Subgroups — C₂×D₁₇×M₆₄ #17192428

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762643	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762642	C₂	1	2	yes	maximal, complemented
4	D₁₇×M₆₄	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₂×D₁₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₇	C₄	1	1	yes	characteristic
8	C₂²×#63214	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₁₇	C₄	1	4	yes	—
10	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
12	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
14	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
15	#11762643	C₂²	1	4	yes	complemented
16	#11762642	C₂²	1	8	yes	complemented
17	C₂×#10587465	C₂²	1	2	yes	complemented
18	C₃₂×D₁₇	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×M₆₄	C₂²	1	4	yes	complemented
21	C₂×C₁₇×C₃₂	C₂²	1	2	yes	complemented
22	C₂²×C₄×D₁₇	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₈	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₈	1	4	yes	—
26	C₂×C₈×D₁₇	C₈	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₈	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
34	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×#63214	C₂×C₄	1	1	yes	characteristic
36	C₂×#63214	C₂×C₄	1	4	yes	—
37	C₂×#63214	C₂×C₄	1	1	yes	characteristic
38	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
39	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
40	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
41	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
42	C₁₆×D₁₇	C₂³	1	2	yes	—
43	C₁₆×D₁₇	C₂³	1	2	yes	—
44	C₁₇×C₃₂	C₂³	1	4	yes	complemented
45	#10587465	C₂³	1	4	yes	complemented
46	C₂³×D₁₇	C₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₁₇	C₁₆	1	1	yes	characteristic
49	C₂×C₄×D₁₇	C₁₆	1	1	yes	characteristic
50	C₂×C₄×D₁₇	C₁₆	1	4	yes	—
51	C₂×C₄×D₁₇	C₁₆	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×#3622	C₂×C₈	1	4	yes	—
60	C₂×#3622	C₂×C₈	1	1	yes	characteristic
61	C₂×#3622	C₂×C₈	1	1	yes	characteristic
62	C₈×D₁₇	C₂×C₈	1	4	yes	—
63	C₈×D₁₇	C₂×C₈	1	2	yes	—
64	C₈×D₁₇	C₂×C₈	1	2	yes	—
65	C₈×D₁₇	C₂×C₈	1	4	yes	—
66	C₈×D₁₇	C₂²×C₄	1	2	yes	—
67	C₈×D₁₇	C₂²×C₄	1	2	yes	—
68	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
69	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
70	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
71	#63214	C₂²×C₄	1	2	yes	—
72	#63214	C₂²×C₄	1	2	yes	—
73	C₂²×M₆₄	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
75	C₂²×D₁₇	—	2	4	no	—
76	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
77	C₂²×D₁₇	C₂×C₁₆	1	4	yes	—
78	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
84	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
85	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
86	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
87	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
88	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
89	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
90	C₄×D₁₇	C₂²×C₈	1	2	yes	—
91	C₄×D₁₇	C₂²×C₈	1	2	yes	—
92	C₈×C₁₇	C₂²×C₈	1	2	yes	—
93	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
94	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
95	#3622	C₂²×C₈	1	2	yes	—
96	#3622	C₂²×C₈	1	2	yes	—
97	C₂³×C₁₆	—	17	1	no	—
98	C₂×M₆₄	—	17	1	no	—
99	C₂×M₆₄	—	17	8	no	—
100	C₂×M₆₄	—	17	2	no	—
101	C₂×M₆₄	D₁₇	1	1	yes	characteristic, direct product factor
102	C₂²×C₃₂	—	17	2	no	—
103	C₂²×C₁₇	—	2	2	no	—
104	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
105	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
106	C₂×D₁₇	—	2	2	no	—
107	C₂×D₁₇	—	2	8	no	—
108	C₂×D₁₇	M₆₄	1	4	yes	direct product factor
109	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
110	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
111	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
112	C₄×C₁₇	C₂²×C₁₆	1	2	yes	—
113	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
114	Dic₁₇	C₂²×C₁₆	1	2	yes	—
115	Dic₁₇	C₂²×C₁₆	1	2	yes	—
116	C₂³×C₈	—	17	1	no	—
117	C₂²×C₁₆	—	17	1	no	—
118	C₂²×C₁₆	—	17	4	no	—
119	C₂²×C₁₆	—	17	1	no	—
120	C₂²×C₁₆	—	17	1	no	—
121	C₂²×C₁₆	—	17	4	no	—
122	C₂²×C₁₆	—	17	1	no	—
123	C₂²×C₁₆	—	17	1	no	—
124	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
125	M₆₄	—	17	4	no	—
126	M₆₄	—	17	8	no	—
127	M₆₄	C₂×D₁₇	1	4	yes	direct product factor
128	C₂×C₃₂	—	17	2	no	—
129	C₂×C₃₂	—	17	8	no	—
130	C₂×C₃₂	C₂×D₁₇	1	2	yes	complemented
131	C₂×C₁₇	—	2	2	no	—
132	C₂×C₁₇	C₂×M₆₄	1	2	yes	complemented
133	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
134	D₁₇	—	2	2	no	—
135	D₁₇	C₂×M₆₄	1	4	yes	direct product factor
136	C₂³×C₄	—	17	1	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	4	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	17	4	no	—
143	C₂²×C₈	—	17	1	no	—
144	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
145	C₂×C₁₆	—	17	4	no	—
146	C₂×C₁₆	—	17	1	no	—
147	C₂×C₁₆	—	17	1	no	—
148	C₂×C₁₆	—	17	4	no	—
149	C₂×C₁₆	—	17	4	no	—
150	C₂×C₁₆	—	17	2	no	—
151	C₂×C₁₆	—	17	2	no	—
152	C₂×C₁₆	—	17	2	no	—
153	C₂×C₁₆	—	17	2	no	—
154	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
155	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
156	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
157	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
158	C₃₂	—	17	4	no	—
159	C₃₂	C₂²×D₁₇	1	4	yes	complemented
160	C₁₇	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
161	C₂⁴	—	17	1	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	4	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	4	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
170	C₂×C₈	—	17	4	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	17	2	no	—
175	C₂×C₈	—	17	4	no	—
176	C₂×C₈	—	17	4	no	—
177	C₂×C₈	—	17	2	no	—
178	C₂×C₈	—	17	2	no	—
179	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
180	C₂×C₈	C₈×D₁₇	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
183	C₁₆	—	17	2	no	—
184	C₁₆	—	17	2	no	—
185	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
186	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
187	C₁₆	C₂³×D₁₇	1	1	yes	characteristic
188	C₂³	—	17	1	no	—
189	C₂³	—	17	4	no	—
190	C₂³	—	17	1	no	—
191	C₂³	—	34	4	no	—
192	C₂³	C₁₆×D₁₇	1	1	yes	characteristic
193	C₂×C₄	—	17	4	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	4	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	17	4	no	—
200	C₂×C₄	—	17	2	no	—
201	C₂×C₄	—	17	2	no	—
202	C₂×C₄	C₁₆×D₁₇	1	2	yes	—
203	C₂×C₄	C₁₆×D₁₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
205	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
206	C₈	—	17	2	no	—
207	C₈	—	17	2	no	—
208	C₈	C₂×C₈×D₁₇	1	2	yes	—
209	C₈	C₂×C₈×D₁₇	1	1	yes	characteristic
210	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	17	4	no	—
213	C₂²	—	17	2	no	—
214	C₂²	—	17	2	no	—
215	C₂²	—	34	8	no	—
216	C₂²	—	34	2	no	—
217	C₂²	C₂×C₁₆×D₁₇	1	2	yes	—
218	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
219	C₄	—	17	2	no	—
220	C₄	—	17	2	no	—
221	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
222	C₄	C₂×C₁₆×D₁₇	1	2	yes	—
223	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	17	4	no	—
226	C₂	—	34	2	no	—
227	C₂	D₁₇×M₆₄	1	2	yes	direct product factor
228	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₁₇
Fitting	C₂×C₁₇×M₆₄	C₂
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂³×D₁₇	C₁₆
cosocle	C₁₆×C₁₇	C₂⁴
hypercenter	C₂×M₆₄	D₁₇
hypocenter	C₁₇	C₂²×M₆₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#11762643	C₂	semidirect
3. C₂×#11762642	C₂	semidirect
4. D₁₇×M₆₄	C₂	direct (N × H)
5. C₂×C₃₂×D₁₇	C₂	semidirect
6. C₂×C₁₇×M₆₄	C₂	semidirect
15. #11762643	C₂²	semidirect
16. #11762642	C₂²	semidirect
17. C₂×#10587465	C₂²	semidirect
18. C₃₂×D₁₇	C₂²	semidirect
20. C₁₇×M₆₄	C₂²	semidirect
21. C₂×C₁₇×C₃₂	C₂²	semidirect
44. C₁₇×C₃₂	C₂³	semidirect
45. #10587465	C₂³	semidirect
101. C₂×M₆₄	D₁₇	direct (N × H)
108. C₂×D₁₇	M₆₄	direct (N × H)
127. M₆₄	C₂×D₁₇	direct (N × H)
130. C₂×C₃₂	C₂×D₁₇	semidirect
132. C₂×C₁₇	C₂×M₆₄	semidirect
135. D₁₇	C₂×M₆₄	direct (N × H)
159. C₃₂	C₂²×D₁₇	semidirect
160. C₁₇	C₂²×M₆₄	semidirect
227. C₂	D₁₇×M₆₄	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×C₁₆×D₁₇	2
C₂×#11762643	2
C₂×#11762642	2
D₁₇×M₆₄	2
C₂×C₃₂×D₁₇	2
C₂×C₁₇×M₆₄	2
C₂²×M₆₄	17

Subgroups — C2×D17×M64 #17192428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₇×M₆₄ #17192428