Subgroups — #347×Dic₆₇ #30770657

Order 17152.

418 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863174	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863173	C₂	1	2	yes	maximal
3	M₃₂×Dic₆₇	C₂	1	2	yes	maximal
4	#97×Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#347×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523954	C₄	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
13	#16863174	C₄	1	4	yes	complemented
14	#16863173	C₄	1	8	yes	complemented
15	#16863171	C₄	1	4	yes	complemented
16	C₁₆×Dic₆₇	C₄	1	4	yes	complemented
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	#347×C₆₇	C₄	1	4	yes	complemented
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	—	2	2	no	—
23	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
25	#14523983	C₂×C₄	1	4	yes	—
26	#14523983	C₂×C₄	1	4	yes	—
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
28	#14523954	—	2	2	no	—
29	#14523953	—	2	2	no	—
30	#14523953	C₂×C₄	1	4	yes	—
31	C₈×Dic₆₇	—	2	2	no	—
32	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
33	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
38	M₃₂×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
40	#97×C₆₇	C₂×C₄	1	2	yes	—
41	#97×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×Dic₆₇	—	2	1	no	—
43	C₂²×Dic₆₇	M₁₆	1	2	yes	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	1	no	—
46	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
47	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
48	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
49	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
50	C₄×Dic₆₇	—	2	1	no	—
51	C₄×Dic₆₇	—	4	4	no	—
52	C₄×Dic₆₇	C₄²	1	2	yes	—
53	C₄×Dic₆₇	C₄²	1	1	yes	characteristic
54	C₄×Dic₆₇	C₄²	1	1	yes	characteristic
55	C₄×Dic₆₇	M₁₆	1	2	yes	—
56	C₆₇⋊C₁₆	C₄²	1	8	yes	complemented
57	C₂²×C₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	C₄²	1	2	yes	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₆₇	—	2	1	no	—
67	C₄²×C₆₇	C₄²	1	2	yes	—
68	C₁₆×C₆₇	C₄²	1	8	yes	complemented
69	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₆₇	—	2	4	no	—
79	C₂×Dic₆₇	—	2	2	no	—
80	C₂×Dic₆₇	—	4	2	no	—
81	C₂×Dic₆₇	#97	1	2	yes	—
82	C₂×Dic₆₇	#97	1	2	yes	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	C₂×C₄²	1	2	yes	—
85	C₈×C₆₇	C₂×C₄²	1	2	yes	—
86	C₆₇⋊C₈	—	2	2	no	—
87	C₆₇⋊C₈	#97	1	4	yes	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
90	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₆₇	—	4	2	no	—
93	C₄×C₆₇	C₄³	1	1	yes	characteristic
94	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
95	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
97	Dic₆₇	—	2	2	no	—
98	Dic₆₇	—	4	2	no	—
99	Dic₆₇	#347	1	4	yes	direct product factor
100	C₄×#347	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	C₄×#97	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#347	1	2	yes	—
105	C₂×#347	—	67	2	no	—
106	C₂×#347	—	67	1	no	—
107	C₂×#347	D₆₇	1	1	yes	characteristic
108	C₄×M₃₂	—	67	2	no	—
109	C₄×#97	—	67	1	no	—
110	C₆₇	C₄×#347	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	67	2	no	—
112	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
113	C₂×#97	—	67	1	no	—
114	C₂×#97	—	67	1	no	—
115	C₂×#97	—	67	1	no	—
116	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
117	C₂×C₄×C₈	—	67	1	no	—
118	C₂×C₄×C₈	—	67	1	no	—
119	C₄³	—	67	1	no	—
120	#347	—	67	8	no	—
121	#347	—	67	4	no	—
122	#347	Dic₆₇	1	4	yes	direct product factor
123	#346	—	67	4	no	—
124	C₄×C₁₆	—	67	4	no	—
125	C₂²×C₈	—	67	1	no	—
126	C₂²×C₈	—	67	1	no	—
127	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
129	C₂×C₄²	—	67	1	no	—
130	C₂×C₄²	—	67	1	no	—
131	C₂×C₄²	—	134	2	no	—
132	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
133	M₃₂	—	67	4	no	—
134	M₃₂	—	67	4	no	—
135	M₃₂	C₄×D₆₇	1	4	yes	—
136	M₃₂	C₂×Dic₆₇	1	4	yes	—
137	C₂×C₁₆	—	67	4	no	—
138	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
139	#97	—	67	4	no	—
140	#97	—	134	2	no	—
141	#97	—	134	2	no	—
142	#97	C₄×D₆₇	1	2	yes	—
143	#97	C₂×Dic₆₇	1	2	yes	—
144	C₄×C₈	—	67	2	no	—
145	C₄×C₈	—	67	2	no	—
146	C₄×C₈	—	134	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	67	2	no	—
149	C₂²×C₄	—	134	1	no	—
150	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	67	2	no	—
155	C₂×C₈	—	67	2	no	—
156	C₂×C₈	—	67	2	no	—
157	C₂×C₈	—	134	1	no	—
158	C₂×C₈	—	134	1	no	—
159	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
163	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
164	C₄²	—	2	1	no	—
165	C₄²	—	268	4	no	—
166	C₄²	—	67	1	no	—
167	C₄²	—	67	1	no	—
168	C₄²	—	67	2	no	—
169	C₄²	—	67	2	no	—
170	C₄²	—	134	1	no	—
171	C₄²	C₄×Dic₆₇	1	2	yes	—
172	C₁₆	—	67	8	no	—
173	C₁₆	C₄×Dic₆₇	1	8	yes	complemented
174	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	268	2	no	—
178	C₂×C₄	—	67	2	no	—
179	C₂×C₄	—	67	2	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	—	134	4	no	—
182	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
183	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	67	4	no	—
190	C₈	—	134	2	no	—
191	C₈	C₄²×D₆₇	1	2	yes	—
192	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	#97×D₆₇	1	1	yes	characteristic
195	C₂²	M₁₆×Dic₆₇	1	1	yes	characteristic
196	C₂²	#16863348	1	1	yes	characteristic
197	C₄	—	4	2	no	—
198	C₄	—	268	2	no	—
199	C₄	—	67	4	no	—
200	C₄	—	134	2	no	—
201	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
202	C₄	#97×D₆₇	1	1	yes	characteristic
203	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
204	C₄	#16863348	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#97×Dic₆₇	1	1	yes	characteristic
208	C₂	#347×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₆₇
commutator	C₄×C₆₇	C₄³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#347×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#347	D₆₇
hypocenter	C₆₇	C₄×#347
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
13. #16863174	C₄	semidirect
14. #16863173	C₄	semidirect
15. #16863171	C₄	semidirect
16. C₁₆×Dic₆₇	C₄	semidirect
19. #347×C₆₇	C₄	semidirect
56. C₆₇⋊C₁₆	C₄²	semidirect
68. C₁₆×C₆₇	C₄²	semidirect
99. Dic₆₇	#347	direct (N × H)
110. C₆₇	C₄×#347	semidirect
122. #347	Dic₆₇	direct (N × H)
173. C₁₆	C₄×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863174	2
C₂×#16863173	2
M₃₂×Dic₆₇	2
#97×Dic₆₇	2
C₂×#347×C₆₇	2
C₄×#347	67

Subgroups — #347×Dic67 #30770657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #347×Dic₆₇ #30770657