Subgroups — C₂²×#11762586 #17186977

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 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	C₂²×#10587486	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762586	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂³×#3622	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂×#10587483	C₄	1	6	yes	complemented
8	C₂×#10587483	C₂²	1	6	yes	—
9	C₂²×#63214	C₄	1	2	yes	complemented
10	C₂×#10587486	C₄	1	6	yes	—
11	C₂×#10587486	C₂²	1	6	yes	—
12	#11762586	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×M₃₂	C₄	1	6	yes	complemented
15	C₂×C₁₇×M₃₂	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₁₇	C₄	1	2	yes	complemented
17	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×Dic₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
25	C₂×C₄×Dic₁₇	—	2	3	no	—
26	C₂×C₄×Dic₁₇	C₂×C₄	1	6	yes	—
27	C₂²×#3622	—	2	3	no	—
28	C₂²×#3622	—	2	1	no	—
29	C₂²×#3622	C₂×C₄	1	6	yes	—
30	C₁₇×M₃₂	C₂×C₄	1	12	yes	complemented
31	C₁₇×M₃₂	C₂³	1	4	yes	—
32	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	complemented
33	#10587486	C₂×C₄	1	12	yes	—
34	#10587486	C₂³	1	4	yes	—
35	#10587483	C₂×C₄	1	12	yes	complemented
36	#10587483	C₂³	1	4	yes	—
37	C₂×#63214	C₂×C₄	1	12	yes	complemented
38	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₄²	1	3	yes	—
40	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	M₁₆	1	3	yes	—
42	C₂²×C₄×C₁₇	M₁₆	1	3	yes	—
43	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₁₇	—	2	6	no	—
46	C₂²×Dic₁₇	—	4	4	no	—
47	C₂×C₈×C₁₇	C₄²	1	3	yes	—
48	C₂×C₈×C₁₇	C₄²	1	3	yes	—
49	C₂×C₈×C₁₇	C₄²	1	6	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₄×Dic₁₇	—	2	6	no	—
57	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
58	C₂×#3622	—	2	6	no	—
59	C₂×#3622	—	2	6	no	—
60	C₂×#3622	C₂²×C₄	1	4	yes	—
61	C₁₆×C₁₇	C₂²×C₄	1	8	yes	complemented
62	#63214	C₂²×C₄	1	8	yes	complemented
63	C₂²×#347	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	—	2	4	no	—
65	C₂³×C₁₇	#97	1	3	yes	—
66	C₂³×C₁₇	#97	1	1	yes	characteristic
67	C₂³×C₁₇	C₂×M₁₆	1	3	yes	—
68	C₂×C₄×C₁₇	#97	1	6	yes	—
69	C₂×C₄×C₁₇	#97	1	3	yes	—
70	C₂×C₄×C₁₇	#97	1	3	yes	—
71	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
73	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
74	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
76	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	4	no	—
78	C₂×Dic₁₇	—	4	12	no	—
79	C₈×C₁₇	C₂×C₄²	1	3	yes	—
80	C₈×C₁₇	C₂×C₄²	1	3	yes	—
81	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
83	#3622	—	2	4	no	—
84	C₂²×M₃₂	—	17	1	no	—
85	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
86	C₂²×#97	—	17	1	no	—
87	C₂×#347	—	17	12	no	—
88	C₂²×C₁₇	—	2	6	no	—
89	C₂²×C₁₇	—	2	6	no	—
90	C₂²×C₁₇	#347	1	4	yes	complemented
91	C₂²×C₁₇	C₂×#97	1	3	yes	—
92	C₂²×C₁₇	C₂×#97	1	3	yes	—
93	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
94	C₄×C₁₇	C₂×#97	1	3	yes	—
95	C₄×C₁₇	C₂×#97	1	3	yes	—
96	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
98	Dic₁₇	—	4	4	no	—
99	C₂³×C₈	—	17	1	no	—
100	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
101	C₂²×C₄²	—	17	1	no	—
102	C₂×M₃₂	—	17	6	no	—
103	C₂×M₃₂	—	17	6	no	—
104	C₂×M₃₂	Dic₁₇	1	6	yes	complemented
105	C₂×M₃₂	C₂×D₁₇	1	6	yes	—
106	C₂²×C₁₆	—	17	2	no	—
107	C₂²×C₁₆	Dic₁₇	1	2	yes	complemented
108	C₂×#97	—	17	6	no	—
109	C₂×#97	—	17	6	no	—
110	#347	—	17	16	no	—
111	C₂×C₁₇	—	2	1	no	—
112	C₂×C₁₇	—	2	3	no	—
113	C₂×C₁₇	C₂×#347	1	6	yes	complemented
114	C₂×C₁₇	C₂²×#97	1	1	yes	characteristic
115	C₂³×C₄	—	34	1	no	—
116	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
117	C₂²×C₈	—	17	6	no	—
118	C₂²×C₈	—	34	1	no	—
119	C₂²×C₈	—	34	3	no	—
120	C₂²×C₈	C₄×D₁₇	1	3	yes	—
121	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₁₇	1	3	yes	—
123	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
124	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
126	C₂×C₄²	—	17	6	no	—
127	C₂×C₄²	—	34	3	no	—
128	M₃₂	—	17	12	no	—
129	M₃₂	—	17	4	no	—
130	M₃₂	C₂×Dic₁₇	1	12	yes	complemented
131	M₃₂	C₂²×D₁₇	1	4	yes	—
132	C₂×C₁₆	—	17	12	no	—
133	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	complemented
134	#97	—	17	12	no	—
135	#97	—	17	4	no	—
136	C₁₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	#63218	1	1	yes	characteristic
138	C₂²×C₄	—	34	6	no	—
139	C₂²×C₄	—	68	4	no	—
140	C₂²×C₄	#63218	1	3	yes	—
141	C₂²×C₄	#63218	1	1	yes	characteristic
142	C₂²×C₄	#63218	1	3	yes	—
143	C₂²×C₄	C₄×Dic₁₇	1	3	yes	—
144	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
146	C₂×C₈	—	17	4	no	—
147	C₂×C₈	—	34	6	no	—
148	C₂×C₈	—	34	6	no	—
149	C₂×C₈	C₄×Dic₁₇	1	6	yes	—
150	C₂×C₈	C₄×Dic₁₇	1	3	yes	—
151	C₂×C₈	C₄×Dic₁₇	1	3	yes	—
152	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
153	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
154	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
155	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
156	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
157	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
158	C₄²	—	17	4	no	—
159	C₄²	—	34	6	no	—
160	C₁₆	—	17	8	no	—
161	C₁₆	C₂²×Dic₁₇	1	8	yes	complemented
162	C₂³	—	2	4	no	—
163	C₂³	#10587486	1	3	yes	—
164	C₂³	#10587486	1	1	yes	characteristic
165	C₂³	C₂×#63218	1	3	yes	—
166	C₂×C₄	—	34	4	no	—
167	C₂×C₄	—	68	12	no	—
168	C₂×C₄	#10587486	1	6	yes	—
169	C₂×C₄	#10587486	1	3	yes	—
170	C₂×C₄	#10587486	1	3	yes	—
171	C₂×C₄	C₂×#63218	1	3	yes	—
172	C₂×C₄	C₂×#63218	1	3	yes	—
173	C₂×C₄	C₂×#63218	1	3	yes	—
174	C₂×C₄	C₂×C₄×Dic₁₇	1	3	yes	—
175	C₂×C₄	C₂×C₄×Dic₁₇	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
177	C₈	—	34	4	no	—
178	C₈	C₂×C₄×Dic₁₇	1	3	yes	—
179	C₈	C₂×C₄×Dic₁₇	1	3	yes	—
180	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
181	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#11762586	1	4	yes	direct product factor
185	C₂²	C₂×#10587486	1	3	yes	—
186	C₂²	C₂×#10587486	1	3	yes	—
187	C₂²	C₂²×#63218	1	1	yes	characteristic
188	C₄	—	68	4	no	—
189	C₄	C₂×#10587486	1	3	yes	—
190	C₄	C₂×#10587486	1	3	yes	—
191	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
192	C₄	C₂²×#63218	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#11762586	1	6	yes	direct product factor
196	C₂	C₂²×#10587486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂³×C₁₇	#97
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	C₂²×#347
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#11762586	C₂	direct (N × H)
7. C₂×#10587483	C₄	semidirect
9. C₂²×#63214	C₄	semidirect
12. #11762586	C₂²	direct (N × H)
14. C₂×C₁₇×M₃₂	C₄	semidirect
16. C₂²×C₁₆×C₁₇	C₄	semidirect
30. C₁₇×M₃₂	C₂×C₄	semidirect
32. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
35. #10587483	C₂×C₄	semidirect
37. C₂×#63214	C₂×C₄	semidirect
61. C₁₆×C₁₇	C₂²×C₄	semidirect
62. #63214	C₂²×C₄	semidirect
90. C₂²×C₁₇	#347	semidirect
104. C₂×M₃₂	Dic₁₇	semidirect
107. C₂²×C₁₆	Dic₁₇	semidirect
113. C₂×C₁₇	C₂×#347	semidirect
130. M₃₂	C₂×Dic₁₇	semidirect
133. C₂×C₁₆	C₂×Dic₁₇	semidirect
136. C₁₇	C₂²×#347	semidirect
161. C₁₆	C₂²×Dic₁₇	semidirect
184. C₂²	#11762586	direct (N × H)
195. C₂	C₂×#11762586	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
C₂²×#10587486	2
C₂×#11762586	2
C₂²×C₁₇×M₃₂	2
C₂²×#347	17

Subgroups — C22×#11762586 #17186977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762586 #17186977