₈×D₆₇	C₂²	1	3	yes	—
17	C₄×C₈×D₆₇	C₂²	1	3	yes	—
18	#16863118	C₂²	1	6	yes	complemented
19	#16863117	C₂²	1	3	yes	complemented
20	C₈×C₆₇⋊C₈	C₂²	1	3	yes	complemented
21	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
22	#322×C₆₇	C₂²	1	3	yes	complemented
23	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₄×D₆₇	C₂×C₄	1	3	yes	—
25	C₂×#11761612	C₂×C₄	1	6	yes	—
26	C₂×#11761611	C₂×C₄	1	6	yes	—
27	C₂×#11761611	C₂×C₄	1	6	yes	—
28	M₁₆×D₆₇	—	2	24	no	—
29	C₂×C₈×D₆₇	C₂×C₄	1	6	yes	—
30	C₂×C₈×D₆₇	C₂×C₄	1	6	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	3	yes	—
32	C₄²×D₆₇	C₂×C₄	1	3	yes	—
33	C₄²×D₆₇	C₂×C₄	1	3	yes	—
34	C₄²×D₆₇	C₂³	1	1	yes	characteristic
35	C₄²×D₆₇	C₂³	1	1	yes	characteristic
36	#14523954	C₂×C₄	1	6	yes	—
37	#14523953	C₂×C₄	1	6	yes	—
38	#14523953	C₂×C₄	1	6	yes	—
39	C₄×C₆₇⋊C₈	C₂³	1	3	yes	—
40	C₄×C₆₇⋊C₈	C₂³	1	3	yes	—
41	C₈×Dic₆₇	C₂×C₄	1	6	yes	—
42	C₈×Dic₆₇	C₂×C₄	1	6	yes	—
43	C₂×M₁₆×C₆₇	C₂×C₄	1	6	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#97×C₆₇	C₂×C₄	1	6	yes	—
46	C₄×C₈×C₆₇	C₂³	1	3	yes	—
47	C₄×C₈×C₆₇	C₂³	1	3	yes	—
48	C₂³×D₆₇	C₄²	1	1	yes	characteristic
49	C₂²×Dic₆₇	C₄²	1	3	yes	—
50	C₂×C₄×D₆₇	—	2	12	no	—
51	C₂×C₄×D₆₇	C₄²	1	3	yes	—
52	C₂×C₄×D₆₇	C₄²	1	3	yes	—
53	C₂×C₄×D₆₇	C₂²×C₄	1	3	yes	—
54	C₂×C₄×D₆₇	C₂²×C₄	1	3	yes	—
55	#11761612	—	2	12	no	—
56	#11761611	—	2	12	no	—
57	#11761611	—	2	12	no	—
58	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
59	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
60	C₈×D₆₇	—	2	12	no	—
61	C₈×D₆₇	M₁₆	1	24	yes	complemented
62	C₄×Dic₆₇	C₄²	1	6	yes	—
63	C₄×Dic₆₇	C₂²×C₄	1	3	yes	—
64	C₄×Dic₆₇	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
66	M₁₆×C₆₇	—	2	12	no	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
72	C₂²×D₆₇	—	2	6	no	—
73	C₂²×D₆₇	C₂×C₄²	1	1	yes	characteristic
74	C₂²×D₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	6	no	—
76	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
77	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
78	C₂×Dic₆₇	—	2	6	no	—
79	C₂×Dic₆₇	C₂×C₄²	1	3	yes	—
80	C₂×Dic₆₇	C₂×C₄²	1	3	yes	—
81	C₄×D₆₇	—	2	6	no	—
82	C₄×D₆₇	—	2	6	no	—
83	C₄×D₆₇	—	2	6	no	—
84	C₄×D₆₇	C₂×M₁₆	1	12	yes	—
85	C₈×C₆₇	—	2	6	no	—
86	C₈×C₆₇	C₂×M₁₆	1	12	yes	complemented
87	C₆₇⋊C₈	—	2	6	no	—
88	C₆₇⋊C₈	C₂×M₁₆	1	12	yes	complemented
89	C₂²×C₆₇	—	2	3	no	—
90	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
91	C₂×D₆₇	—	2	3	no	—
92	C₂×D₆₇	—	4	4	no	—
93	C₂×D₆₇	C₄×M₁₆	1	6	yes	—
94	C₄×C₆₇	—	2	3	no	—
95	C₄×C₆₇	C₂²×M₁₆	1	6	yes	—
96	Dic₆₇	—	2	3	no	—
97	Dic₆₇	C₄×M₁₆	1	6	yes	—
98	C₂×#1455	—	67	1	no	maximal, Sylow
99	C₂×C₆₇	—	4	1	no	—
100	C₂×C₆₇	C₂×C₄×M₁₆	1	3	yes	—
101	D₆₇	—	4	1	no	—
102	D₆₇	#1455	1	4	yes	direct product factor
103	C₂×C₄×M₁₆	—	67	3	no	—
104	#1455	—	67	3	no	—
105	#1455	—	67	3	no	—
106	#1455	—	67	1	no	—
107	#1455	D₆₇	1	1	yes	characteristic, direct product factor
108	C₂×#322	—	67	3	no	—
109	C₂×C₈²	—	67	1	no	—
110	C₆₇	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
111	C₂²×M₁₆	—	67	6	no	—
112	C₂²×C₄²	—	67	1	no	—
113	C₄×M₁₆	—	67	6	no	—
114	C₄×M₁₆	—	67	3	no	—
115	C₄×M₁₆	—	67	6	no	—
116	C₄×M₁₆	—	67	3	no	—
117	C₄×M₁₆	—	67	3	no	—
118	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
119	C₂×#97	—	67	6	no	—
120	C₂×C₄×C₈	—	67	3	no	—
121	C₂×C₄×C₈	—	67	3	no	—
122	#322	—	67	3	no	—
123	#322	—	67	6	no	—
124	#322	C₂×D₆₇	1	3	yes	complemented
125	C₈²	—	67	3	no	—
126	C₈²	C₂×D₆₇	1	1	yes	characteristic, complemented
127	C₂³×C₄	—	67	3	no	—
128	C₂×M₁₆	—	67	6	no	—
129	C₂×M₁₆	—	67	6	no	—
130	C₂×M₁₆	—	67	6	no	—
131	C₂×M₁₆	—	134	24	no	—
132	C₂×M₁₆	C₄×D₆₇	1	6	yes	—
133	C₂²×C₈	—	67	6	no	—
134	C₂²×C₈	—	67	6	no	—
135	C₂×C₄²	—	67	3	no	—
136	C₂×C₄²	—	67	3	no	—
137	C₂×C₄²	—	67	3	no	—
138	C₂×C₄²	—	67	1	no	—
139	C₂×C₄²	—	67	1	no	—
140	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
141	#97	—	67	6	no	—
142	#97	—	67	6	no	—
143	#97	—	67	6	no	—
144	#97	C₄×D₆₇	1	6	yes	—
145	C₄×C₈	—	67	6	no	—
146	C₄×C₈	—	67	6	no	—
147	C₄×C₈	—	67	3	no	—
148	C₄×C₈	—	67	3	no	—
149	C₄×C₈	C₂²×D₆₇	1	3	yes	—
150	C₄×C₈	C₂²×D₆₇	1	3	yes	—
151	C₂⁴	—	67	1	no	—
152	C₂²×C₄	—	67	3	no	—
153	C₂²×C₄	—	67	3	no	—
154	C₂²×C₄	—	67	3	no	—
155	C₂²×C₄	—	67	3	no	—
156	C₂²×C₄	—	67	3	no	—
157	C₂²×C₄	—	134	12	no	—
158	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
159	M₁₆	—	2	12	no	—
160	M₁₆	—	134	12	no	—
161	M₁₆	—	134	12	no	—
162	M₁₆	—	134	12	no	—
163	C₂×C₈	—	67	6	no	—
164	C₂×C₈	—	67	24	no	—
165	C₂×C₈	—	67	6	no	—
166	C₂×C₈	—	134	12	no	—
167	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
168	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
169	C₄²	—	67	6	no	—
170	C₄²	—	67	3	no	—
171	C₄²	—	67	3	no	—
172	C₄²	C₂×C₄×D₆₇	1	3	yes	—
173	C₄²	C₂³×D₆₇	1	1	yes	characteristic
174	C₂³	—	67	1	no	—
175	C₂³	—	67	1	no	—
176	C₂³	—	134	6	no	—
177	C₂³	C₄²×D₆₇	1	1	yes	characteristic
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	67	3	no	—
180	C₂×C₄	—	67	12	no	—
181	C₂×C₄	—	67	3	no	—
182	C₂×C₄	—	134	6	no	—
183	C₂×C₄	—	134	6	no	—
184	C₂×C₄	—	134	6	no	—
185	C₂×C₄	—	134	6	no	—
186	C₂×C₄	C₄²×D₆₇	1	3	yes	—
187	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
188	C₈	—	2	6	no	—
189	C₈	—	67	12	no	—
190	C₈	—	134	6	no	—
191	C₈	M₁₆×D₆₇	1	12	yes	complemented
192	C₂²	—	2	3	no	—
193	C₂²	—	268	4	no	—
194	C₂²	—	67	6	no	—
195	C₂²	—	134	3	no	—
196	C₂²	C₂×C₄²×D₆₇	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	67	6	no	—
199	C₄	—	134	3	no	—
200	C₄	C₂×M₁₆×D₆₇	1	6	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	268	1	no	—
203	C₂	—	67	4	no	—
204	C₂	C₄×M₁₆×D₆₇	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₄²
Fitting	C₆₇×#1455	C₂
prime radical	C₂³×D₆₇	C₄²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1455	D₆₇
hypocenter	C₆₇	C₂×#1455
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
2. #22558513	C₂	semidirect
3. #22558512	C₂	semidirect
4. #22558511	C₂	semidirect
5. #322×D₆₇	C₂	semidirect
6. C₈²×D₆₇	C₂	semidirect
7. C₆₇×#1455	C₂	semidirect
18. #16863118	C₂²	semidirect
19. #16863117	C₂²	semidirect
20. C₈×C₆₇⋊C₈	C₂²	semidirect
22. #322×C₆₇	C₂²	semidirect
23. C₈²×C₆₇	C₂²	semidirect
61. C₈×D₆₇	M₁₆	semidirect
86. C₈×C₆₇	C₂×M₁₆	semidirect
88. C₆₇⋊C₈	C₂×M₁₆	semidirect
102. D₆₇	#1455	direct (N × H)
107. #1455	D₆₇	direct (N × H)
110. C₆₇	C₂×#1455	semidirect
124. #322	C₂×D₆₇	semidirect
126. C₈²	C₂×D₆₇	semidirect
191. C₈	M₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₁₆×D₆₇	2
#22558513	2
#22558512	2
#22558511	2
#322×D₆₇	2
C₈²×D₆₇	2
C₆₇×#1455	2
C₂×#1455	67

Subgroups — #1455×D67 #30821535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1455×D₆₇ #30821535