Subgroups — C₂×#22559965 #30853335

Order 17152.

930 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863354	C₂	1	3	yes	maximal
2	#22559965	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863153	C₂	1	3	yes	maximal
4	C₂×#338×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×#14524002	C₂²	1	3	yes	—
7	#16863354	C₄	1	12	yes	complemented
8	#16863354	C₂²	1	12	yes	—
9	C₂×#14523953	C₄	1	6	yes	complemented
10	C₂×C₈×Dic₆₇	C₄	1	6	yes	complemented
11	#16863153	C₂²	1	12	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	3	yes	—
13	#338×C₆₇	C₂²	1	4	yes	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	3	yes	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	3	yes	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
17	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
18	#14524002	—	2	3	no	—
19	#14524002	—	2	3	no	—
20	#14524002	C₂×C₄	1	6	yes	—
21	#14524002	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₆₇	—	2	3	no	—
23	C₂×C₂²⋊Dic₆₇	—	2	3	no	—
24	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
25	#14523953	C₂×C₄	1	24	yes	complemented
26	C₈×Dic₆₇	C₂×C₄	1	24	yes	complemented
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	6	no	—
29	C₂×M₁₆×C₆₇	—	2	6	no	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	6	yes	—
31	C₂×M₁₆×C₆₇	C₂³	1	6	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
33	C₂²×Dic₆₇	—	2	6	no	—
34	C₂×C₆₇⋊C₈	C₄²	1	6	yes	—
35	C₂×C₆₇⋊C₈	C₄²	1	6	yes	—
36	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
37	C₂×C₆₇⋊C₈	C₄²	1	3	yes	—
38	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
39	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
40	C₄⋊Dic₆₇	—	2	6	no	—
41	C₄⋊Dic₆₇	—	2	6	no	—
42	C₂²⋊Dic₆₇	—	2	6	no	—
43	C₂²⋊Dic₆₇	—	2	6	no	—
44	C₄×Dic₆₇	—	2	6	no	—
45	C₄×Dic₆₇	C₂²×C₄	1	12	yes	—
46	C₂⁴×C₆₇	#41	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₆₇	#41	1	3	yes	—
50	C₂²×C₄×C₆₇	#41	1	3	yes	—
51	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	12	no	—
54	M₁₆×C₆₇	—	2	12	no	—
55	C₂×C₈×C₆₇	—	2	6	no	—
56	C₂×C₈×C₆₇	—	2	6	no	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
58	C₂³×C₆₇	—	4	2	no	—
59	C₂³×C₆₇	#117	1	3	yes	—
60	C₂³×C₆₇	#117	1	3	yes	—
61	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#117	1	3	yes	—
65	C₂×C₄×C₆₇	#117	1	3	yes	—
66	C₂×C₄×C₆₇	#117	1	6	yes	—
67	C₂×C₄×C₆₇	#117	1	3	yes	—
68	C₂×C₄×C₆₇	#117	1	3	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
70	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
71	C₂×Dic₆₇	—	2	12	no	—
72	C₂×Dic₆₇	—	4	12	no	—
73	C₈×C₆₇	—	2	12	no	—
74	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
75	C₆₇⋊C₈	C₂×C₄²	1	6	yes	—
76	C₂²×C₆₇	—	2	6	no	—
77	C₂²×C₆₇	—	4	3	no	—
78	C₂²×C₆₇	—	4	1	no	—
79	C₂²×C₆₇	#376	1	3	yes	—
80	C₂²×C₆₇	#376	1	1	yes	characteristic
81	C₂²×C₆₇	C₂×#117	1	3	yes	—
82	C₄×C₆₇	#376	1	3	yes	—
83	C₄×C₆₇	#376	1	1	yes	characteristic
84	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
85	C₄×C₆₇	C₂×#117	1	3	yes	—
86	Dic₆₇	—	4	12	no	—
87	C₂×#1747	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	3	no	—
89	C₂×C₆₇	—	2	3	no	—
90	C₂×C₆₇	#1747	1	2	yes	complemented
91	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
92	C₂×#405	—	67	3	no	—
93	#1747	—	67	8	no	—
94	C₂×#338	D₆₇	1	1	yes	characteristic
95	C₂×H(3,ℤ/4)	—	67	3	no	—
96	C₆₇	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
97	C₂²×M₁₆	C₂×D₆₇	1	3	yes	—
98	C₂³×C₈	—	67	1	no	—
99	C₂×#117	—	67	3	no	—
100	#405	—	67	12	no	—
101	#405	—	67	12	no	—
102	C₂×#97	—	67	6	no	—
103	C₂×C₄×C₈	—	67	6	no	—
104	#338	C₂×D₆₇	1	4	yes	—
105	H(3,ℤ/4)	—	67	12	no	—
106	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×M₁₆	—	2	6	no	—
108	C₂×M₁₆	—	2	6	no	—
109	C₂×M₁₆	C₄×D₆₇	1	6	yes	—
110	C₂×M₁₆	C₂²×D₆₇	1	6	yes	—
111	C₂²×C₈	—	67	3	no	—
112	C₂²×C₈	—	67	3	no	—
113	C₂²×C₈	—	67	6	no	—
114	C₂²×C₈	—	67	2	no	—
115	C₂²×C₈	C₄×D₆₇	1	6	yes	—
116	#117	—	67	6	no	—
117	#117	—	67	6	no	—
118	#117	—	134	3	no	—
119	#117	—	134	3	no	—
120	C₂×C₄⋊C₄	—	134	3	no	—
121	C₂×C₂²⋊C₄	—	134	3	no	—
122	C₂×C₄²	—	67	6	no	—
123	#97	—	67	24	no	—
124	C₄×C₈	—	67	24	no	—
125	C₂⁴	#11761629	1	1	yes	characteristic
126	C₂²×C₄	—	134	6	no	—
127	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
128	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
129	C₂²×C₄	#11761627	1	3	yes	—
130	C₂²×C₄	#11761627	1	3	yes	—
131	C₂²×C₄	#11761629	1	1	yes	characteristic
132	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
133	M₁₆	—	2	12	no	—
134	M₁₆	—	2	12	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	67	6	no	—
138	C₂×C₈	—	67	6	no	—
139	C₂×C₈	—	67	1	no	—
140	C₂×C₈	—	67	3	no	—
141	C₂×C₈	—	67	6	no	—
142	C₂×C₈	—	67	6	no	—
143	C₂×C₈	C₂×C₄×D₆₇	1	12	yes	—
144	C₄⋊C₄	—	134	6	no	—
145	C₄⋊C₄	—	134	6	no	—
146	C₂²⋊C₄	—	134	6	no	—
147	C₂²⋊C₄	—	134	6	no	—
148	C₄²	—	67	12	no	—
149	C₄²	—	134	6	no	—
150	C₂³	—	4	2	no	—
151	C₂³	#14524001	1	3	yes	—
152	C₂³	#14524001	1	3	yes	—
153	C₂³	C₂×#11761629	1	1	yes	characteristic
154	C₂×C₄	—	268	12	no	—
155	C₂×C₄	—	134	12	no	—
156	C₂×C₄	C₄²×D₆₇	1	3	yes	—
157	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
158	C₂×C₄	#14523999	1	3	yes	—
159	C₂×C₄	#14523999	1	3	yes	—
160	C₂×C₄	#14523999	1	6	yes	—
161	C₂×C₄	#14524001	1	3	yes	—
162	C₂×C₄	#14524001	1	3	yes	—
163	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
164	C₂×C₄	C₂×#11761627	1	3	yes	—
165	C₈	—	2	12	no	—
166	C₈	—	67	2	no	—
167	C₈	—	67	6	no	—
168	C₂²	—	2	6	no	—
169	C₂²	—	4	3	no	—
170	C₂²	—	4	1	no	—
171	C₂²	#16863241	1	3	yes	—
172	C₂²	#16863241	1	1	yes	characteristic
173	C₂²	C₂×#14524001	1	3	yes	—
174	C₄	—	268	12	no	—
175	C₄	#16863241	1	3	yes	—
176	C₄	#16863241	1	1	yes	characteristic
177	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
178	C₄	C₂×#14523999	1	3	yes	—
179	C₂	—	2	3	no	—
180	C₂	—	2	3	no	—
181	C₂	#22559965	1	2	yes	direct product factor
182	C₂	C₂×#16863241	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#338×C₆₇	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#338	D₆₇
hypocenter	C₆₇	C₂×#1747
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
2. #22559965	C₂	direct (N × H)
7. #16863354	C₄	semidirect
9. C₂×#14523953	C₄	semidirect
10. C₂×C₈×Dic₆₇	C₄	semidirect
25. #14523953	C₂×C₄	semidirect
26. C₈×Dic₆₇	C₂×C₄	semidirect
90. C₂×C₆₇	#1747	semidirect
96. C₆₇	C₂×#1747	semidirect
181. C₂	#22559965	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×#16863354	2
#22559965	2
C₂×#16863153	2
C₂×#338×C₆₇	2
C₂×#1747	67

Subgroups — C2×#22559965 #30853335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559965 #30853335