Subgroups — C₂²×#16863226 #31008883

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal
2	C₂×#16863226	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	—	2	2	no	—
5	C₂×#14523979	—	2	6	no	—
6	C₂×#14523979	C₂²	1	12	yes	—
7	#16863226	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₆₇	C₄	1	6	yes	—
10	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
12	C₂×#11761612	—	2	12	no	—
13	C₂×#11761612	—	4	6	no	—
14	C₂²×C₆₇⋊C₈	—	4	2	no	—
15	#14523979	—	2	12	no	—
16	#14523979	C₂³	1	8	yes	—
17	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	D₄	1	3	yes	—
21	C₂²×C₈×C₆₇	Q₈	1	3	yes	—
22	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
24	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
25	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
26	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
27	#11761612	—	2	8	no	—
28	#11761612	—	4	12	no	—
29	C₂×C₆₇⋊C₈	—	4	12	no	—
30	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	D₈	1	3	yes	—
34	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	Q₁₆	1	3	yes	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
37	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
38	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
39	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
43	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
45	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
47	C₂³×C₆₇	—	2	4	no	—
48	C₂³×C₆₇	#107	1	3	yes	—
49	C₂³×C₆₇	#107	1	1	yes	characteristic
50	C₂³×C₆₇	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄×C₆₇	#107	1	6	yes	—
52	C₂×C₄×C₆₇	#107	1	3	yes	—
53	C₂×C₄×C₆₇	#107	1	3	yes	—
54	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₆₇	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
61	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
64	C₆₇⋊C₈	—	4	8	no	—
65	C₂²×C₆₇	—	2	6	no	—
66	C₂²×C₆₇	—	2	6	no	—
67	C₂²×C₆₇	#368	1	4	yes	complemented
68	C₂²×C₆₇	C₂×#107	1	3	yes	—
69	C₂²×C₆₇	C₂×#107	1	3	yes	—
70	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
71	C₄×C₆₇	C₂×#107	1	3	yes	—
72	C₄×C₆₇	C₂×#107	1	3	yes	—
73	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
75	C₂²×#368	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	—	2	1	no	—
77	C₂×C₆₇	—	2	3	no	—
78	C₂×C₆₇	C₂×#368	1	6	yes	complemented
79	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
80	C₂³×C₁₆	D₆₇	1	1	yes	characteristic
81	C₂²×#108	—	67	2	no	—
82	C₂×#368	—	67	12	no	—
83	C₆₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	—	134	2	no	—
85	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₆₇	1	2	yes	—
87	C₂²×C₁₆	Dic₆₇	1	6	yes	—
88	C₂²×C₁₆	C₂×D₆₇	1	6	yes	—
89	C₂×#108	—	67	12	no	—
90	C₂×#108	—	134	6	no	—
91	#368	—	67	16	no	—
92	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
93	C₂×M₁₆	—	268	6	no	—
94	C₂×M₁₆	—	134	12	no	—
95	C₂²×C₈	—	268	2	no	—
96	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
97	C₂²×C₈	Dic₁₃₄	1	3	yes	—
98	C₂²×C₈	D₂₆₈	1	3	yes	—
99	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
100	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
101	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
102	C₂×C₁₆	C₂×Dic₆₇	1	12	yes	—
103	C₂×C₁₆	C₂×Dic₆₇	1	12	yes	—
104	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
105	#108	—	67	8	no	—
106	#108	—	134	12	no	—
107	C₂⁴	Dic₂₆₈	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₆₇	1	3	yes	—
110	C₂²×C₄	D₅₃₆	1	3	yes	—
111	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
112	C₂²×C₄	Dic₂₆₈	1	3	yes	—
113	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
114	M₁₆	—	268	12	no	—
115	M₁₆	—	134	8	no	—
116	C₂×C₈	—	268	12	no	—
117	C₂×C₈	C₄⋊Dic₆₇	1	6	yes	—
118	C₂×C₈	C₄⋊Dic₆₇	1	3	yes	—
119	C₂×C₈	C₄⋊Dic₆₇	1	3	yes	—
120	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
121	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
122	C₂×C₈	C₂×D₂₆₈	1	3	yes	—
123	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
124	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
125	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₆₇	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#14523977	1	3	yes	—
129	C₂³	#14523977	1	1	yes	characteristic
130	C₂³	C₂×Dic₂₆₈	1	3	yes	—
131	C₂×C₄	#14523977	1	6	yes	—
132	C₂×C₄	#14523977	1	3	yes	—
133	C₂×C₄	#14523977	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
136	C₂×C₄	C₂×D₅₃₆	1	3	yes	—
137	C₂×C₄	C₂×D₅₃₆	1	3	yes	—
138	C₂×C₄	C₂×Dic₂₆₈	1	3	yes	—
139	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
140	C₈	—	268	8	no	—
141	C₈	C₂×C₄⋊Dic₆₇	1	3	yes	—
142	C₈	C₂×C₄⋊Dic₆₇	1	3	yes	—
143	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
144	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#16863226	1	4	yes	direct product factor
148	C₂²	C₂×#14523977	1	3	yes	—
149	C₂²	C₂×#14523977	1	3	yes	—
150	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
151	C₄	C₂×#14523977	1	3	yes	—
152	C₄	C₂×#14523977	1	3	yes	—
153	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	3	no	—
157	C₂	C₂×#16863226	1	6	yes	direct product factor
158	C₂	C₂²×#14523977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂³×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	C₂²×#368
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863226	C₂	direct (N × H)
7. #16863226	C₂²	direct (N × H)
67. C₂²×C₆₇	#368	semidirect
78. C₂×C₆₇	C₂×#368	semidirect
83. C₆₇	C₂²×#368	semidirect
147. C₂²	#16863226	direct (N × H)
157. C₂	C₂×#16863226	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
C₂×#16863226	2
C₂³×C₁₆×C₆₇	2
C₂²×#368	67

Subgroups — C22×#16863226 #31008883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863226 #31008883