Subgroups — C₂²×C₁₇×#436 #16917876

Order 4352.

3262 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₁₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#436	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₇	—	2	16	no	—
10	C₂²×M₁₆×C₁₇	C₂²	1	24	yes	complemented
11	C₂³×C₈×C₁₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₇	C₄	1	12	yes	—
15	C₂×C₄×D₄×C₁₇	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₁₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#436	C₂²	1	64	yes	direct product factor
20	C₂×C₁₇×C₄⋊C₈	C₂²	1	12	yes	complemented
21	C₂×C₁₇×C₂²⋊C₈	C₂²	1	24	yes	complemented
22	C₂×C₄×C₈×C₁₇	C₂²	1	12	yes	complemented
23	C₂⁵×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×D₄×C₁₇	—	2	8	no	—
25	C₂²×D₄×C₁₇	C₂×C₄	1	12	yes	—
26	C₂³×C₄×C₁₇	—	2	8	no	—
27	C₂³×C₄×C₁₇	—	2	8	no	—
28	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₇	C₂×C₄	1	12	yes	—
31	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₇	C₂³	1	12	yes	—
33	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₁₇	—	2	96	no	—
35	C₂×M₁₆×C₁₇	C₂³	1	32	yes	complemented
36	C₂²×C₈×C₁₇	—	2	8	no	—
37	C₂²×C₈×C₁₇	—	2	4	no	—
38	C₂²×C₈×C₁₇	D₄	1	8	yes	complemented
39	C₂²×C₈×C₁₇	C₂³	1	12	yes	—
40	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
41	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
42	C₄×D₄×C₁₇	C₂×C₄	1	48	yes	—
43	C₄×D₄×C₁₇	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	12	yes	—
45	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	24	yes	—
46	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₁₇	C₂³	1	6	yes	—
48	C₁₇×C₄⋊C₈	C₂³	1	16	yes	complemented
49	C₁₇×C₂²⋊C₈	C₂³	1	32	yes	complemented
50	C₄×C₈×C₁₇	C₂³	1	16	yes	complemented
51	C₂⁴×C₁₇	—	2	8	no	—
52	C₂⁴×C₁₇	—	2	8	no	—
53	C₂⁴×C₁₇	—	2	8	no	—
54	C₂⁴×C₁₇	C₂²×C₄	1	12	yes	—
55	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄×C₁₇	—	2	48	no	—
57	C₂×D₄×C₁₇	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₁₇	—	2	8	no	—
59	C₂²×C₄×C₁₇	—	2	4	no	—
60	C₂²×C₄×C₁₇	—	2	48	no	—
61	C₂²×C₄×C₁₇	—	2	24	no	—
62	C₂²×C₄×C₁₇	—	2	24	no	—
63	C₂²×C₄×C₁₇	M₁₆	1	8	yes	complemented
64	C₂²×C₄×C₁₇	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₇	C₂²×C₄	1	24	yes	—
66	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₁₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₁₇	#41	1	4	yes	—
71	C₂²×C₄×C₁₇	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₁₇	C₂⁴	1	3	yes	—
73	M₁₆×C₁₇	—	2	64	no	—
74	C₂×C₈×C₁₇	—	2	24	no	—
75	C₂×C₈×C₁₇	—	2	12	no	—
76	C₂×C₈×C₁₇	C₂×D₄	1	24	yes	complemented
77	C₂×C₈×C₁₇	C₂⁴	1	8	yes	—
78	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
79	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
80	C₄⋊C₄×C₁₇	C₂²×C₄	1	16	yes	—
81	C₂²⋊C₄×C₁₇	C₂²×C₄	1	32	yes	—
82	C₄²×C₁₇	C₂²×C₄	1	12	yes	—
83	C₄²×C₁₇	C₂⁴	1	4	yes	—
84	C₂²×#436	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₇	—	2	24	no	—
86	C₂³×C₁₇	—	2	24	no	—
87	C₂³×C₁₇	—	2	24	no	—
88	C₂³×C₁₇	—	4	32	no	—
89	C₂³×C₁₇	C₄×D₄	1	4	yes	—
90	C₂³×C₁₇	C₂×M₁₆	1	4	yes	—
91	C₂³×C₁₇	#131	1	4	yes	—
92	C₂³×C₁₇	C₂³×C₄	1	8	yes	—
93	C₂³×C₁₇	C₂³×C₄	1	3	yes	—
94	D₄×C₁₇	—	2	32	no	—
95	C₂×C₄×C₁₇	—	2	24	no	—
96	C₂×C₄×C₁₇	—	2	12	no	—
97	C₂×C₄×C₁₇	—	2	48	no	—
98	C₂×C₄×C₁₇	—	2	8	no	—
99	C₂×C₄×C₁₇	—	2	8	no	—
100	C₂×C₄×C₁₇	C₄×D₄	1	12	yes	—
101	C₂×C₄×C₁₇	C₂×M₁₆	1	24	yes	complemented
102	C₂×C₄×C₁₇	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
104	C₂×C₄×C₁₇	C₂³×C₄	1	3	yes	—
105	C₂×C₄×C₁₇	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
107	C₂×C₄×C₁₇	C₂×#41	1	6	yes	—
108	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
109	C₈×C₁₇	—	2	8	no	—
110	C₈×C₁₇	—	2	4	no	—
111	C₈×C₁₇	C₂²×D₄	1	8	yes	complemented
112	C₂³×M₁₆	C₂×C₁₇	1	2	yes	complemented
113	C₂²×C₄×D₄	C₂×C₁₇	1	1	yes	characteristic
114	C₂×#436	C₂×C₁₇	1	24	yes	direct product factor
115	C₂²×C₄⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
116	C₂²×C₂²⋊C₈	C₂×C₁₇	1	2	yes	complemented
117	C₂²×C₄×C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
118	C₂²×C₁₇	—	2	8	no	—
119	C₂²×C₁₇	—	2	8	no	—
120	C₂²×C₁₇	—	2	8	no	—
121	C₂²×C₁₇	—	4	48	no	—
122	C₂²×C₁₇	#436	1	16	yes	direct product factor
123	C₂²×C₁₇	C₂×C₄×D₄	1	6	yes	—
124	C₂²×C₁₇	C₂²×M₁₆	1	6	yes	—
125	C₂²×C₁₇	C₂×#131	1	6	yes	—
126	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
127	C₄×C₁₇	—	2	8	no	—
128	C₄×C₁₇	—	2	4	no	—
129	C₄×C₁₇	C₂×C₄×D₄	1	6	yes	—
130	C₄×C₁₇	C₂²×M₁₆	1	8	yes	complemented
131	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₁₇	C₂²×#41	1	1	yes	characteristic
133	C₂³×D₄	C₄×C₁₇	1	1	yes	characteristic
134	C₂⁴×C₄	C₂²×C₁₇	1	2	yes	—
135	C₂²×M₁₆	—	2	16	no	—
136	C₂²×M₁₆	C₂²×C₁₇	1	24	yes	complemented
137	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
138	C₂³×C₈	C₂²×C₁₇	1	2	yes	—
139	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
140	C₂×C₄×D₄	C₄×C₁₇	1	12	yes	—
141	C₂×C₄×D₄	C₂²×C₁₇	1	12	yes	—
142	C₂²×C₄⋊C₄	C₄×C₁₇	1	1	yes	characteristic
143	C₂²×C₂²⋊C₄	C₄×C₁₇	1	2	yes	—
144	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
145	#436	C₂²×C₁₇	1	64	yes	direct product factor
146	C₂×C₄⋊C₈	C₂²×C₁₇	1	12	yes	complemented
147	C₂×C₂²⋊C₈	C₂²×C₁₇	1	24	yes	complemented
148	C₂×C₄×C₈	C₂²×C₁₇	1	12	yes	complemented
149	C₂×C₁₇	—	4	8	no	—
150	C₂×C₁₇	C₂×#436	1	12	yes	direct product factor
151	C₂×C₁₇	C₂²×C₄×D₄	1	1	yes	characteristic
152	C₂×C₁₇	C₂³×M₁₆	1	1	yes	characteristic
153	C₂×C₁₇	C₂²×#131	1	1	yes	characteristic
154	C₂⁵	C₂×C₄×C₁₇	1	2	yes	—
155	C₂²×D₄	—	2	8	no	—
156	C₂²×D₄	C₂×C₄×C₁₇	1	12	yes	—
157	C₂³×C₄	—	2	8	no	—
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₁₇	1	2	yes	—
161	C₂³×C₄	C₂×C₄×C₁₇	1	12	yes	—
162	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₁₇	1	12	yes	—
164	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	96	no	—
166	C₂×M₁₆	C₂³×C₁₇	1	32	yes	complemented
167	C₂²×C₈	—	2	8	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	D₄×C₁₇	1	8	yes	complemented
170	C₂²×C₈	C₂³×C₁₇	1	12	yes	—
171	C₂²×C₈	C₂³×C₁₇	1	6	yes	—
172	C₂²×C₈	C₂³×C₁₇	1	6	yes	—
173	C₄×D₄	C₂×C₄×C₁₇	1	48	yes	—
174	C₄×D₄	C₂³×C₁₇	1	16	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	12	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	24	yes	—
177	C₂×C₄²	C₂×C₄×C₁₇	1	6	yes	—
178	C₂×C₄²	C₂³×C₁₇	1	6	yes	—
179	C₄⋊C₈	C₂³×C₁₇	1	16	yes	complemented
180	C₂²⋊C₈	C₂³×C₁₇	1	32	yes	complemented
181	C₄×C₈	C₂³×C₁₇	1	16	yes	complemented
182	C₁₇	C₂²×#436	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	2	8	no	—
184	C₂⁴	—	2	8	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	C₂²×C₄×C₁₇	1	12	yes	—
187	C₂⁴	C₂²×C₄×C₁₇	1	1	yes	characteristic
188	C₂×D₄	—	2	48	no	—
189	C₂×D₄	C₂²×C₄×C₁₇	1	16	yes	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	48	no	—
195	C₂²×C₄	M₁₆×C₁₇	1	8	yes	complemented
196	C₂²×C₄	C₂²×C₄×C₁₇	1	6	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₇	1	12	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₇	1	24	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₇	1	6	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₇	1	3	yes	—
201	C₂²×C₄	C₂×D₄×C₁₇	1	4	yes	—
202	C₂²×C₄	#41×C₁₇	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₁₇	1	8	yes	—
204	C₂²×C₄	C₂⁴×C₁₇	1	3	yes	—
205	M₁₆	—	2	64	no	—
206	C₂×C₈	—	2	24	no	—
207	C₂×C₈	—	2	12	no	—
208	C₂×C₈	C₂×D₄×C₁₇	1	24	yes	complemented
209	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
210	C₂×C₈	C₂⁴×C₁₇	1	8	yes	—
211	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
212	C₄⋊C₄	C₂²×C₄×C₁₇	1	16	yes	—
213	C₂²⋊C₄	C₂²×C₄×C₁₇	1	32	yes	—
214	C₄²	C₂²×C₄×C₁₇	1	12	yes	—
215	C₄²	C₂⁴×C₁₇	1	4	yes	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	—	4	32	no	—
220	C₂³	C₄×D₄×C₁₇	1	4	yes	—
221	C₂³	C₂×M₁₆×C₁₇	1	4	yes	—
222	C₂³	C₁₇×#131	1	4	yes	—
223	C₂³	C₂³×C₄×C₁₇	1	8	yes	—
224	C₂³	C₂³×C₄×C₁₇	1	3	yes	—
225	D₄	—	2	32	no	—
226	C₂×C₄	—	2	24	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	C₄×D₄×C₁₇	1	12	yes	—
232	C₂×C₄	C₂×M₁₆×C₁₇	1	24	yes	complemented
233	C₂×C₄	C₂³×C₄×C₁₇	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₁₇	1	8	yes	—
235	C₂×C₄	C₂³×C₄×C₁₇	1	3	yes	—
236	C₂×C₄	C₂³×C₄×C₁₇	1	4	yes	—
237	C₂×C₄	C₂²×D₄×C₁₇	1	6	yes	—
238	C₂×C₄	C₂×#41×C₁₇	1	6	yes	—
239	C₂×C₄	C₂⁵×C₁₇	1	1	yes	characteristic
240	C₈	—	2	8	no	—
241	C₈	—	2	4	no	—
242	C₈	C₂²×D₄×C₁₇	1	8	yes	complemented
243	C₂²	—	2	8	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	48	no	—
247	C₂²	C₁₇×#436	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₄×C₁₇	1	6	yes	—
249	C₂²	C₂²×M₁₆×C₁₇	1	6	yes	—
250	C₂²	C₂×C₁₇×#131	1	6	yes	—
251	C₂²	C₂⁴×C₄×C₁₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	2	8	no	—
254	C₄	C₂×C₄×D₄×C₁₇	1	6	yes	—
255	C₄	C₂²×M₁₆×C₁₇	1	8	yes	complemented
256	C₄	C₂³×D₄×C₁₇	1	1	yes	characteristic
257	C₄	C₂²×#41×C₁₇	1	1	yes	characteristic
258	C₂	—	4	8	no	—
259	C₂	C₂×C₁₇×#436	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₄×C₁₇	1	1	yes	characteristic
261	C₂	C₂³×M₁₆×C₁₇	1	1	yes	characteristic
262	C₂	C₂²×C₁₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₁₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₇
socle	C₂⁴×C₁₇	C₂²×C₄
prime radical	C₂³×D₄×C₁₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×M₁₆×C₁₇	C₂	semidirect
3. C₂×C₁₇×#436	C₂	direct (N × H)
4. C₂²×C₁₇×C₄⋊C₈	C₂	semidirect
5. C₂²×C₁₇×C₂²⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈×C₁₇	C₂	semidirect
10. C₂²×M₁₆×C₁₇	C₂²	semidirect
19. C₁₇×#436	C₂²	direct (N × H)
20. C₂×C₁₇×C₄⋊C₈	C₂²	semidirect
21. C₂×C₁₇×C₂²⋊C₈	C₂²	semidirect
22. C₂×C₄×C₈×C₁₇	C₂²	semidirect
35. C₂×M₁₆×C₁₇	C₂³	semidirect
38. C₂²×C₈×C₁₇	D₄	semidirect
48. C₁₇×C₄⋊C₈	C₂³	semidirect
49. C₁₇×C₂²⋊C₈	C₂³	semidirect
50. C₄×C₈×C₁₇	C₂³	semidirect
63. C₂²×C₄×C₁₇	M₁₆	semidirect
76. C₂×C₈×C₁₇	C₂×D₄	semidirect
84. C₂²×#436	C₁₇	direct (N × H)
101. C₂×C₄×C₁₇	C₂×M₁₆	semidirect
111. C₈×C₁₇	C₂²×D₄	semidirect
112. C₂³×M₁₆	C₂×C₁₇	semidirect
114. C₂×#436	C₂×C₁₇	direct (N × H)
115. C₂²×C₄⋊C₈	C₂×C₁₇	semidirect
116. C₂²×C₂²⋊C₈	C₂×C₁₇	semidirect
117. C₂²×C₄×C₈	C₂×C₁₇	semidirect
122. C₂²×C₁₇	#436	direct (N × H)
130. C₄×C₁₇	C₂²×M₁₆	semidirect
136. C₂²×M₁₆	C₂²×C₁₇	semidirect
145. #436	C₂²×C₁₇	direct (N × H)
146. C₂×C₄⋊C₈	C₂²×C₁₇	semidirect
147. C₂×C₂²⋊C₈	C₂²×C₁₇	semidirect
148. C₂×C₄×C₈	C₂²×C₁₇	semidirect
150. C₂×C₁₇	C₂×#436	direct (N × H)
166. C₂×M₁₆	C₂³×C₁₇	semidirect
169. C₂²×C₈	D₄×C₁₇	semidirect
179. C₄⋊C₈	C₂³×C₁₇	semidirect
180. C₂²⋊C₈	C₂³×C₁₇	semidirect
181. C₄×C₈	C₂³×C₁₇	semidirect
182. C₁₇	C₂²×#436	direct (N × H)
195. C₂²×C₄	M₁₆×C₁₇	semidirect
208. C₂×C₈	C₂×D₄×C₁₇	semidirect
232. C₂×C₄	C₂×M₁₆×C₁₇	semidirect
242. C₈	C₂²×D₄×C₁₇	semidirect
247. C₂²	C₁₇×#436	direct (N × H)
255. C₄	C₂²×M₁₆×C₁₇	semidirect
259. C₂	C₂×C₁₇×#436	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×M₁₆×C₁₇	2
C₂²×C₄×D₄×C₁₇	2
C₂×C₁₇×#436	2
C₂²×C₁₇×C₄⋊C₈	2
C₂²×C₁₇×C₂²⋊C₈	2
C₂²×C₄×C₈×C₁₇	2
C₂²×#436	17

Subgroups — C22×C17×#436 #16917876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₇×#436 #16917876