Subgroups — C₂×#11762563 #14528474

Order 2176.

528 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	C₂³×#3622	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762563	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₇	C₄	1	2	yes	—
7	C₂²×C₈×C₁₇	—	2	2	no	—
8	C₂²×C₈×C₁₇	—	2	2	no	—
9	C₂²×C₈×C₁₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₇	—	2	2	no	—
12	C₂×C₄×Dic₁₇	—	2	2	no	—
13	C₂×C₄×Dic₁₇	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
15	C₂²×#3622	—	2	2	no	—
16	C₂²×#3622	—	2	2	no	—
17	C₂²×#3622	C₄	1	4	yes	complemented
18	C₂²×#3622	C₂²	1	2	yes	—
19	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₇	—	2	4	no	—
29	C₂²×Dic₁₇	—	2	4	no	—
30	C₂²×Dic₁₇	C₈	1	8	yes	complemented
31	C₂²×Dic₁₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₁₇	—	2	8	no	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	complemented
36	C₄×Dic₁₇	—	2	4	no	—
37	C₄×Dic₁₇	—	2	2	no	—
38	C₄×Dic₁₇	—	2	2	no	—
39	C₂×#3622	—	2	8	no	—
40	C₂×#3622	—	2	2	no	—
41	C₂×#3622	—	2	2	no	—
42	C₂×#3622	C₂×C₄	1	4	yes	complemented
43	C₂³×C₁₇	C₄²	1	2	yes	—
44	C₂³×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
46	C₂³×C₁₇	C₂×C₈	1	4	yes	—
47	C₂³×C₁₇	M₁₆	1	4	yes	—
48	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₄²	1	2	yes	—
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×Dic₁₇	—	2	16	no	—
63	C₂×Dic₁₇	—	2	4	no	—
64	C₂×Dic₁₇	—	2	4	no	—
65	C₂×Dic₁₇	C₂×C₈	1	8	yes	complemented
66	C₈×C₁₇	—	2	4	no	—
67	#3622	—	2	4	no	—
68	C₂×#336	—	17	1	no	maximal, Sylow
69	C₂²×C₁₇	#95	1	4	yes	—
70	C₂²×C₁₇	C₄×C₈	1	4	yes	—
71	C₂²×C₁₇	#97	1	4	yes	—
72	C₂²×C₁₇	C₂²⋊C₈	1	8	yes	—
73	C₂²×C₁₇	C₄⋊C₈	1	8	yes	—
74	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
75	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₁₇	C₂²×C₈	1	2	yes	—
78	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
79	C₄×C₁₇	#95	1	4	yes	—
80	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	Dic₁₇	—	2	8	no	—
85	C₂³×C₈	—	17	1	no	—
86	C₂³×C₈	D₁₇	1	1	yes	characteristic
87	C₂²×C₄²	—	17	1	no	—
88	#336	—	17	4	no	—
89	C₂×C₁₇	#336	1	8	yes	complemented
90	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
91	C₂×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂×C₁₇	C₂×#97	1	1	yes	characteristic
93	C₂×C₁₇	C₂×C₂²⋊C₈	1	2	yes	—
94	C₂×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
95	C₂³×C₄	—	17	2	no	—
96	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	17	4	no	—
100	C₂²×C₈	—	17	2	no	—
101	C₂²×C₈	—	34	2	no	—
102	C₂²×C₈	—	34	2	no	—
103	C₂²×C₈	Dic₁₇	1	4	yes	complemented
104	C₂²×C₈	C₂×D₁₇	1	2	yes	—
105	C₂×C₄²	—	17	2	no	—
106	C₂×C₄²	—	17	2	no	—
107	C₂×C₄²	—	34	2	no	—
108	C₂×C₄²	—	34	2	no	—
109	C₁₇	C₂×#336	1	1	yes	characteristic, complemented, Sylow
110	C₂⁴	C₄×D₁₇	1	1	yes	characteristic
111	C₂²×C₄	—	17	4	no	—
112	C₂²×C₄	—	17	8	no	—
113	C₂²×C₄	—	34	4	no	—
114	C₂²×C₄	—	34	4	no	—
115	C₂²×C₄	Dic₃₄	1	2	yes	—
116	C₂²×C₄	C₄×D₁₇	1	2	yes	—
117	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
118	C₂²×C₄	D₆₈	1	2	yes	—
119	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
120	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
121	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
123	C₂×C₈	—	2	8	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	17	4	no	—
127	C₂×C₈	—	34	2	no	—
128	C₂×C₈	—	34	2	no	—
129	C₂×C₈	—	34	8	no	—
130	C₂×C₈	C₂×Dic₁₇	1	4	yes	complemented
131	C₄²	—	34	2	no	—
132	C₄²	—	34	2	no	—
133	C₄²	—	34	4	no	—
134	C₂³	C₈×D₁₇	1	4	yes	—
135	C₂³	#63218	1	4	yes	—
136	C₂³	C₄×Dic₁₇	1	2	yes	—
137	C₂³	#63225	1	2	yes	—
138	C₂³	#63227	1	2	yes	—
139	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
140	C₂×C₄	—	17	8	no	—
141	C₂×C₄	—	34	16	no	—
142	C₂×C₄	—	34	4	no	—
143	C₂×C₄	—	34	4	no	—
144	C₂×C₄	C₄×Dic₁₇	1	2	yes	—
145	C₂×C₄	#63225	1	2	yes	—
146	C₂×C₄	#63225	1	4	yes	—
147	C₂×C₄	C₄⋊Dic₁₇	1	4	yes	—
148	C₂×C₄	#63227	1	2	yes	—
149	C₂×C₄	#63227	1	4	yes	—
150	C₂×C₄	C₂²⋊Dic₁₇	1	4	yes	—
151	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
153	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	34	4	no	—
159	C₂²	C₈×Dic₁₇	1	4	yes	—
160	C₂²	#10587485	1	8	yes	—
161	C₂²	#10587486	1	4	yes	—
162	C₂²	#10587491	1	8	yes	—
163	C₂²	#10587502	1	4	yes	—
164	C₂²	C₂×C₈×D₁₇	1	2	yes	—
165	C₂²	C₂×#63218	1	2	yes	—
166	C₂²	C₂×C₄×Dic₁₇	1	1	yes	characteristic
167	C₂²	C₂×#63225	1	1	yes	characteristic
168	C₂²	C₂×#63227	1	1	yes	characteristic
169	C₄	—	34	8	no	—
170	C₄	#10587502	1	4	yes	—
171	C₄	C₂×#63225	1	1	yes	characteristic
172	C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
173	C₄	C₂×#63227	1	1	yes	characteristic
174	C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
175	C₂	#11762563	1	8	yes	direct product factor
176	C₂	C₂×#10587502	1	1	yes	characteristic
177	C₂	C₂×C₈×Dic₁₇	1	1	yes	characteristic
178	C₂	C₂×#10587486	1	1	yes	characteristic
179	C₂	C₂×#10587491	1	2	yes	—
180	C₂	C₂×#10587485	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂³×C₈×C₁₇	C₂
socle, prime radical	C₂⁴×C₁₇	C₂×C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂³×C₈	D₁₇
hypocenter	C₁₇	C₂×#336
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #11762563	C₂	direct (N × H)
9. C₂²×C₈×C₁₇	C₄	semidirect
17. C₂²×#3622	C₄	semidirect
30. C₂²×Dic₁₇	C₈	semidirect
35. C₂×C₈×C₁₇	C₂×C₄	semidirect
42. C₂×#3622	C₂×C₄	semidirect
65. C₂×Dic₁₇	C₂×C₈	semidirect
89. C₂×C₁₇	#336	semidirect
103. C₂²×C₈	Dic₁₇	semidirect
109. C₁₇	C₂×#336	semidirect
130. C₂×C₈	C₂×Dic₁₇	semidirect
175. C₂	#11762563	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#3622	2
C₂²×C₄×Dic₁₇	2
#11762563	2
C₂³×C₈×C₁₇	2
C₂×#336	17

Subgroups — C2×#11762563 #14528474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762563 #14528474