Subgroups — #1726×D₆₇ #30853156

Order 17152.

1614 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×D₆₇	C₂	1	3	yes	maximal
2	C₂×C₄×C₈×D₆₇	C₂	1	3	yes	maximal
3	#376×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22559897	C₂	1	3	yes	maximal, complemented
5	#22559896	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559895	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₆₇	C₂²	1	3	yes	—
10	C₂×C₄²×D₆₇	C₂²	1	3	yes	—
11	C₂×#14523954	C₂²	1	3	yes	—
12	C₂×#14523953	C₂²	1	6	yes	—
13	#97×D₆₇	C₄	1	24	yes	complemented
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
15	C₂×C₈×Dic₆₇	C₂²	1	3	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	3	yes	—
17	C₄×C₈×D₆₇	C₄	1	24	yes	complemented
18	#16863242	C₂²	1	3	yes	—
19	#16863241	C₄	1	4	yes	—
20	#95×D₆₇	C₄	1	4	yes	—
21	C₂×#97×C₆₇	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
23	#376×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₆₇	C₂×C₄	1	6	yes	—
25	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
28	C₂×C₈×D₆₇	C₂×C₄	1	12	yes	—
29	C₂×C₈×D₆₇	C₂×C₄	1	12	yes	—
30	C₂×C₈×D₆₇	C₂×C₄	1	24	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
32	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
33	C₄²×D₆₇	C₂×C₄	1	12	yes	—
34	C₄²×D₆₇	C₂×C₄	1	12	yes	—
35	#14523954	C₂×C₄	1	12	yes	complemented
36	#14523953	C₂×C₄	1	24	yes	complemented
37	C₄×C₆₇⋊C₈	C₂×C₄	1	12	yes	complemented
38	C₈×Dic₆₇	C₂×C₄	1	12	yes	complemented
39	C₈×Dic₆₇	C₂×C₄	1	12	yes	complemented
40	C₆₇⋊#95	C₂×C₄	1	12	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
44	#97×C₆₇	C₂×C₄	1	12	yes	complemented
45	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	complemented
46	#95×C₆₇	C₂×C₄	1	4	yes	—
47	C₂³×D₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
49	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₆₇	—	2	24	no	—
51	C₂×C₄×D₆₇	C₄²	1	24	yes	—
52	C₂×C₄×D₆₇	C₂²×C₄	1	6	yes	—
53	C₂×C₄×D₆₇	C₂²×C₄	1	6	yes	—
54	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
55	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	12	yes	—
57	C₈×D₆₇	C₄²	1	16	yes	—
58	C₈×D₆₇	C₄²	1	48	yes	—
59	C₄×Dic₆₇	C₄²	1	24	yes	—
60	C₄×Dic₆₇	C₂²×C₄	1	6	yes	—
61	C₄×Dic₆₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
66	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
67	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
68	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
69	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂²×D₆₇	C₂×C₄²	1	6	yes	—
71	C₂²×D₆₇	#131	1	8	yes	—
72	C₂×C₄×C₆₇	—	2	12	no	—
73	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
75	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
76	C₂×Dic₆₇	—	2	12	no	—
77	C₂×Dic₆₇	C₂×C₄²	1	12	yes	—
78	C₂×Dic₆₇	C₂×C₄²	1	6	yes	—
79	C₄×D₆₇	—	2	48	no	—
80	C₄×D₆₇	C₂×C₄²	1	4	yes	—
81	C₄×D₆₇	C₂×C₄²	1	12	yes	—
82	C₈×C₆₇	C₂×C₄²	1	4	yes	—
83	C₈×C₆₇	C₂×C₄²	1	12	yes	—
84	C₆₇⋊C₈	C₂×C₄²	1	4	yes	—
85	C₆₇⋊C₈	C₂×C₄²	1	12	yes	—
86	C₂²×C₆₇	C₂²×C₄²	1	3	yes	—
87	C₂²×C₆₇	C₂×#131	1	4	yes	—
88	C₂×D₆₇	C₄³	1	4	yes	—
89	C₂×D₆₇	#405	1	24	yes	—
90	C₄×C₆₇	—	2	12	no	—
91	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
92	C₄×C₆₇	C₂²×C₄²	1	3	yes	—
93	Dic₆₇	—	2	12	no	—
94	Dic₆₇	C₄³	1	4	yes	—
95	C₂×#1726	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
97	C₂×C₆₇	C₂×#405	1	6	yes	—
98	D₆₇	#1726	1	8	yes	direct product factor
99	C₂²×#97	—	67	3	no	—
100	C₂²×C₄×C₈	—	67	3	no	—
101	C₂×#376	—	67	1	no	—
102	#1726	—	67	3	no	—
103	#1726	—	67	3	no	—
104	#1726	—	67	1	no	—
105	#1726	D₆₇	1	1	yes	characteristic, direct product factor
106	C₆₇	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	—	67	3	no	—
108	C₂³×C₈	—	67	1	no	—
109	C₂²×C₄²	—	67	3	no	—
110	C₂×#97	—	67	6	no	—
111	C₂×#97	—	67	3	no	—
112	C₂×#97	—	67	24	no	—
113	C₂×#97	C₂×D₆₇	1	3	yes	—
114	C₂×C₄×C₈	—	67	3	no	—
115	C₂×C₄×C₈	—	67	3	no	—
116	C₂×C₄×C₈	—	67	3	no	—
117	C₂×C₄×C₈	—	67	24	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
119	#376	—	67	3	no	—
120	#376	—	67	4	no	—
121	#376	C₂×D₆₇	1	1	yes	characteristic
122	C₂×#95	—	67	4	no	—
123	C₂³×C₄	—	67	1	no	—
124	C₂³×C₄	—	67	6	no	—
125	C₂²×C₈	—	67	1	no	—
126	C₂²×C₈	—	67	3	no	—
127	C₂²×C₈	—	67	12	no	—
128	C₂²×C₈	—	67	12	no	—
129	C₂²×C₈	—	67	24	no	—
130	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
131	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
132	C₂×C₄²	—	67	6	no	—
133	C₂×C₄²	—	67	3	no	—
134	C₂×C₄²	—	67	12	no	—
135	C₂×C₄²	—	67	12	no	—
136	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
137	#97	—	67	24	no	—
138	#97	—	67	12	no	—
139	#97	C₄×D₆₇	1	12	yes	complemented
140	C₄×C₈	—	67	12	no	—
141	C₄×C₈	—	67	12	no	—
142	C₄×C₈	—	67	12	no	—
143	C₄×C₈	C₄×D₆₇	1	12	yes	complemented
144	#95	—	67	12	no	—
145	#95	C₄×D₆₇	1	4	yes	—
146	C₂⁴	—	67	1	no	—
147	C₂²×C₄	—	67	1	no	—
148	C₂²×C₄	—	67	6	no	—
149	C₂²×C₄	—	67	6	no	—
150	C₂²×C₄	—	67	6	no	—
151	C₂²×C₄	—	67	24	no	—
152	C₂²×C₄	—	134	24	no	—
153	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
154	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
155	C₂×C₈	—	67	6	no	—
156	C₂×C₈	—	67	6	no	—
157	C₂×C₈	—	67	12	no	—
158	C₂×C₈	—	67	16	no	—
159	C₂×C₈	—	67	48	no	—
160	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
161	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
162	C₂×C₈	C₂×C₄×D₆₇	1	12	yes	—
163	C₄²	—	67	24	no	—
164	C₄²	—	67	6	no	—
165	C₄²	—	67	6	no	—
166	C₄²	C₂×C₄×D₆₇	1	6	yes	—
167	C₄²	C₂×C₄×D₆₇	1	6	yes	—
168	C₂³	—	67	6	no	—
169	C₂³	—	67	8	no	—
170	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₄	—	2	12	no	—
172	C₂×C₄	—	67	6	no	—
173	C₂×C₄	—	67	4	no	—
174	C₂×C₄	—	67	12	no	—
175	C₂×C₄	—	67	12	no	—
176	C₂×C₄	—	134	48	no	—
177	C₂×C₄	—	134	12	no	—
178	C₂×C₄	C₄²×D₆₇	1	12	yes	—
179	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
180	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
181	C₈	—	67	4	no	—
182	C₈	—	67	12	no	—
183	C₈	C₄²×D₆₇	1	12	yes	—
184	C₈	C₄²×D₆₇	1	4	yes	—
185	C₂²	—	67	24	no	—
186	C₂²	—	67	4	no	—
187	C₂²	C₂×C₄²×D₆₇	1	3	yes	—
188	C₂²	#131×D₆₇	1	4	yes	—
189	C₄	—	2	12	no	—
190	C₄	—	67	4	no	—
191	C₄	—	134	12	no	—
192	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
193	C₄	C₂×C₄²×D₆₇	1	3	yes	—
194	C₂	—	67	8	no	—
195	C₂	C₄³×D₆₇	1	1	yes	characteristic
196	C₂	#405×D₆₇	1	6	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂³×D₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	C₂×#1726
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
4. #22559897	C₂	semidirect
5. #22559896	C₂	semidirect
6. #22559895	C₂	semidirect
7. C₆₇×#1726	C₂	semidirect
13. #97×D₆₇	C₄	semidirect
17. C₄×C₈×D₆₇	C₄	semidirect
35. #14523954	C₂×C₄	semidirect
36. #14523953	C₂×C₄	semidirect
37. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
38. C₈×Dic₆₇	C₂×C₄	semidirect
39. C₈×Dic₆₇	C₂×C₄	semidirect
44. #97×C₆₇	C₂×C₄	semidirect
45. C₄×C₈×C₆₇	C₂×C₄	semidirect
98. D₆₇	#1726	direct (N × H)
105. #1726	D₆₇	direct (N × H)
106. C₆₇	C₂×#1726	semidirect
139. #97	C₄×D₆₇	semidirect
143. C₄×C₈	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×D₆₇	2
C₂×C₄×C₈×D₆₇	2
#376×D₆₇	2
#22559897	2
#22559896	2
#22559895	2
C₆₇×#1726	2
C₂×#1726	67

Subgroups — #1726×D67 #30853156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1726×D₆₇ #30853156