Subgroups — C₂²×#16863796 #31725423

Order 17152.

1966 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524034	C₂	1	2	yes	maximal
2	C₂×#16863796	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523976	C₂	1	2	yes	maximal
4	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14524034	—	2	12	no	—
9	C₂×#14524034	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₆₇	—	2	2	no	—
11	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂²×Dic₆₇⋊C₄	—	2	2	no	—
13	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863796	C₂²	1	64	yes	direct product factor
15	C₂×#14523976	C₂²	1	24	yes	—
16	C₂×#14523975	C₂²	1	12	yes	—
17	C₂×C₈×Dic₆₇	C₂²	1	12	yes	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂×#106×C₆₇	C₂²	1	12	yes	—
21	C₂³×Dic₆₇	—	2	2	no	—
22	C₂³×Dic₆₇	D₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
25	#14524034	—	2	48	no	—
26	#14524034	C₂³	1	32	yes	—
27	C₂×C₄⋊Dic₆₇	—	2	24	no	—
28	C₂×C₄⋊Dic₆₇	—	2	6	no	—
29	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
30	C₂×Dic₆₇⋊C₄	—	2	24	no	—
31	C₂×C₄×Dic₆₇	D₄	1	6	yes	—
32	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
33	#14523976	C₂³	1	32	yes	—
34	#14523975	C₂³	1	16	yes	—
35	C₈×Dic₆₇	C₂³	1	16	yes	—
36	C₂³×C₄×C₆₇	—	2	2	no	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	Q₈	1	8	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
42	#106×C₆₇	C₂³	1	16	yes	—
43	C₂²×Dic₆₇	—	2	8	no	—
44	C₂²×Dic₆₇	—	2	12	no	—
45	C₂²×Dic₆₇	—	4	8	no	—
46	C₂²×Dic₆₇	C₂×D₄	1	12	yes	—
47	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	—
48	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
49	C₄⋊Dic₆₇	—	2	12	no	—
50	C₄⋊Dic₆₇	—	2	32	no	—
51	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
52	Dic₆₇⋊C₄	—	2	32	no	—
53	C₄×Dic₆₇	C₂×D₄	1	12	yes	—
54	C₄×Dic₆₇	C₂⁴	1	4	yes	—
55	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	4	8	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
62	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	—
63	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
64	C₄⋊C₄×C₆₇	—	2	12	no	—
65	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
66	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
67	C₂³×C₆₇	#135	1	8	yes	—
68	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
69	C₂×C₄×C₆₇	—	2	8	no	—
70	C₂×C₄×C₆₇	—	4	24	no	—
71	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
72	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
73	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
74	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
75	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
76	C₂×Dic₆₇	—	2	24	no	—
77	C₂×Dic₆₇	—	2	8	no	—
78	C₂×Dic₆₇	—	4	24	no	—
79	C₂×Dic₆₇	C₂²×D₄	1	8	yes	—
80	C₈×C₆₇	C₂²×Q₈	1	8	yes	—
81	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	—
82	C₂²×C₆₇	#499	1	16	yes	complemented
83	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
84	C₂²×C₆₇	C₂×#135	1	12	yes	—
85	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
86	C₄×C₆₇	—	4	8	no	—
87	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
88	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
89	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
90	Dic₆₇	—	2	8	no	—
91	Dic₆₇	—	4	8	no	—
92	C₂²×#499	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	C₂×#499	1	12	yes	complemented
94	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₆₇	C₂²×#135	1	2	yes	—
96	C₂²×#125	—	67	2	no	—
97	C₂×#499	—	67	24	no	—
98	C₂²×#107	—	67	2	no	—
99	C₂²×#106	—	67	1	no	—
100	C₂²×#106	D₆₇	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	67	1	no	—
102	C₆₇	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₈	—	67	1	no	—
104	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#125	—	67	24	no	—
106	C₂×#125	—	134	12	no	—
107	C₂²×C₄⋊C₄	—	67	2	no	—
108	C₂²×C₄⋊C₄	—	134	2	no	—
109	C₂²×C₄⋊C₄	—	134	2	no	—
110	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
111	C₂²×C₄²	—	67	1	no	—
112	#499	—	67	64	no	—
113	C₂×#107	—	67	24	no	—
114	C₂×#106	—	67	12	no	—
115	C₂×#106	C₂×D₆₇	1	12	yes	—
116	C₂×C₄×C₈	—	67	12	no	—
117	C₂³×C₄	—	2	2	no	—
118	C₂³×C₄	—	67	2	no	—
119	C₂³×C₄	—	134	2	no	—
120	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
121	C₂²×C₈	—	67	8	no	—
122	C₂²×C₈	—	67	6	no	—
123	C₂²×C₈	Dic₁₃₄	1	8	yes	—
124	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
125	#125	—	67	32	no	—
126	#125	—	134	48	no	—
127	C₂×C₄⋊C₄	—	2	6	no	—
128	C₂×C₄⋊C₄	—	67	12	no	—
129	C₂×C₄⋊C₄	—	134	24	no	—
130	C₂×C₄⋊C₄	—	134	24	no	—
131	C₂×C₄⋊C₄	—	134	6	no	—
132	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
133	C₂×C₄²	—	67	6	no	—
134	C₂×C₄²	—	67	6	no	—
135	#107	—	67	32	no	—
136	#106	—	67	16	no	—
137	#106	C₂²×D₆₇	1	16	yes	—
138	C₄×C₈	—	67	16	no	—
139	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₄	—	2	12	no	—
141	C₂²×C₄	—	4	8	no	—
142	C₂²×C₄	—	268	8	no	—
143	C₂²×C₄	—	67	12	no	—
144	C₂²×C₄	—	134	12	no	—
145	C₂²×C₄	—	134	8	no	—
146	C₂²×C₄	D₄×D₆₇	1	3	yes	—
147	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
148	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
150	C₂×C₈	—	67	24	no	—
151	C₂×C₈	—	67	4	no	—
152	C₂×C₈	C₂×Dic₁₃₄	1	24	yes	—
153	C₂×C₈	C₂³×D₆₇	1	4	yes	—
154	C₄⋊C₄	—	2	12	no	—
155	C₄⋊C₄	—	67	8	no	—
156	C₄⋊C₄	—	134	32	no	—
157	C₄⋊C₄	—	134	32	no	—
158	C₄⋊C₄	—	134	12	no	—
159	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
160	C₄²	—	67	12	no	—
161	C₄²	—	67	4	no	—
162	C₂³	#14524042	1	4	yes	—
163	C₂³	#14524064	1	8	yes	—
164	C₂³	C₂×D₄×D₆₇	1	3	yes	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	24	no	—
167	C₂×C₄	—	268	24	no	—
168	C₂×C₄	—	67	8	no	—
169	C₂×C₄	—	134	8	no	—
170	C₂×C₄	—	134	24	no	—
171	C₂×C₄	#14524042	1	12	yes	—
172	C₂×C₄	C₂×D₄×D₆₇	1	3	yes	—
173	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
174	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
175	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
176	C₈	—	67	8	no	—
177	C₈	C₂²×Dic₁₃₄	1	8	yes	—
178	C₂²	#16863796	1	16	yes	direct product factor
179	C₂²	C₂×#14524042	1	6	yes	—
180	C₂²	C₂×#14524064	1	12	yes	—
181	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
182	C₄	—	4	8	no	—
183	C₄	—	268	8	no	—
184	C₄	—	134	8	no	—
185	C₄	C₂×#14524042	1	6	yes	—
186	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
187	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
188	C₂	C₂×#16863796	1	12	yes	direct product factor
189	C₂	C₂²×#14524042	1	1	yes	characteristic
190	C₂	C₂²×#14524064	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#106×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×#106	D₆₇
hypocenter	C₆₇	C₂²×#499
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#16863796	C₂	direct (N × H)
14. #16863796	C₂²	direct (N × H)
82. C₂²×C₆₇	#499	semidirect
93. C₂×C₆₇	C₂×#499	semidirect
102. C₆₇	C₂²×#499	semidirect
178. C₂²	#16863796	direct (N × H)
188. C₂	C₂×#16863796	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×#14524034	2
C₂×#16863796	2
C₂²×#14523976	2
C₂²×#14523975	2
C₂²×C₈×Dic₆₇	2
C₂²×#106×C₆₇	2
C₂²×#499	67

Subgroups — C22×#16863796 #31725423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863796 #31725423