Subgroups — C₂×#14530380 #17186958

Order 4352.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762764	C₂	1	1	yes	maximal, characteristic
3	#14530380	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762584	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₁₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63223	C₄	1	1	yes	characteristic
8	C₂×#10587612	C₄	1	1	yes	characteristic
9	C₂×#10587483	C₄	1	8	yes	complemented
10	C₂×#10587483	C₂²	1	4	yes	—
11	C₂²×#63214	C₂²	1	2	yes	—
12	#11762764	C₄	1	4	yes	—
13	#11762764	C₂²	1	4	yes	—
14	C₂×#10587486	C₄	1	2	yes	—
15	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
16	#11762584	C₂²	1	8	yes	complemented
17	C₁₆×Dic₁₇	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₁₇	C₄	1	8	yes	complemented
20	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₁₇	C₂²	1	4	yes	—
23	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
30	C₂×C₂²⋊Dic₁₇	C₈	1	2	yes	—
31	#10587612	C₈	1	4	yes	—
32	#10587612	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₁₇	C₈	1	2	yes	—
34	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×#63223	C₈	1	8	yes	—
36	C₂×#63223	C₂×C₄	1	4	yes	—
37	C₂²×#3622	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₁₇	C₂×C₄	1	16	yes	complemented
41	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
42	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
43	#10587486	C₂×C₄	1	8	yes	—
44	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
45	C₈×Dic₁₇	C₂³	1	4	yes	—
46	#10587483	C₂×C₄	1	16	yes	complemented
47	C₂×#63214	C₂×C₄	1	8	yes	—
48	C₂×#63214	C₂³	1	4	yes	—
49	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₄²	1	4	yes	—
51	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
56	C₂²×Dic₁₇	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₁₇	C₄²	1	4	yes	—
58	C₂×C₈×C₁₇	C₄²	1	8	yes	—
59	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
66	C₂²⋊Dic₁₇	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₁₇	C₂×C₈	1	8	yes	—
68	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
69	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
70	#63223	C₂×C₈	1	16	yes	—
71	C₂×#3622	C₂×C₈	1	8	yes	—
72	C₂×#3622	C₂²×C₄	1	4	yes	—
73	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
74	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
75	#63214	C₂²×C₄	1	8	yes	—
76	C₂×#2108	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	—	2	4	no	—
78	C₂³×C₁₇	C₄×C₈	1	4	yes	—
79	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
80	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
83	C₂×C₄×C₁₇	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
87	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₁₇	—	2	8	no	—
91	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
92	C₈×C₁₇	C₂×C₄²	1	2	yes	—
93	C₈×C₁₇	C₂×C₄²	1	4	yes	—
94	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
96	#3622	C₂²×C₈	1	8	yes	—
97	C₂²×M₃₂	—	17	1	no	—
98	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
99	C₂×#405	—	17	1	no	—
100	#2108	—	17	8	no	—
101	C₂×#346	—	17	2	no	—
102	C₂×C₄×C₁₆	—	17	2	no	—
103	C₂²×C₁₇	—	2	8	no	—
104	C₂²×C₁₇	—	2	4	no	—
105	C₂²×C₁₇	C₂×C₄×C₈	1	4	yes	—
106	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
107	C₂²×C₁₇	#504	1	4	yes	—
108	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
109	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
110	C₄×C₁₇	C₂×C₄×C₈	1	4	yes	—
111	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
113	Dic₁₇	—	2	8	no	—
114	C₂²×M₁₆	—	17	1	no	—
115	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
116	C₂×#117	—	17	1	no	—
117	C₂×M₃₂	—	17	8	no	—
118	C₂×M₃₂	—	17	4	no	—
119	C₂²×C₁₆	—	17	2	no	—
120	C₂²×C₁₆	Dic₁₇	1	8	yes	complemented
121	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₁₇	1	4	yes	—
123	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
124	#405	—	17	4	no	—
125	#405	—	17	4	no	—
126	C₂×#97	—	17	2	no	—
127	C₂×C₄×C₈	—	17	2	no	—
128	#346	—	17	8	no	—
129	C₄×C₁₆	—	17	8	no	—
130	C₂×C₁₇	—	2	4	no	—
131	C₂×C₁₇	#2108	1	4	yes	complemented
132	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
133	C₂×C₁₇	C₂×#504	1	2	yes	—
134	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
135	C₂×M₁₆	—	17	4	no	—
136	C₂×M₁₆	—	17	8	no	—
137	C₂²×C₈	—	17	2	no	—
138	C₂²×C₈	C₄×D₁₇	1	4	yes	—
139	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₁₇	1	2	yes	—
141	C₂²×C₈	C₂×Dic₁₇	1	4	yes	—
142	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
144	#117	—	17	4	no	—
145	#117	—	17	4	no	—
146	C₂×C₄⋊C₄	—	17	2	no	—
147	C₂×C₂²⋊C₄	—	17	2	no	—
148	C₂×C₄²	—	17	2	no	—
149	M₃₂	—	17	16	no	—
150	C₂×C₁₆	—	17	8	no	—
151	C₂×C₁₆	—	17	4	no	—
152	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
153	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
154	C₂×C₁₆	C₂×Dic₁₇	1	16	yes	complemented
155	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
156	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
157	#97	—	17	8	no	—
158	C₄×C₈	—	17	4	no	—
159	C₄×C₈	—	17	4	no	—
160	C₁₇	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
161	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
162	C₂²×C₄	—	17	4	no	—
163	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
164	C₂²×C₄	C₈×D₁₇	1	2	yes	—
165	C₂²×C₄	C₈×D₁₇	1	4	yes	—
166	C₂²×C₄	C₄×Dic₁₇	1	4	yes	—
167	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₁₇	1	2	yes	—
169	M₁₆	—	17	16	no	—
170	C₂×C₈	—	17	4	no	—
171	C₂×C₈	—	17	8	no	—
172	C₂×C₈	C₄×Dic₁₇	1	4	yes	—
173	C₂×C₈	C₄×Dic₁₇	1	8	yes	—
174	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
176	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
178	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
179	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
180	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
181	C₄⋊C₄	—	17	8	no	—
182	C₂²⋊C₄	—	17	8	no	—
183	C₄²	—	17	4	no	—
184	C₄²	—	17	4	no	—
185	C₁₆	—	17	8	no	—
186	C₁₆	C₂²×Dic₁₇	1	4	yes	—
187	C₁₆	C₂²×Dic₁₇	1	4	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₈×Dic₁₇	1	4	yes	—
190	C₂³	C₂×C₈×D₁₇	1	2	yes	—
191	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
192	C₂×C₄	—	17	8	no	—
193	C₂×C₄	—	34	8	no	—
194	C₂×C₄	C₈×Dic₁₇	1	8	yes	—
195	C₂×C₄	C₈×Dic₁₇	1	4	yes	—
196	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
198	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
199	C₂×C₄	C₂×C₈×D₁₇	1	4	yes	—
200	C₂×C₄	C₂×C₄×Dic₁₇	1	4	yes	—
201	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
202	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
203	C₈	—	17	8	no	—
204	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
205	C₈	C₂×C₄×Dic₁₇	1	4	yes	—
206	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
207	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₈×Dic₁₇	1	4	yes	—
211	C₂²	C₂×C₈×Dic₁₇	1	2	yes	—
212	C₂²	#11763225	1	4	yes	—
213	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
214	C₄	—	34	8	no	—
215	C₄	C₂×C₈×Dic₁₇	1	2	yes	—
216	C₄	C₂×C₈×Dic₁₇	1	4	yes	—
217	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
218	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#14530380	1	4	yes	direct product factor
221	C₂	C₂²×C₈×Dic₁₇	1	1	yes	characteristic
222	C₂	C₂×#11763225	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₂×#2108
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14530380	C₂	direct (N × H)
4. C₂×#11762584	C₂	semidirect
5. C₂×C₁₆×Dic₁₇	C₂	semidirect
9. C₂×#10587483	C₄	semidirect
16. #11762584	C₂²	semidirect
17. C₁₆×Dic₁₇	C₂²	semidirect
19. C₂²×C₁₆×C₁₇	C₄	semidirect
40. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
46. #10587483	C₂×C₄	semidirect
120. C₂²×C₁₆	Dic₁₇	semidirect
131. C₂×C₁₇	#2108	semidirect
154. C₂×C₁₆	C₂×Dic₁₇	semidirect
160. C₁₇	C₂×#2108	semidirect
220. C₂	#14530380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10587483	2
C₂×#11762764	2
#14530380	2
C₂×#11762584	2
C₂×C₁₆×Dic₁₇	2
C₂³×C₁₆×C₁₇	2
C₂×#2108	17

Subgroups — C2×#14530380 #17186958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530380 #17186958