Subgroups — C₂×C₆₇×#2244 #30712950

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2244	C₂	1	8	yes	maximal, direct product factor
4	C₂×#357×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₆₇	—	2	1	no	—
7	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
8	C₂×#127×C₆₇	—	2	1	no	—
9	C₂×#125×C₆₇	—	2	1	no	—
10	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
11	#499×C₆₇	—	2	2	no	—
12	#499×C₆₇	C₂²	1	4	yes	complemented
13	#493×C₆₇	—	2	2	no	—
14	#493×C₆₇	C₂²	1	4	yes	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#106×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#357×C₆₇	C₂²	1	16	yes	complemented
19	#346×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×D₄×C₆₇	—	2	2	no	—
21	C₂²×D₄×C₆₇	—	4	2	no	—
22	C₂×D₈×C₆₇	—	2	2	no	—
23	C₂×D₈×C₆₇	—	2	4	no	—
24	C₂×D₈×C₆₇	—	2	8	no	—
25	C₂×D₈×C₆₇	—	4	4	no	—
26	C₂×D₈×C₆₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	#127×C₆₇	—	2	4	no	—
30	#127×C₆₇	—	4	2	no	—
31	#125×C₆₇	—	2	4	no	—
32	#125×C₆₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₆₇	—	2	4	no	—
37	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
38	#107×C₆₇	—	2	2	no	—
39	#107×C₆₇	C₂³	1	4	yes	—
40	#106×C₆₇	—	2	4	no	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂⁴×C₆₇	—	4	2	no	—
44	C₂×D₄×C₆₇	—	2	4	no	—
45	C₂×D₄×C₆₇	—	4	4	no	—
46	C₂×D₄×C₆₇	—	4	8	no	—
47	C₂×D₄×C₆₇	—	4	2	no	—
48	C₂×D₄×C₆₇	—	4	4	no	—
49	C₂×D₄×C₆₇	—	8	8	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	D₈	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	D₈×C₆₇	—	2	8	no	—
54	D₈×C₆₇	—	4	4	no	—
55	D₈×C₆₇	—	4	8	no	—
56	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	4	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	8	no	—
64	C₄²×C₆₇	D₈	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	2	yes	—
66	C₁₆×C₆₇	—	2	4	no	—
67	C₂³×C₆₇	—	4	4	no	—
68	C₂³×C₆₇	—	8	4	no	—
69	C₂³×C₆₇	—	8	8	no	—
70	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
71	D₄×C₆₇	—	4	4	no	—
72	D₄×C₆₇	—	4	4	no	—
73	D₄×C₆₇	—	8	16	no	—
74	D₄×C₆₇	—	8	4	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	8	4	no	—
78	C₂×C₄×C₆₇	#124	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
80	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₆₇	#124	1	2	yes	—
84	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
85	C₈×C₆₇	C₂×#41	1	2	yes	—
86	C₂²×C₆₇	—	8	4	no	—
87	C₂²×C₆₇	—	8	8	no	—
88	C₂²×C₆₇	—	16	8	no	—
89	C₂²×C₆₇	#486	1	2	yes	—
90	C₂²×C₆₇	#509	1	4	yes	—
91	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	—	8	4	no	—
94	C₄×C₆₇	#486	1	2	yes	—
95	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
96	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
97	C₂×#2244	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₆₇	—	16	4	no	—
99	C₂×C₆₇	#2244	1	4	yes	direct product factor
100	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#509	1	2	yes	—
102	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
103	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
104	#2244	C₂×C₆₇	1	8	yes	direct product factor
105	C₂×#357	C₂×C₆₇	1	4	yes	complemented
106	C₂×#346	C₂×C₆₇	1	1	yes	characteristic, complemented
107	C₆₇	C₂×#2244	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×D₈	—	2	1	no	—
109	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
110	C₂×#127	—	2	1	no	—
111	C₂×#125	—	2	1	no	—
112	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
113	#499	—	2	2	no	—
114	#499	C₂²×C₆₇	1	4	yes	complemented
115	#493	—	2	2	no	—
116	#493	C₂²×C₆₇	1	4	yes	—
117	C₂×#107	C₂²×C₆₇	1	2	yes	—
118	C₂×#106	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
120	#357	C₂²×C₆₇	1	16	yes	complemented
121	#346	C₂²×C₆₇	1	4	yes	complemented
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	—	4	2	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	2	8	no	—
127	C₂×D₈	—	4	4	no	—
128	C₂×D₈	C₂³×C₆₇	1	4	yes	—
129	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
131	#127	—	2	4	no	—
132	#127	—	4	2	no	—
133	#125	—	2	4	no	—
134	#125	—	4	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
140	#107	—	2	2	no	—
141	#107	C₂³×C₆₇	1	4	yes	—
142	#106	—	2	4	no	—
143	C₄×C₈	D₄×C₆₇	1	2	yes	—
144	C₄×C₈	C₂³×C₆₇	1	2	yes	—
145	C₂⁴	—	4	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	8	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	8	8	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	D₈×C₆₇	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
155	D₈	—	2	8	no	—
156	D₈	—	4	4	no	—
157	D₈	—	4	8	no	—
158	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
159	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
161	C₂×C₈	#41×C₆₇	1	4	yes	—
162	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	8	no	—
166	C₄²	D₈×C₆₇	1	2	yes	—
167	C₄²	C₂×D₄×C₆₇	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	8	4	no	—
171	C₂³	—	8	8	no	—
172	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	8	16	no	—
176	D₄	—	8	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#124×C₆₇	1	2	yes	—
181	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
182	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
184	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
185	C₈	#124×C₆₇	1	2	yes	—
186	C₈	C₂×SD₁₆×C₆₇	1	4	yes	—
187	C₈	C₂×#41×C₆₇	1	2	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	#486×C₆₇	1	2	yes	—
192	C₂²	#509×C₆₇	1	4	yes	—
193	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	#486×C₆₇	1	2	yes	—
197	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
198	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
199	C₂	—	16	4	no	—
200	C₂	C₆₇×#2244	1	4	yes	direct product factor
201	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
202	C₂	C₂×#509×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#493×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₆₇	C₂	semidirect
3. C₆₇×#2244	C₂	direct (N × H)
4. C₂×#357×C₆₇	C₂	semidirect
5. C₂×#346×C₆₇	C₂	semidirect
12. #499×C₆₇	C₂²	semidirect
18. #357×C₆₇	C₂²	semidirect
19. #346×C₆₇	C₂²	semidirect
97. C₂×#2244	C₆₇	direct (N × H)
99. C₂×C₆₇	#2244	direct (N × H)
102. C₂×#499	C₂×C₆₇	semidirect
104. #2244	C₂×C₆₇	direct (N × H)
105. C₂×#357	C₂×C₆₇	semidirect
106. C₂×#346	C₂×C₆₇	semidirect
107. C₆₇	C₂×#2244	direct (N × H)
114. #499	C₂²×C₆₇	semidirect
120. #357	C₂²×C₆₇	semidirect
121. #346	C₂²×C₆₇	semidirect
200. C₂	C₆₇×#2244	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×#499×C₆₇	2
C₂×#493×C₆₇	2
C₆₇×#2244	2
C₂×#357×C₆₇	2
C₂×#346×C₆₇	2
C₂×#2244	67

Subgroups — C2×C67×#2244 #30712950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2244 #30712950