Subgroups — C₂×#14530689 #17187808

Order 4352.

542 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×C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	#14530689	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762627	C₂	1	2	yes	maximal
4	C₂×#11762584	C₂	1	2	yes	maximal
5	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₃₄	C₄	1	1	yes	characteristic
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₈×Dic₃₄	C₄	1	4	yes	—
11	C₈×Dic₃₄	C₂²	1	4	yes	—
12	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
13	C₂×#10587485	C₄	1	2	yes	—
14	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
15	#11762627	C₂²	1	8	yes	—
16	#11762584	C₂²	1	8	yes	—
17	C₄×#63214	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₂²×Dic₃₄	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
26	C₂×#63225	C₈	1	2	yes	—
27	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
28	C₂²×#3622	C₂×C₄	1	2	yes	—
29	C₄×Dic₃₄	C₈	1	4	yes	—
30	C₄×Dic₃₄	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₁₇	—	2	4	no	—
32	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	C₂³	1	2	yes	—
35	#10587485	C₂×C₄	1	8	yes	—
36	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
37	C₈×Dic₁₇	C₂³	1	4	yes	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	Q₈	1	4	yes	complemented
40	C₂×#63214	C₂³	1	2	yes	—
41	C₂×#63214	C₂³	1	2	yes	—
42	C₄⋊#3622	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
47	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
48	C₂×Dic₃₄	—	2	4	no	—
49	C₂×Dic₃₄	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₇	#41	1	2	yes	—
55	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₄²×C₁₇	C₂×C₈	1	2	yes	—
57	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
58	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
59	#63225	C₂×C₈	1	8	yes	—
60	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
61	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
62	C₂×#3622	—	2	4	no	—
63	C₂×#3622	C₂²×C₄	1	4	yes	—
64	C₁₆×C₁₇	—	2	4	no	—
65	#63214	—	2	2	no	—
66	#63214	C₂×Q₈	1	4	yes	complemented
67	C₂×#2183	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
71	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₇	#131	1	2	yes	—
76	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	4	no	—
78	C₂×Dic₁₇	—	2	4	no	—
79	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
80	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
81	Dic₃₄	—	2	8	no	—
82	C₈×C₁₇	—	2	2	no	—
83	C₈×C₁₇	C₄×Q₈	1	2	yes	—
84	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
86	#3622	—	2	4	no	—
87	C₂×C₈×Q₈	—	17	1	no	—
88	#2183	—	17	8	no	—
89	C₂×C₄⋊C₁₆	—	17	2	no	—
90	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic
91	C₂×#346	—	17	2	no	—
92	C₂×C₄×C₁₆	—	17	1	no	—
93	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
94	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
95	C₂²×C₁₇	#504	1	2	yes	—
96	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₁₇	—	2	2	no	—
98	C₄×C₁₇	C₈×Q₈	1	2	yes	—
99	C₄×C₁₇	C₂×M₃₂	1	4	yes	—
100	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
102	Dic₁₇	—	2	4	no	—
103	Dic₁₇	—	2	4	no	—
104	C₂×C₄×Q₈	—	17	1	no	—
105	C₂²×C₁₆	—	17	1	no	—
106	C₂²×C₁₆	—	17	1	no	—
107	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
108	C₈×Q₈	—	17	4	no	—
109	C₈×Q₈	—	17	4	no	—
110	C₂×C₄⋊C₈	—	17	2	no	—
111	C₂×C₄⋊C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	2	no	—
113	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
114	C₄⋊C₁₆	—	17	8	no	—
115	C₄⋊C₁₆	C₂×D₁₇	1	4	yes	—
116	#346	—	17	8	no	—
117	C₄×C₁₆	—	17	4	no	—
118	C₂×C₁₇	#2183	1	4	yes	complemented
119	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
120	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
121	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
122	C₂²×Q₈	—	17	1	no	—
123	C₂²×C₈	—	17	2	no	—
124	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	C₄×Q₈	—	17	4	no	—
127	C₄×Q₈	—	17	4	no	—
128	C₂×C₄⋊C₄	—	17	2	no	—
129	C₂×C₄⋊C₄	—	17	1	no	—
130	C₂×C₄²	—	17	2	no	—
131	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	—	17	4	no	—
134	C₂×C₁₆	—	17	2	no	—
135	C₂×C₁₆	—	17	2	no	—
136	C₂×C₁₆	—	34	2	no	—
137	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
138	C₄⋊C₈	—	17	8	no	—
139	C₄⋊C₈	—	17	4	no	—
140	C₄×C₈	—	17	4	no	—
141	C₄×C₈	—	17	4	no	—
142	C₄×C₈	C₄×D₁₇	1	2	yes	—
143	C₄×C₈	C₂²×D₁₇	1	2	yes	—
144	C₁₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	17	4	no	—
146	C₂×Q₈	—	34	4	no	—
147	C₂²×C₄	—	17	2	no	—
148	C₂²×C₄	—	17	2	no	—
149	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
150	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	17	4	no	—
154	C₂×C₈	—	34	4	no	—
155	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
157	C₂×C₈	#63254	1	2	yes	—
158	C₂×C₈	Q₈×D₁₇	1	2	yes	—
159	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
160	C₄⋊C₄	—	17	8	no	—
161	C₄⋊C₄	—	17	4	no	—
162	C₄²	—	17	4	no	—
163	C₄²	—	17	4	no	—
164	C₄²	C₈×D₁₇	1	2	yes	—
165	C₄²	C₂×C₄×D₁₇	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	17	4	no	—
168	C₁₆	—	34	2	no	—
169	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
170	Q₈	—	34	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	17	4	no	—
173	C₂×C₄	—	17	4	no	—
174	C₂×C₄	—	34	4	no	—
175	C₂×C₄	—	34	4	no	—
176	C₂×C₄	#10587469	1	4	yes	—
177	C₂×C₄	#10587573	1	2	yes	—
178	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
180	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
181	C₂×C₄	#10587593	1	2	yes	—
182	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	34	4	no	—
185	C₈	#10587573	1	2	yes	—
186	C₈	C₂×#63254	1	1	yes	characteristic
187	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
188	C₂²	#11762933	1	2	yes	—
189	C₂²	C₂×#10587469	1	2	yes	—
190	C₂²	#11763227	1	2	yes	—
191	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	34	4	no	—
194	C₄	—	34	4	no	—
195	C₄	#11762933	1	2	yes	—
196	C₄	C₂×#10587469	1	4	yes	—
197	C₄	C₂×#10587573	1	1	yes	characteristic
198	C₄	C₂×#10587593	1	1	yes	characteristic
199	C₂	#14530689	1	4	yes	direct product factor
200	C₂	C₂×#11762933	1	1	yes	characteristic
201	C₂	C₂²×#10587469	1	1	yes	characteristic
202	C₂	C₂×#11763227	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	C₂×#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #14530689	C₂	direct (N × H)
39. C₂×#63214	Q₈	semidirect
66. #63214	C₂×Q₈	semidirect
118. C₂×C₁₇	#2183	semidirect
144. C₁₇	C₂×#2183	semidirect
199. C₂	#14530689	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×C₈×Dic₃₄	2
#14530689	2
C₂×#11762627	2
C₂×#11762584	2
C₂×C₄×#63214	2
C₂×C₁₇×C₄⋊C₁₆	2
C₂×#2183	17

Subgroups — C2×#14530689 #17187808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530689 #17187808