Subgroups — C₂²×#16863793 #31725419

Order 17152.

2398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524042	C₂	1	2	yes	maximal
2	C₂×#16863793	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14523974	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	—	2	2	no	—
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524042	—	2	12	no	—
10	C₂×#14524042	C₂²	1	24	yes	—
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	#16863793	C₂²	1	64	yes	direct product factor
15	C₂×#14523975	C₂²	1	12	yes	—
16	C₂×#14523974	C₂²	1	24	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	C₂²×Dic₁₃₄	—	2	24	no	—
22	C₂²×Dic₁₃₄	—	4	16	no	—
23	C₂³×Dic₆₇	—	2	2	no	—
24	C₂³×Dic₆₇	D₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
26	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
27	#14524042	—	2	48	no	—
28	#14524042	C₂³	1	32	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	6	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
31	C₂×Dic₆₇⋊C₄	—	2	24	no	—
32	C₂×C₄×Dic₆₇	D₄	1	6	yes	—
33	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
34	#14523975	C₂³	1	16	yes	—
35	#14523974	C₂³	1	32	yes	—
36	C₈×Dic₆₇	C₂³	1	16	yes	—
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	Q₈	1	8	yes	complemented
40	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
43	#106×C₆₇	C₂³	1	16	yes	—
44	C₂×Dic₁₃₄	—	2	32	no	—
45	C₂×Dic₁₃₄	—	4	96	no	—
46	C₂²×Dic₆₇	—	2	12	no	—
47	C₂²×Dic₆₇	—	4	8	no	—
48	C₂²×Dic₆₇	SD₁₆	1	16	yes	—
49	C₂²×Dic₆₇	C₂×D₄	1	12	yes	—
50	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	complemented
51	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
52	C₄⋊Dic₆₇	—	2	12	no	—
53	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
54	Dic₆₇⋊C₄	—	2	32	no	—
55	C₄×Dic₆₇	C₂×D₄	1	12	yes	—
56	C₄×Dic₆₇	C₂⁴	1	4	yes	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	12	no	—
59	C₂²×C₄×C₆₇	—	4	8	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	complemented
65	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
66	C₄⋊C₄×C₆₇	—	2	12	no	—
67	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
68	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
69	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
70	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₆₇	—	2	8	no	—
72	C₂×C₄×C₆₇	—	4	24	no	—
73	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
74	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
76	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
77	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₆₇	—	2	8	no	—
79	C₂×Dic₆₇	—	4	24	no	—
80	C₂×Dic₆₇	C₂×SD₁₆	1	48	yes	—
81	C₂×Dic₆₇	C₂²×D₄	1	8	yes	—
82	Dic₁₃₄	—	4	64	no	—
83	C₈×C₆₇	C₂²×Q₈	1	8	yes	complemented
84	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	complemented
85	C₂²×C₆₇	#498	1	16	yes	complemented
86	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
87	C₂²×C₆₇	C₂²×SD₁₆	1	12	yes	—
88	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
89	C₄×C₆₇	—	4	8	no	—
90	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
91	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
92	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
93	Dic₆₇	—	4	8	no	—
94	Dic₆₇	C₂²×SD₁₆	1	16	yes	—
95	C₂²×#498	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	C₂×#498	1	12	yes	complemented
97	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₆₇	C₂³×SD₁₆	1	2	yes	—
99	C₂²×C₄⋊Q₈	—	67	2	no	—
100	C₂×#498	—	67	24	no	—
101	C₂²×#106	—	67	1	no	—
102	C₂²×#106	—	67	2	no	—
103	C₂²×#106	D₆₇	1	1	yes	characteristic
104	C₂²×C₄×C₈	—	67	1	no	—
105	C₆₇	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
106	C₂³×Q₈	—	134	2	no	—
107	C₂³×C₈	—	67	1	no	—
108	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	—	67	24	no	—
110	C₂×C₄⋊Q₈	—	134	12	no	—
111	C₂²×C₄⋊C₄	—	67	2	no	—
112	C₂²×C₄⋊C₄	—	134	2	no	—
113	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
114	C₂²×C₄²	—	67	1	no	—
115	#498	—	67	64	no	—
116	C₂×#106	—	67	12	no	—
117	C₂×#106	—	67	24	no	—
118	C₂×#106	C₂×D₆₇	1	12	yes	—
119	C₂×C₄×C₈	—	67	12	no	—
120	C₂²×Q₈	—	268	16	no	—
121	C₂²×Q₈	—	134	24	no	—
122	C₂³×C₄	—	2	2	no	—
123	C₂³×C₄	—	67	2	no	—
124	C₂³×C₄	—	134	2	no	—
125	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
126	C₂²×C₈	—	67	8	no	—
127	C₂²×C₈	—	67	6	no	—
128	C₂²×C₈	Dic₁₃₄	1	8	yes	complemented
129	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
130	C₄⋊Q₈	—	67	32	no	—
131	C₄⋊Q₈	—	134	48	no	—
132	C₂×C₄⋊C₄	—	2	6	no	—
133	C₂×C₄⋊C₄	—	67	12	no	—
134	C₂×C₄⋊C₄	—	134	24	no	—
135	C₂×C₄⋊C₄	—	134	6	no	—
136	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
137	C₂×C₄²	—	67	6	no	—
138	C₂×C₄²	—	67	6	no	—
139	#106	—	67	16	no	—
140	#106	—	67	32	no	—
141	#106	C₂²×D₆₇	1	16	yes	—
142	C₄×C₈	—	67	16	no	—
143	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
144	C₂×Q₈	—	268	96	no	—
145	C₂×Q₈	—	134	32	no	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	4	8	no	—
148	C₂²×C₄	—	268	8	no	—
149	C₂²×C₄	—	67	16	no	—
150	C₂²×C₄	—	67	12	no	—
151	C₂²×C₄	—	134	12	no	—
152	C₂²×C₄	D₄×D₆₇	1	3	yes	—
153	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
154	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
155	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
156	C₂×C₈	—	67	24	no	—
157	C₂×C₈	—	67	4	no	—
158	C₂×C₈	C₂×Dic₁₃₄	1	24	yes	complemented
159	C₂×C₈	C₂³×D₆₇	1	4	yes	—
160	C₄⋊C₄	—	2	12	no	—
161	C₄⋊C₄	—	67	8	no	—
162	C₄⋊C₄	—	134	32	no	—
163	C₄⋊C₄	—	134	12	no	—
164	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
165	C₄²	—	67	12	no	—
166	C₄²	—	67	4	no	—
167	C₂³	#14524042	1	4	yes	—
168	C₂³	SD₁₆×D₆₇	1	8	yes	—
169	C₂³	C₂×D₄×D₆₇	1	3	yes	—
170	Q₈	—	268	64	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	24	no	—
173	C₂×C₄	—	268	24	no	—
174	C₂×C₄	—	67	48	no	—
175	C₂×C₄	—	67	8	no	—
176	C₂×C₄	—	134	8	no	—
177	C₂×C₄	#14524042	1	12	yes	—
178	C₂×C₄	C₂×D₄×D₆₇	1	3	yes	—
179	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
180	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
181	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
182	C₈	—	67	8	no	—
183	C₈	C₂²×Dic₁₃₄	1	8	yes	complemented
184	C₂²	#16863793	1	16	yes	direct product factor
185	C₂²	C₂×#14524042	1	6	yes	—
186	C₂²	C₂×SD₁₆×D₆₇	1	12	yes	—
187	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
188	C₄	—	4	8	no	—
189	C₄	—	268	8	no	—
190	C₄	—	67	16	no	—
191	C₄	C₂×#14524042	1	6	yes	—
192	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
193	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
194	C₂	C₂×#16863793	1	12	yes	direct product factor
195	C₂	C₂²×#14524042	1	1	yes	characteristic
196	C₂	C₂²×SD₁₆×D₆₇	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×#498
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×#16863793	C₂	direct (N × H)
14. #16863793	C₂²	direct (N × H)
25. C₂²×C₆₇⋊C₈	Q₈	semidirect
39. C₂²×C₈×C₆₇	Q₈	semidirect
50. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
64. C₂×C₈×C₆₇	C₂×Q₈	semidirect
83. C₈×C₆₇	C₂²×Q₈	semidirect
84. C₆₇⋊C₈	C₂²×Q₈	semidirect
85. C₂²×C₆₇	#498	semidirect
96. C₂×C₆₇	C₂×#498	semidirect
105. C₆₇	C₂²×#498	semidirect
128. C₂²×C₈	Dic₁₃₄	semidirect
158. C₂×C₈	C₂×Dic₁₃₄	semidirect
183. C₈	C₂²×Dic₁₃₄	semidirect
184. C₂²	#16863793	direct (N × H)
194. C₂	C₂×#16863793	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×#14524042	2
C₂×#16863793	2
C₂²×#14523975	2
C₂²×#14523974	2
C₂²×C₈×Dic₆₇	2
C₂²×#106×C₆₇	2
C₂²×#498	67

Subgroups — C22×#16863793 #31725419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863793 #31725419