Subgroups — C₂×#14528381 #17021973

Order 4352.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763207	C₂	1	2	yes	maximal
3	C₂×#11763189	C₂	1	1	yes	maximal, characteristic
4	C₂×#11763160	C₂	1	2	yes	maximal
5	#14528381	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₈	—	2	2	no	—
8	C₂×#10587555	C₂²	1	2	yes	—
9	C₂×#10587555	C₂²	1	2	yes	—
10	#11763214	—	2	2	no	—
11	#11763214	C₂²	1	4	yes	—
12	#11763207	—	2	4	no	—
13	#11763207	C₂²	1	8	yes	—
14	#11763189	—	2	2	no	—
15	#11763189	C₂²	1	4	yes	—
16	#11763160	—	2	4	no	—
17	#11763160	C₂²	1	8	yes	—
18	C₂×#10587489	—	2	2	no	—
19	C₂×#10587488	—	2	4	no	—
20	C₂×#10587487	—	2	4	no	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₈²×C₁₇	C₂²	1	4	yes	—
25	C₂²×Dic₃₄	—	2	2	no	—
26	C₂²×Dic₃₄	—	2	2	no	—
27	C₂²×C₈×C₁₇	D₄	1	2	yes	—
28	C₂²×C₈×C₁₇	D₄	1	2	yes	—
29	C₂²×C₈×C₁₇	D₄	1	2	yes	—
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₇	—	2	2	no	—
32	C₂×C₄⋊Dic₁₇	—	2	2	no	—
33	C₂×C₄⋊Dic₁₇	—	2	2	no	—
34	C₂×C₄⋊Dic₁₇	—	2	2	no	—
35	C₂×Dic₆₈	—	2	8	no	—
36	C₂×Dic₆₈	—	4	8	no	—
37	#10587555	—	2	2	no	—
38	#10587555	—	2	2	no	—
39	#10587555	—	2	2	no	—
40	#10587555	—	2	2	no	—
41	#10587555	—	2	2	no	—
42	#10587555	—	2	2	no	—
43	#10587555	C₂³	1	4	yes	—
44	#10587555	C₂³	1	4	yes	—
45	C₄×C₈×C₁₇	D₄	1	2	yes	—
46	C₄×C₈×C₁₇	D₄	1	2	yes	—
47	C₄×C₈×C₁₇	D₄	1	2	yes	—
48	C₄×C₈×C₁₇	C₂³	1	2	yes	—
49	C₄×C₈×C₁₇	C₂³	1	2	yes	—
50	C₄×C₈×C₁₇	C₂³	1	2	yes	—
51	#10587489	—	2	8	no	—
52	#10587488	—	2	16	no	—
53	#10587487	—	2	16	no	—
54	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₇	—	4	2	no	—
58	C₂²×Dic₁₇	—	4	2	no	—
59	C₂×Dic₃₄	—	2	4	no	—
60	C₂×Dic₃₄	—	2	4	no	—
61	C₂×Dic₃₄	—	4	8	no	—
62	C₂×Dic₃₄	—	4	8	no	—
63	C₂×Dic₃₄	—	4	2	no	—
64	C₂×Dic₃₄	—	4	2	no	—
65	C₂×C₈×C₁₇	—	2	4	no	—
66	C₂×C₈×C₁₇	SD₁₆	1	8	yes	—
67	C₂×C₈×C₁₇	Q₁₆	1	8	yes	complemented
68	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
71	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊Dic₁₇	—	2	4	no	—
76	C₄⋊Dic₁₇	—	2	4	no	—
77	C₄⋊Dic₁₇	—	2	4	no	—
78	C₄⋊Dic₁₇	—	2	4	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	C₄⋊Dic₁₇	—	4	2	no	—
81	C₄⋊Dic₁₇	—	4	2	no	—
82	C₄⋊Dic₁₇	—	4	2	no	—
83	Dic₆₈	—	4	16	no	—
84	C₂×#1707	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×SD₁₆	1	4	yes	—
90	C₂×C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₁₇	#135	1	4	yes	—
92	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×Dic₁₇	—	4	4	no	—
96	C₂×Dic₁₇	—	4	4	no	—
97	C₂×Dic₁₇	—	8	4	no	—
98	C₂×Dic₁₇	—	8	4	no	—
99	Dic₃₄	—	4	8	no	—
100	Dic₃₄	—	4	8	no	—
101	Dic₃₄	—	8	4	no	—
102	Dic₃₄	—	8	4	no	—
103	C₈×C₁₇	—	2	4	no	—
104	C₈×C₁₇	C₂×SD₁₆	1	8	yes	—
105	C₈×C₁₇	C₂×Q₁₆	1	8	yes	complemented
106	C₂×#500	—	17	1	no	—
107	C₂×#498	—	17	2	no	—
108	C₂×#494	—	17	1	no	—
109	C₂×#487	—	17	2	no	—
110	#1707	—	17	8	no	—
111	C₂×C₈²	D₁₇	1	1	yes	characteristic
112	C₂²×C₁₇	#492	1	2	yes	—
113	C₂²×C₁₇	#494	1	2	yes	—
114	C₂²×C₁₇	#495	1	2	yes	—
115	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
116	C₄×C₁₇	#492	1	2	yes	—
117	C₄×C₁₇	#494	1	2	yes	—
118	C₄×C₁₇	#495	1	2	yes	—
119	C₄×C₁₇	C₂²×SD₁₆	1	2	yes	—
120	C₄×C₁₇	C₂²×Q₁₆	1	2	yes	—
121	C₄×C₁₇	C₂×#135	1	2	yes	—
122	Dic₁₇	—	8	4	no	—
123	Dic₁₇	—	8	4	no	—
124	C₂²×Q₁₆	—	34	2	no	—
125	C₂×C₄⋊Q₈	—	17	2	no	—
126	C₂×C₄⋊Q₈	—	17	2	no	—
127	#500	—	17	4	no	—
128	#500	—	34	2	no	—
129	#498	—	17	8	no	—
130	#498	—	34	4	no	—
131	#494	—	17	4	no	—
132	#494	—	34	2	no	—
133	#487	—	17	8	no	—
134	#487	—	34	4	no	—
135	C₂×#107	—	34	2	no	—
136	C₂×#106	—	34	4	no	—
137	C₂×#103	—	34	4	no	—
138	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
141	C₈²	C₂×D₁₇	1	4	yes	—
142	C₂×C₁₇	#1707	1	4	yes	complemented
143	C₂×C₁₇	C₂×#492	1	1	yes	characteristic
144	C₂×C₁₇	C₂×#494	1	1	yes	characteristic
145	C₂×C₁₇	C₂×#495	1	1	yes	characteristic
146	C₂²×Q₈	—	34	2	no	—
147	C₂²×Q₈	—	34	2	no	—
148	C₂×Q₁₆	—	34	8	no	—
149	C₂×Q₁₆	—	68	8	no	—
150	C₂²×C₈	D₆₈	1	2	yes	—
151	C₂²×C₈	D₆₈	1	2	yes	—
152	C₂²×C₈	D₆₈	1	2	yes	—
153	C₄⋊Q₈	—	17	4	no	—
154	C₄⋊Q₈	—	17	4	no	—
155	C₄⋊Q₈	—	34	2	no	—
156	C₄⋊Q₈	—	34	2	no	—
157	C₄⋊Q₈	—	34	2	no	—
158	C₄⋊Q₈	—	34	2	no	—
159	C₄⋊Q₈	—	34	2	no	—
160	C₄⋊Q₈	—	34	2	no	—
161	C₂×C₄⋊C₄	—	34	2	no	—
162	C₂×C₄⋊C₄	—	34	2	no	—
163	C₂×C₄⋊C₄	—	34	2	no	—
164	C₂×C₄⋊C₄	—	34	2	no	—
165	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
166	#107	—	34	8	no	—
167	#106	—	34	16	no	—
168	#103	—	34	16	no	—
169	C₄×C₈	D₆₈	1	2	yes	—
170	C₄×C₈	D₆₈	1	2	yes	—
171	C₄×C₈	D₆₈	1	2	yes	—
172	C₄×C₈	C₂²×D₁₇	1	2	yes	—
173	C₄×C₈	C₂²×D₁₇	1	2	yes	—
174	C₄×C₈	C₂²×D₁₇	1	2	yes	—
175	C₁₇	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
176	C₂×Q₈	—	34	4	no	—
177	C₂×Q₈	—	34	4	no	—
178	C₂×Q₈	—	68	2	no	—
179	C₂×Q₈	—	68	2	no	—
180	C₂×Q₈	—	68	8	no	—
181	C₂×Q₈	—	68	8	no	—
182	C₂²×C₄	—	68	2	no	—
183	C₂²×C₄	—	68	2	no	—
184	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
187	Q₁₆	—	68	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	#63219	1	8	yes	—
190	C₂×C₈	Dic₆₈	1	8	yes	complemented
191	C₂×C₈	C₂×D₆₈	1	4	yes	—
192	C₂×C₈	C₂×D₆₈	1	4	yes	—
193	C₂×C₈	C₂×D₆₈	1	4	yes	—
194	C₄⋊C₄	—	34	4	no	—
195	C₄⋊C₄	—	34	4	no	—
196	C₄⋊C₄	—	34	4	no	—
197	C₄⋊C₄	—	34	4	no	—
198	C₄⋊C₄	—	68	2	no	—
199	C₄⋊C₄	—	68	2	no	—
200	C₄⋊C₄	—	68	2	no	—
201	C₄⋊C₄	—	68	2	no	—
202	C₄²	C₂×D₆₈	1	1	yes	characteristic
203	C₄²	C₂×D₆₈	1	1	yes	characteristic
204	C₄²	C₂×D₆₈	1	1	yes	characteristic
205	C₄²	C₂³×D₁₇	1	1	yes	characteristic
206	C₂³	#10587560	1	1	yes	characteristic
207	Q₈	—	68	8	no	—
208	Q₈	—	68	8	no	—
209	Q₈	—	136	4	no	—
210	Q₈	—	136	4	no	—
211	C₂×C₄	—	68	4	no	—
212	C₂×C₄	—	68	4	no	—
213	C₂×C₄	—	136	4	no	—
214	C₂×C₄	—	136	4	no	—
215	C₂×C₄	#10587560	1	1	yes	characteristic
216	C₂×C₄	#10587560	1	1	yes	characteristic
217	C₂×C₄	#10587560	1	1	yes	characteristic
218	C₂×C₄	C₂×#63219	1	4	yes	—
219	C₂×C₄	#10587590	1	4	yes	—
220	C₂×C₄	C₂×Dic₆₈	1	4	yes	—
221	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×#63219	1	8	yes	—
226	C₈	C₂×Dic₆₈	1	8	yes	complemented
227	C₂²	#11763182	1	2	yes	—
228	C₂²	#11763189	1	2	yes	—
229	C₂²	#11763193	1	2	yes	—
230	C₂²	C₂×#10587560	1	1	yes	characteristic
231	C₄	—	136	4	no	—
232	C₄	—	136	4	no	—
233	C₄	#11763182	1	2	yes	—
234	C₄	#11763189	1	2	yes	—
235	C₄	#11763193	1	2	yes	—
236	C₄	C₂²×#63219	1	2	yes	—
237	C₄	C₂²×Dic₆₈	1	2	yes	—
238	C₄	C₂×#10587590	1	2	yes	—
239	C₂	#14528381	1	4	yes	direct product factor
240	C₂	C₂×#11763182	1	1	yes	characteristic
241	C₂	C₂×#11763189	1	1	yes	characteristic
242	C₂	C₂×#11763193	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587560
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₈²×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	#127
hypercenter	C₂×C₈²	D₁₇
hypocenter	C₁₇	C₂×#1707
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
5. #14528381	C₂	direct (N × H)
67. C₂×C₈×C₁₇	Q₁₆	semidirect
105. C₈×C₁₇	C₂×Q₁₆	semidirect
142. C₂×C₁₇	#1707	semidirect
175. C₁₇	C₂×#1707	semidirect
190. C₂×C₈	Dic₆₈	semidirect
226. C₈	C₂×Dic₆₈	semidirect
239. C₂	#14528381	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×#11763214	2
C₂×#11763207	2
C₂×#11763189	2
C₂×#11763160	2
#14528381	2
C₂×C₈²×C₁₇	2
C₂×#1707	17

Subgroups — C2×#14528381 #17021973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528381 #17021973