Subgroups — C₂×C₁₇×#3391 #16919131

Order 4352.

2582 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#550	C₂	1	2	yes	maximal, complemented
2	C₁₇×#3391	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₇×#493	C₂	1	3	yes	maximal, complemented
4	C₂×C₁₇×#459	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₁₇	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₁₇	—	2	2	no	—
8	C₂²×D₈×C₁₇	C₂²	1	3	yes	—
9	C₂²×D₈×C₁₇	C₂²	1	6	yes	—
10	C₁₇×#550	—	2	8	no	—
11	C₁₇×#550	C₂²	1	16	yes	complemented
12	C₂×C₁₇×#127	C₂²	1	6	yes	—
13	C₂×C₁₇×#121	—	2	6	no	—
14	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₇	C₂²	1	6	yes	—
16	C₁₇×#493	C₂²	1	24	yes	complemented
17	C₁₇×#459	C₂²	1	48	yes	complemented
18	C₈×Q₈×C₁₇	C₂²	1	8	yes	complemented
19	C₄×D₈×C₁₇	C₂²	1	24	yes	complemented
20	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×C₄⋊C₈	C₂²	1	3	yes	—
22	C₂×C₁₇×#102	C₂²	1	6	yes	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
24	C₂×#41×C₁₇	—	2	16	no	—
25	C₂×#41×C₁₇	—	4	16	no	—
26	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₇	—	2	6	no	—
28	C₂²×D₄×C₁₇	—	2	6	no	—
29	C₂²×D₄×C₁₇	C₂³	1	6	yes	—
30	C₂³×C₄×C₁₇	—	2	6	no	—
31	C₂×D₈×C₁₇	—	2	12	no	—
32	C₂×D₈×C₁₇	—	4	12	no	—
33	C₂×D₈×C₁₇	C₂³	1	12	yes	—
34	C₂×D₈×C₁₇	C₂³	1	24	yes	—
35	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
36	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
37	C₁₇×#127	—	2	12	no	—
38	C₁₇×#127	C₂³	1	24	yes	—
39	C₁₇×#121	—	2	48	no	—
40	C₄×Q₈×C₁₇	D₄	1	4	yes	—
41	C₄×Q₈×C₁₇	C₂³	1	4	yes	—
42	C₄×D₄×C₁₇	—	2	12	no	—
43	C₄×D₄×C₁₇	C₂³	1	24	yes	—
44	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₇	—	2	6	no	—
47	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
48	C₁₇×#107	C₂³	1	4	yes	—
49	C₁₇×C₄⋊C₈	C₂³	1	12	yes	—
50	C₁₇×#102	C₂³	1	24	yes	—
51	C₄×C₈×C₁₇	C₂³	1	12	yes	—
52	C₂⁴×C₁₇	—	2	6	no	—
53	#41×C₁₇	—	4	64	no	—
54	C₂×Q₈×C₁₇	D₈	1	8	yes	complemented
55	C₂×Q₈×C₁₇	C₂×D₄	1	4	yes	—
56	C₂×D₄×C₁₇	—	2	6	no	—
57	C₂×D₄×C₁₇	—	2	24	no	—
58	C₂×D₄×C₁₇	—	2	24	no	—
59	C₂×D₄×C₁₇	—	2	12	no	—
60	C₂×D₄×C₁₇	—	2	12	no	—
61	C₂×D₄×C₁₇	—	4	24	no	—
62	C₂×D₄×C₁₇	—	8	12	no	—
63	C₂×D₄×C₁₇	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₁₇	—	2	2	no	—
65	C₂²×C₄×C₁₇	—	2	24	no	—
66	C₂²×C₄×C₁₇	—	4	24	no	—
67	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
70	D₈×C₁₇	—	2	24	no	—
71	D₈×C₁₇	—	4	24	no	—
72	C₂×C₈×C₁₇	—	2	6	no	—
73	C₂×C₈×C₁₇	#41	1	4	yes	complemented
74	C₂×C₈×C₁₇	C₂⁴	1	6	yes	—
75	C₂×C₈×C₁₇	C₂⁴	1	2	yes	—
76	C₄⋊C₄×C₁₇	—	2	2	no	—
77	C₄⋊C₄×C₁₇	C₂×D₄	1	12	yes	—
78	C₄⋊C₄×C₁₇	C₂⁴	1	4	yes	—
79	C₂²⋊C₄×C₁₇	—	2	24	no	—
80	C₄²×C₁₇	C₂×D₄	1	6	yes	—
81	C₄²×C₁₇	C₂⁴	1	6	yes	—
82	C₂×#3391	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₇	—	2	12	no	—
84	C₂³×C₁₇	—	4	12	no	—
85	C₂³×C₁₇	—	4	12	no	—
86	C₂³×C₁₇	—	4	12	no	—
87	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₁₇	C₂×D₈	1	16	yes	complemented
89	D₄×C₁₇	—	2	12	no	—
90	D₄×C₁₇	—	2	12	no	—
91	D₄×C₁₇	—	4	12	no	—
92	D₄×C₁₇	—	4	48	no	—
93	D₄×C₁₇	—	8	24	no	—
94	C₂×C₄×C₁₇	—	2	6	no	—
95	C₂×C₄×C₁₇	—	2	4	no	—
96	C₂×C₄×C₁₇	—	4	4	no	—
97	C₂×C₄×C₁₇	—	4	48	no	—
98	C₂×C₄×C₁₇	C₂×D₈	1	12	yes	—
99	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₁₇	H(5,𝔽₂)	1	2	yes	—
104	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
105	C₈×C₁₇	—	2	6	no	—
106	C₈×C₁₇	C₂×#41	1	4	yes	complemented
107	C₂×#550	C₂×C₁₇	1	2	yes	complemented
108	#3391	C₂×C₁₇	1	16	yes	direct product factor
109	C₂×#493	C₂×C₁₇	1	3	yes	complemented
110	C₂×#459	C₂×C₁₇	1	6	yes	complemented
111	C₂×C₈×Q₈	C₂×C₁₇	1	1	yes	characteristic, complemented
112	C₂×C₄×D₈	C₂×C₁₇	1	3	yes	complemented
113	C₂²×C₁₇	—	4	12	no	—
114	C₂²×C₁₇	—	4	12	no	—
115	C₂²×C₁₇	—	4	12	no	—
116	C₂²×C₁₇	—	8	24	no	—
117	C₂²×C₁₇	#550	1	2	yes	—
118	C₂²×C₁₇	C₂²×D₈	1	2	yes	—
119	C₂²×C₁₇	#576	1	2	yes	—
120	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
121	C₄×C₁₇	—	2	6	no	—
122	C₄×C₁₇	—	4	4	no	—
123	C₄×C₁₇	#550	1	2	yes	—
124	C₄×C₁₇	C₂²×D₈	1	12	yes	—
125	C₄×C₁₇	C₂²×#41	1	1	yes	characteristic
126	C₄×C₁₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×#41	—	2	2	no	—
128	C₂²×D₈	C₂²×C₁₇	1	3	yes	—
129	C₂²×D₈	C₂²×C₁₇	1	6	yes	—
130	#550	—	2	8	no	—
131	#550	C₂²×C₁₇	1	16	yes	complemented
132	C₂×#127	C₂²×C₁₇	1	6	yes	—
133	C₂×#121	—	2	6	no	—
134	C₂×C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
135	C₂×C₄×D₄	C₂²×C₁₇	1	6	yes	—
136	#493	C₂²×C₁₇	1	24	yes	complemented
137	#459	C₂²×C₁₇	1	48	yes	complemented
138	C₈×Q₈	C₂²×C₁₇	1	8	yes	complemented
139	C₄×D₈	C₂²×C₁₇	1	24	yes	complemented
140	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
141	C₂×C₄⋊C₈	C₂²×C₁₇	1	3	yes	—
142	C₂×#102	C₂²×C₁₇	1	6	yes	—
143	C₂×C₄×C₈	C₂²×C₁₇	1	3	yes	—
144	C₂×C₁₇	—	8	12	no	—
145	C₂×C₁₇	#3391	1	4	yes	direct product factor
146	C₂×C₁₇	C₂×#550	1	1	yes	characteristic
147	C₂×C₁₇	C₂³×D₈	1	1	yes	characteristic
148	C₂×C₁₇	C₂×#576	1	1	yes	characteristic
149	C₂×#41	—	2	16	no	—
150	C₂×#41	—	4	16	no	—
151	C₂²×Q₈	D₄×C₁₇	1	1	yes	characteristic
152	C₂²×D₄	—	2	6	no	—
153	C₂²×D₄	—	2	6	no	—
154	C₂²×D₄	C₂³×C₁₇	1	6	yes	—
155	C₂³×C₄	—	2	6	no	—
156	C₂×D₈	—	2	12	no	—
157	C₂×D₈	—	4	12	no	—
158	C₂×D₈	C₂³×C₁₇	1	12	yes	—
159	C₂×D₈	C₂³×C₁₇	1	24	yes	—
160	C₂²×C₈	C₂³×C₁₇	1	3	yes	—
161	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
162	#127	—	2	12	no	—
163	#127	C₂³×C₁₇	1	24	yes	—
164	#121	—	2	48	no	—
165	C₄×Q₈	D₄×C₁₇	1	4	yes	—
166	C₄×Q₈	C₂³×C₁₇	1	4	yes	—
167	C₄×D₄	—	2	12	no	—
168	C₄×D₄	C₂³×C₁₇	1	24	yes	—
169	C₂×C₄⋊C₄	D₄×C₁₇	1	3	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	6	no	—
172	C₂×C₄²	C₂³×C₁₇	1	3	yes	—
173	#107	C₂³×C₁₇	1	4	yes	—
174	C₄⋊C₈	C₂³×C₁₇	1	12	yes	—
175	#102	C₂³×C₁₇	1	24	yes	—
176	C₄×C₈	C₂³×C₁₇	1	12	yes	—
177	C₁₇	C₂×#3391	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	—	2	6	no	—
179	#41	—	4	64	no	—
180	C₂×Q₈	D₈×C₁₇	1	8	yes	complemented
181	C₂×Q₈	C₂×D₄×C₁₇	1	4	yes	—
182	C₂×D₄	—	2	6	no	—
183	C₂×D₄	—	2	24	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	—	2	12	no	—
186	C₂×D₄	—	2	12	no	—
187	C₂×D₄	—	4	24	no	—
188	C₂×D₄	—	8	12	no	—
189	C₂×D₄	C₂⁴×C₁₇	1	12	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	24	no	—
192	C₂²×C₄	—	4	24	no	—
193	C₂²×C₄	C₂×D₄×C₁₇	1	3	yes	—
194	C₂²×C₄	C₂×D₄×C₁₇	1	3	yes	—
195	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
196	D₈	—	2	24	no	—
197	D₈	—	4	24	no	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	#41×C₁₇	1	4	yes	complemented
200	C₂×C₈	C₂⁴×C₁₇	1	6	yes	—
201	C₂×C₈	C₂⁴×C₁₇	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂×D₄×C₁₇	1	12	yes	—
204	C₄⋊C₄	C₂⁴×C₁₇	1	4	yes	—
205	C₂²⋊C₄	—	2	24	no	—
206	C₄²	C₂×D₄×C₁₇	1	6	yes	—
207	C₄²	C₂⁴×C₁₇	1	6	yes	—
208	C₂³	—	2	12	no	—
209	C₂³	—	4	12	no	—
210	C₂³	—	4	12	no	—
211	C₂³	—	4	12	no	—
212	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
213	Q₈	C₂×D₈×C₁₇	1	16	yes	complemented
214	D₄	—	2	12	no	—
215	D₄	—	2	12	no	—
216	D₄	—	4	12	no	—
217	D₄	—	4	48	no	—
218	D₄	—	8	24	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	48	no	—
223	C₂×C₄	C₂×D₈×C₁₇	1	12	yes	—
224	C₂×C₄	C₂²×D₄×C₁₇	1	6	yes	—
225	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄×C₁₇	1	6	yes	—
227	C₂×C₄	C₂×#41×C₁₇	1	2	yes	—
228	C₂×C₄	C₁₇×H(5,𝔽₂)	1	2	yes	—
229	C₂×C₄	C₂⁵×C₁₇	1	1	yes	characteristic
230	C₈	—	2	6	no	—
231	C₈	C₂×#41×C₁₇	1	4	yes	complemented
232	C₂²	—	4	12	no	—
233	C₂²	—	4	12	no	—
234	C₂²	—	4	12	no	—
235	C₂²	—	8	24	no	—
236	C₂²	C₁₇×#550	1	2	yes	—
237	C₂²	C₂²×D₈×C₁₇	1	2	yes	—
238	C₂²	C₁₇×#576	1	2	yes	—
239	C₂²	C₂³×D₄×C₁₇	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	4	4	no	—
242	C₄	C₁₇×#550	1	2	yes	—
243	C₄	C₂²×D₈×C₁₇	1	12	yes	—
244	C₄	C₂²×#41×C₁₇	1	1	yes	characteristic
245	C₄	C₂×C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂	—	8	12	no	—
247	C₂	C₁₇×#3391	1	4	yes	direct product factor
248	C₂	C₂×C₁₇×#550	1	1	yes	characteristic
249	C₂	C₂³×D₈×C₁₇	1	1	yes	characteristic
250	C₂	C₂×C₁₇×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄×Q₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#550	C₂	semidirect
2. C₁₇×#3391	C₂	direct (N × H)
3. C₂×C₁₇×#493	C₂	semidirect
4. C₂×C₁₇×#459	C₂	semidirect
5. C₂×C₈×Q₈×C₁₇	C₂	semidirect
6. C₂×C₄×D₈×C₁₇	C₂	semidirect
11. C₁₇×#550	C₂²	semidirect
16. C₁₇×#493	C₂²	semidirect
17. C₁₇×#459	C₂²	semidirect
18. C₈×Q₈×C₁₇	C₂²	semidirect
19. C₄×D₈×C₁₇	C₂²	semidirect
54. C₂×Q₈×C₁₇	D₈	semidirect
73. C₂×C₈×C₁₇	#41	semidirect
82. C₂×#3391	C₁₇	direct (N × H)
88. Q₈×C₁₇	C₂×D₈	semidirect
106. C₈×C₁₇	C₂×#41	semidirect
107. C₂×#550	C₂×C₁₇	semidirect
108. #3391	C₂×C₁₇	direct (N × H)
109. C₂×#493	C₂×C₁₇	semidirect
110. C₂×#459	C₂×C₁₇	semidirect
111. C₂×C₈×Q₈	C₂×C₁₇	semidirect
112. C₂×C₄×D₈	C₂×C₁₇	semidirect
131. #550	C₂²×C₁₇	semidirect
136. #493	C₂²×C₁₇	semidirect
137. #459	C₂²×C₁₇	semidirect
138. C₈×Q₈	C₂²×C₁₇	semidirect
139. C₄×D₈	C₂²×C₁₇	semidirect
145. C₂×C₁₇	#3391	direct (N × H)
177. C₁₇	C₂×#3391	direct (N × H)
180. C₂×Q₈	D₈×C₁₇	semidirect
199. C₂×C₈	#41×C₁₇	semidirect
213. Q₈	C₂×D₈×C₁₇	semidirect
231. C₈	C₂×#41×C₁₇	semidirect
247. C₂	C₁₇×#3391	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×C₁₇×#550	2
C₁₇×#3391	2
C₂×C₁₇×#493	2
C₂×C₁₇×#459	2
C₂×C₈×Q₈×C₁₇	2
C₂×C₄×D₈×C₁₇	2
C₂×#3391	17

Subgroups — C2×C17×#3391 #16919131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#3391 #16919131