Subgroups — C₄×#16863222 #30771230

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863222	C₂	1	2	yes	maximal
2	C₄×#14523977	C₂	1	2	yes	maximal
3	#22558301	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523977	C₂²	1	4	yes	—
6	C₄×C₄⋊Dic₆₇	—	2	2	no	—
7	#16863222	C₄	1	8	yes	direct product factor
8	#16863159	—	2	2	no	—
9	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₆₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₆₇	—	2	4	no	—
14	C₂×C₄×Dic₆₇	—	4	2	no	—
15	#14523977	—	2	4	no	—
16	#14523977	C₂×C₄	1	8	yes	—
17	C₆₇⋊#95	—	4	2	no	—
18	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₇	—	2	2	no	—
22	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₆₇	Q₈	1	2	yes	—
27	C₂²×Dic₆₇	—	4	4	no	—
28	C₄⋊Dic₆₇	—	2	8	no	—
29	C₄⋊Dic₆₇	—	4	4	no	—
30	C₄×Dic₆₇	—	8	4	no	—
31	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	#41	1	2	yes	—
40	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₄²×C₆₇	Q₁₆	1	2	yes	—
42	C₁₆×C₆₇	—	2	2	no	—
43	C₁₆×C₆₇	C₄²	1	4	yes	complemented
44	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	#107	1	2	yes	—
47	C₂×C₄×C₆₇	#107	1	2	yes	—
48	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
49	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#135	1	2	yes	—
54	C₂×Dic₆₇	—	4	8	no	—
55	C₂×Dic₆₇	—	8	4	no	—
56	C₂×Dic₆₇	—	8	4	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₆₇	#117	1	1	yes	characteristic
60	C₈×C₆₇	C₄×Q₈	1	2	yes	—
61	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
62	C₂²×C₆₇	C₄×D₈	1	2	yes	—
63	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
65	C₂²×C₆₇	#508	1	2	yes	—
66	C₄×C₆₇	—	2	2	no	—
67	C₄×C₆₇	#366	1	4	yes	complemented
68	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₆₇	#427	1	1	yes	characteristic
70	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
71	Dic₆₇	—	8	8	no	—
72	C₄×#366	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
74	C₂×C₆₇	C₂×#366	1	1	yes	characteristic
75	C₂×C₆₇	#2158	1	1	yes	characteristic
76	C₂×C₆₇	C₄×D₁₆	1	2	yes	—
77	C₂×C₆₇	C₄×Q₃₂	1	2	yes	—
78	C₂×#366	—	67	2	no	—
79	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
80	C₄×#107	—	67	2	no	—
81	#1380	—	67	2	no	—
82	C₆₇	C₄×#366	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
85	C₂×#107	—	67	4	no	—
86	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄×C₄⋊C₄	—	134	2	no	—
88	#366	—	67	8	no	—
89	C₄×C₁₆	Dic₆₇	1	4	yes	complemented
90	#340	—	134	2	no	—
91	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	134	4	no	—
94	C₂×C₄²	—	268	2	no	—
95	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
98	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
99	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
100	#107	—	67	8	no	—
101	#107	—	134	4	no	—
102	C₄×C₈	Dic₁₃₄	1	2	yes	—
103	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
104	#95	—	268	2	no	—
105	C₂²×C₄	—	268	4	no	—
106	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
107	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
111	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
112	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
113	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
114	C₂×C₈	#11761628	1	2	yes	—
115	C₄⋊C₄	—	268	4	no	—
116	C₄⋊C₄	—	134	8	no	—
117	C₄²	—	536	4	no	—
118	C₄²	C₄⋊Dic₆₇	1	2	yes	—
119	C₄²	Dic₂₆₈	1	2	yes	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₄×Dic₆₇	1	4	yes	complemented
122	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	268	8	no	—
125	C₂×C₄	—	536	4	no	—
126	C₂×C₄	—	536	4	no	—
127	C₂×C₄	#14523977	1	2	yes	—
128	C₂×C₄	#14523977	1	2	yes	—
129	C₂×C₄	D₁₀₇₂	1	2	yes	—
130	C₂×C₄	Dic₅₃₆	1	2	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
132	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
133	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
134	C₂×C₄	#14524065	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
137	C₈	#14524002	1	1	yes	characteristic
138	C₈	C₄×Dic₁₃₄	1	2	yes	—
139	C₂²	C₂×#14523977	1	1	yes	characteristic
140	C₂²	C₄×D₅₃₆	1	2	yes	—
141	C₂²	C₂×D₁₀₇₂	1	1	yes	characteristic
142	C₂²	C₂×Dic₅₃₆	1	1	yes	characteristic
143	C₂²	#16863828	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	536	8	no	—
146	C₄	#16863222	1	4	yes	direct product factor
147	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
148	C₄	#16863434	1	1	yes	characteristic
149	C₄	C₄×Dic₂₆₈	1	2	yes	—
150	C₂	C₄×#14523977	1	1	yes	characteristic
151	C₂	C₂×#16863222	1	1	yes	characteristic
152	C₂	#22561993	1	1	yes	characteristic
153	C₂	C₄×D₁₀₇₂	1	2	yes	—
154	C₂	C₄×Dic₅₃₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₅₃₆
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₄×#366
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. #16863222	C₄	direct (N × H)
12. C₄×C₁₆×C₆₇	C₄	semidirect
43. C₁₆×C₆₇	C₄²	semidirect
67. C₄×C₆₇	#366	semidirect
82. C₆₇	C₄×#366	semidirect
89. C₄×C₁₆	Dic₆₇	semidirect
121. C₁₆	C₄×Dic₆₇	semidirect
146. C₄	#16863222	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863222	2
C₄×#14523977	2
#22558301	2
C₂×C₄×C₁₆×C₆₇	2
C₄×#366	67

Subgroups — C4×#16863222 #30771230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863222 #30771230