Subgroups — C₂×#11762566 #14528485

Order 2176.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587612	C₂	1	2	yes	maximal
2	#11762566	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₇	—	2	2	no	—
6	C₂×M₁₆×C₁₇	—	2	2	no	—
7	C₂×M₁₆×C₁₇	C₄	1	2	yes	complemented
8	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₇	C₄	1	2	yes	complemented
10	C₂×C₂²⋊Dic₁₇	—	2	2	no	—
11	#10587612	—	2	2	no	—
12	#10587612	—	2	2	no	—
13	#10587612	C₄	1	4	yes	complemented
14	#10587612	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₁₇	—	2	2	no	—
16	C₂×C₄×Dic₁₇	C₄	1	4	yes	complemented
17	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₇	—	2	4	no	—
29	M₁₆×C₁₇	—	2	4	no	—
30	M₁₆×C₁₇	—	2	4	no	—
31	C₂×C₈×C₁₇	—	2	2	no	—
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
34	C₂²⋊Dic₁₇	—	2	4	no	—
35	C₂²⋊Dic₁₇	—	2	4	no	—
36	C₄⋊Dic₁₇	—	2	4	no	—
37	C₄⋊Dic₁₇	—	2	4	no	—
38	C₄×Dic₁₇	—	2	4	no	—
39	C₄×Dic₁₇	C₂×C₄	1	8	yes	complemented
40	C₂³×C₁₇	—	4	2	no	—
41	C₂³×C₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄²	1	2	yes	—
47	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×Dic₁₇	—	2	8	no	—
65	C₂×Dic₁₇	—	4	8	no	—
66	C₈×C₁₇	—	2	4	no	—
67	C₂×H(3,ℤ/4)	—	17	1	no	maximal, Sylow
68	C₂²×C₁₇	—	2	4	no	—
69	C₂²×C₁₇	—	2	2	no	—
70	C₂²×C₁₇	—	4	2	no	—
71	C₂²×C₁₇	—	4	1	no	—
72	C₂²×C₁₇	—	4	1	no	—
73	C₂²×C₁₇	#95	1	1	yes	characteristic
74	C₂²×C₁₇	#95	1	2	yes	—
75	C₂²×C₁₇	#95	1	1	yes	characteristic
76	C₂²×C₁₇	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₁₇	#95	1	1	yes	characteristic
79	C₄×C₁₇	#95	1	2	yes	—
80	C₄×C₁₇	#95	1	1	yes	characteristic
81	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
82	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
84	Dic₁₇	—	4	8	no	—
85	C₂²×M₁₆	D₁₇	1	1	yes	characteristic
86	C₂×#117	—	17	2	no	—
87	H(3,ℤ/4)	—	17	4	no	—
88	C₂×C₁₇	—	2	2	no	—
89	C₂×C₁₇	—	2	2	no	—
90	C₂×C₁₇	—	2	1	no	—
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	H(3,ℤ/4)	1	2	yes	complemented
93	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
94	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	Dic₁₇	1	2	yes	complemented
98	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
99	C₂²×C₈	Dic₁₇	1	2	yes	complemented
100	#117	—	17	4	no	—
101	#117	—	17	4	no	—
102	#117	—	34	2	no	—
103	#117	—	34	2	no	—
104	C₂×C₄⋊C₄	—	34	2	no	—
105	C₂×C₂²⋊C₄	—	34	2	no	—
106	C₂×C₄²	—	17	4	no	—
107	C₁₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	D₆₈	1	1	yes	characteristic
109	C₂²×C₄	—	34	4	no	—
110	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
111	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
112	C₂²×C₄	C₄×D₁₇	1	2	yes	—
113	C₂²×C₄	C₄×D₁₇	1	2	yes	—
114	C₂²×C₄	D₆₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
118	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	4	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×Dic₁₇	1	4	yes	complemented
125	C₄⋊C₄	—	34	4	no	—
126	C₄⋊C₄	—	34	4	no	—
127	C₂²⋊C₄	—	34	4	no	—
128	C₂²⋊C₄	—	34	4	no	—
129	C₄²	—	17	8	no	—
130	C₄²	—	34	4	no	—
131	C₂³	—	4	2	no	—
132	C₂³	C₄⋊Dic₁₇	1	1	yes	characteristic
133	C₂³	C₄⋊Dic₁₇	1	1	yes	characteristic
134	C₂³	#63227	1	2	yes	—
135	C₂³	#63227	1	2	yes	—
136	C₂³	C₂×D₆₈	1	1	yes	characteristic
137	C₂×C₄	—	34	8	no	—
138	C₂×C₄	—	68	8	no	—
139	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
140	C₂×C₄	C₄×Dic₁₇	1	2	yes	—
141	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
142	C₂×C₄	#63225	1	2	yes	—
143	C₂×C₄	#63225	1	2	yes	—
144	C₂×C₄	#63225	1	2	yes	—
145	C₂×C₄	#63225	1	2	yes	—
146	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
147	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
148	C₂×C₄	#63227	1	2	yes	—
149	C₂×C₄	#63227	1	2	yes	—
150	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
151	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
152	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
153	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
155	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₂²	—	2	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	#10587502	1	1	yes	characteristic
164	C₂²	#10587502	1	2	yes	—
165	C₂²	#10587502	1	1	yes	characteristic
166	C₂²	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
167	C₂²	C₂×#63227	1	2	yes	—
168	C₄	—	68	8	no	—
169	C₄	#10587502	1	1	yes	characteristic
170	C₄	#10587502	1	2	yes	—
171	C₄	#10587502	1	1	yes	characteristic
172	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
173	C₄	C₂×#63225	1	2	yes	—
174	C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#11762566	1	2	yes	direct product factor
180	C₂	C₂×#10587502	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂²×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	#95
prime radical	C₂⁴×C₁₇	D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂²×M₁₆	D₁₇
hypocenter	C₁₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11762566	C₂	direct (N × H)
7. C₂×M₁₆×C₁₇	C₄	semidirect
9. C₂²×C₈×C₁₇	C₄	semidirect
13. #10587612	C₄	semidirect
16. C₂×C₄×Dic₁₇	C₄	semidirect
33. C₂×C₈×C₁₇	C₂×C₄	semidirect
39. C₄×Dic₁₇	C₂×C₄	semidirect
92. C₂×C₁₇	H(3,ℤ/4)	semidirect
97. C₂×M₁₆	Dic₁₇	semidirect
99. C₂²×C₈	Dic₁₇	semidirect
107. C₁₇	C₂×H(3,ℤ/4)	semidirect
124. C₂×C₈	C₂×Dic₁₇	semidirect
179. C₂	#11762566	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×#10587612	2
#11762566	2
C₂²×M₁₆×C₁₇	2
C₂×H(3,ℤ/4)	17

Subgroups — C2×#11762566 #14528485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762566 #14528485