Subgroups — C₂²×#16863223 #31008854

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863223	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	C₄	1	8	yes	complemented
7	C₂²×C₆₇⋊C₁₆	C₂²	1	6	yes	—
8	C₂×#14523977	—	2	3	no	—
9	C₂×#14523977	C₂²	1	6	yes	—
10	#16863223	C₂²	1	16	yes	direct product factor
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
13	C₂×#107×C₆₇	—	2	3	no	—
14	C₂×#107×C₆₇	C₂²	1	6	yes	—
15	C₂³×Dic₆₇	—	4	1	no	—
16	C₂×C₄⋊Dic₆₇	—	2	6	no	—
17	C₂×C₄⋊Dic₆₇	—	4	3	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	24	yes	complemented
19	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
20	#14523977	—	2	6	no	—
21	#14523977	C₂³	1	4	yes	—
22	C₂³×C₄×C₆₇	—	4	1	no	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₇	D₄	1	3	yes	—
26	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
29	C₂×C₄⋊C₄×C₆₇	—	4	3	no	—
30	#107×C₆₇	—	2	6	no	—
31	#107×C₆₇	C₂³	1	4	yes	—
32	C₂²×Dic₆₇	—	4	6	no	—
33	C₂²×Dic₆₇	—	8	4	no	—
34	C₄⋊Dic₆₇	—	2	4	no	—
35	C₄⋊Dic₆₇	—	4	6	no	—
36	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
37	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	4	6	no	—
39	C₂²×C₄×C₆₇	—	8	4	no	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₇	D₈	1	3	yes	—
42	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
47	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
48	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	—	2	4	no	—
50	C₄⋊C₄×C₆₇	—	4	6	no	—
51	C₂³×C₆₇	#107	1	4	yes	—
52	C₂³×C₆₇	SD₃₂	1	4	yes	—
53	C₂³×C₆₇	SD₃₂	1	4	yes	—
54	C₂³×C₆₇	C₂×D₈	1	3	yes	—
55	C₂×C₄×C₆₇	—	4	4	no	—
56	C₂×C₄×C₆₇	—	8	12	no	—
57	C₂×C₄×C₆₇	#107	1	12	yes	—
58	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
59	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
60	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
61	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×Dic₆₇	—	4	4	no	—
63	C₂×Dic₆₇	—	8	12	no	—
64	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₆₇	#367	1	16	yes	complemented
68	C₂²×C₆₇	C₂×#107	1	6	yes	—
69	C₂²×C₆₇	C₂×SD₃₂	1	6	yes	—
70	C₂²×C₆₇	C₂×SD₃₂	1	6	yes	—
71	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
72	C₄×C₆₇	—	8	4	no	—
73	C₄×C₆₇	C₂×#107	1	6	yes	—
74	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
76	Dic₆₇	—	8	4	no	—
77	C₂²×#367	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	C₂×#367	1	12	yes	complemented
79	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
80	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
81	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
82	C₂³×C₁₆	—	67	1	no	—
83	C₂²×#107	—	67	1	no	—
84	C₂²×#107	D₆₇	1	1	yes	characteristic
85	C₂×#367	—	67	12	no	—
86	C₆₇	C₂²×#367	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	2	1	no	—
89	C₂²×C₄⋊C₄	—	134	1	no	—
90	C₂²×C₁₆	—	67	8	no	—
91	C₂²×C₁₆	—	67	6	no	—
92	C₂×#107	—	2	3	no	—
93	C₂×#107	—	67	6	no	—
94	C₂×#107	—	134	3	no	—
95	C₂×#107	C₂×D₆₇	1	6	yes	—
96	#367	—	67	16	no	—
97	C₂³×C₄	—	4	1	no	—
98	C₂³×C₄	—	268	1	no	—
99	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
100	C₂²×C₈	Dic₁₃₄	1	4	yes	—
101	C₂²×C₈	C₄×D₆₇	1	4	yes	—
102	C₂²×C₈	C₂²⋊D₆₇	1	3	yes	—
103	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	—	4	3	no	—
106	C₂×C₄⋊C₄	—	268	3	no	—
107	C₂×C₄⋊C₄	—	134	6	no	—
108	C₂×C₁₆	—	67	24	no	—
109	C₂×C₁₆	—	67	4	no	—
110	#107	—	2	6	no	—
111	#107	—	67	4	no	—
112	#107	—	134	6	no	—
113	#107	C₂²×D₆₇	1	4	yes	—
114	C₂⁴	#11761613	1	1	yes	characteristic
115	C₂²×C₄	—	4	6	no	—
116	C₂²×C₄	—	8	4	no	—
117	C₂²×C₄	—	268	6	no	—
118	C₂²×C₄	—	536	4	no	—
119	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
120	C₂²×C₄	#11761613	1	3	yes	—
121	C₂²×C₄	#11761618	1	4	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
123	C₂×C₈	Dic₆₇⋊C₄	1	12	yes	—
124	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
125	C₂×C₈	C₂×C₂²⋊D₆₇	1	3	yes	—
126	C₂×C₈	C₂×Dic₁₃₄	1	6	yes	—
127	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	268	6	no	—
131	C₄⋊C₄	—	134	4	no	—
132	C₁₆	—	67	8	no	—
133	C₂³	#14523976	1	4	yes	—
134	C₂³	#14523986	1	4	yes	—
135	C₂³	#14523987	1	4	yes	—
136	C₂³	C₂×#11761613	1	3	yes	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	12	no	—
139	C₂×C₄	—	268	4	no	—
140	C₂×C₄	—	536	12	no	—
141	C₂×C₄	#14523976	1	12	yes	—
142	C₂×C₄	C₂×Dic₆₇⋊C₄	1	6	yes	—
143	C₂×C₄	C₂×#11761613	1	3	yes	—
144	C₂×C₄	C₂×#11761618	1	6	yes	—
145	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₈	C₂×Dic₆₇⋊C₄	1	6	yes	—
147	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
148	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
149	C₂²	#16863223	1	16	yes	direct product factor
150	C₂²	C₂×#14523976	1	6	yes	—
151	C₂²	C₂×#14523986	1	6	yes	—
152	C₂²	C₂×#14523987	1	6	yes	—
153	C₂²	C₂²×#11761613	1	1	yes	characteristic
154	C₄	—	8	4	no	—
155	C₄	—	536	4	no	—
156	C₄	C₂×#14523976	1	6	yes	—
157	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
158	C₄	C₂²×#11761618	1	1	yes	characteristic
159	C₂	C₂×#16863223	1	12	yes	direct product factor
160	C₂	C₂²×#14523976	1	1	yes	characteristic
161	C₂	C₂²×#14523986	1	1	yes	characteristic
162	C₂	C₂²×#14523987	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×#107×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×#107	D₆₇
hypocenter	C₆₇	C₂²×#367
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#16863223	C₂	direct (N × H)
6. C₂²×C₆₇⋊C₁₆	C₄	semidirect
10. #16863223	C₂²	direct (N × H)
18. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
36. C₆₇⋊C₁₆	C₂²×C₄	semidirect
67. C₂²×C₆₇	#367	semidirect
78. C₂×C₆₇	C₂×#367	semidirect
86. C₆₇	C₂²×#367	semidirect
149. C₂²	#16863223	direct (N × H)
159. C₂	C₂×#16863223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂²×#14523977	2
C₂×#16863223	2
C₂²×#107×C₆₇	2
C₂²×#367	67

Subgroups — C22×#16863223 #31008854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863223 #31008854