Subgroups — C₂²×#11762588 #17186988

Order 4352.

1470 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×#63214	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762588	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄×D₁₇	C₄	1	1	yes	characteristic
6	C₂³×#3622	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₇	—	2	4	no	—
8	C₂²×C₈×D₁₇	—	2	4	no	—
9	C₂²×C₈×D₁₇	C₄	1	6	yes	—
10	C₂²×C₈×D₁₇	C₂²	1	6	yes	—
11	C₂²×#63214	—	2	4	no	—
12	C₂²×#63214	C₂²	1	6	yes	complemented
13	#11762588	C₂²	1	16	yes	direct product factor
14	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₇	—	2	4	no	—
16	C₂²×C₁₆×C₁₇	C₂²	1	6	yes	complemented
17	C₂⁴×D₁₇	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₁₇	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₁₇	—	2	4	no	—
21	C₂²×C₄×D₁₇	—	2	4	no	—
22	C₂²×C₄×D₁₇	C₈	1	6	yes	—
23	C₂²×C₄×D₁₇	C₂×C₄	1	6	yes	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₁₇	D₄	1	4	yes	—
26	C₂²×C₈×C₁₇	D₄	1	4	yes	—
27	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
28	C₂²×#3622	—	2	4	no	—
29	C₂²×#3622	C₂×C₄	1	6	yes	—
30	C₂×C₈×D₁₇	—	2	24	no	—
31	C₂×C₈×D₁₇	—	2	12	no	—
32	C₂×C₈×D₁₇	—	2	12	no	—
33	C₂×C₈×D₁₇	C₂×C₄	1	12	yes	—
34	C₂×C₈×D₁₇	C₂³	1	4	yes	—
35	C₂×C₁₆×C₁₇	—	2	12	no	—
36	C₂×C₁₆×C₁₇	C₂³	1	4	yes	complemented
37	C₂×#63214	—	2	12	no	—
38	C₂×#63214	C₂³	1	4	yes	complemented
39	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂³×D₁₇	—	2	4	no	—
41	C₂³×D₁₇	—	2	4	no	—
42	C₂³×D₁₇	C₁₆	1	8	yes	complemented
43	C₂³×D₁₇	C₂×C₈	1	6	yes	—
44	C₂²×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
46	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₇	—	2	4	no	—
49	C₂²×Dic₁₇	C₂×C₈	1	6	yes	—
50	C₂×C₄×D₁₇	—	2	24	no	—
51	C₂×C₄×D₁₇	—	2	12	no	—
52	C₂×C₄×D₁₇	—	2	12	no	—
53	C₂×C₄×D₁₇	C₂×C₈	1	12	yes	—
54	C₂×C₄×D₁₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	12	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
59	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
60	C₂×#3622	—	2	12	no	—
61	C₂×#3622	C₂²×C₄	1	4	yes	—
62	C₈×D₁₇	—	2	24	no	—
63	C₈×D₁₇	—	2	4	no	—
64	C₈×D₁₇	—	2	4	no	—
65	C₁₆×C₁₇	—	2	4	no	—
66	#63214	—	2	4	no	—
67	C₂²×C₂²⋊C₁₆	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂²⋊C₈	1	4	yes	—
69	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
70	C₂³×C₁₇	M₃₂	1	4	yes	—
71	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
72	C₂²×D₁₇	—	2	32	no	—
73	C₂²×D₁₇	—	2	12	no	—
74	C₂²×D₁₇	—	2	12	no	—
75	C₂²×D₁₇	C₂×C₁₆	1	24	yes	complemented
76	C₂²×D₁₇	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₁₇	C₂²⋊C₈	1	12	yes	—
78	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	6	yes	—
79	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
80	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
81	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	—	2	12	no	—
83	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
84	C₄×D₁₇	—	2	4	no	—
85	C₄×D₁₇	—	2	4	no	—
86	C₄×D₁₇	—	2	24	no	—
87	C₈×C₁₇	C₂×C₂²⋊C₄	1	6	yes	—
88	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
90	#3622	—	2	4	no	—
91	C₂⁴×C₈	—	17	1	no	—
92	C₂³×C₁₆	—	17	1	no	—
93	C₂³×C₁₆	D₁₇	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊C₁₆	—	17	12	no	—
95	C₂²×C₁₇	C₂²⋊C₁₆	1	16	yes	complemented
96	C₂²×C₁₇	C₂×C₂²⋊C₈	1	6	yes	—
97	C₂²×C₁₇	C₂²×C₁₆	1	6	yes	—
98	C₂²×C₁₇	C₂×M₃₂	1	6	yes	—
99	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
100	C₂×D₁₇	—	2	4	no	—
101	C₂×D₁₇	—	2	48	no	—
102	C₂×D₁₇	—	2	4	no	—
103	C₂×D₁₇	C₂²×C₁₆	1	8	yes	complemented
104	C₄×C₁₇	C₂×C₂²⋊C₈	1	6	yes	—
105	C₄×C₁₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
107	Dic₁₇	—	2	4	no	—
108	C₂⁴×C₄	—	17	1	no	—
109	C₂³×C₈	—	17	1	no	—
110	C₂³×C₈	—	17	6	no	—
111	C₂³×C₈	—	17	6	no	—
112	C₂³×C₈	—	34	4	no	—
113	C₂³×C₈	—	34	4	no	—
114	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
115	C₂²×C₁₆	—	2	4	no	—
116	C₂²×C₁₆	—	17	6	no	—
117	C₂²×C₁₆	—	34	4	no	—
118	C₂²×C₁₆	C₂×D₁₇	1	6	yes	complemented
119	C₂²⋊C₁₆	—	17	16	no	—
120	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	12	yes	complemented
121	C₂×C₁₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
123	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
124	D₁₇	—	2	8	no	—
125	C₂⁵	—	17	1	no	—
126	C₂³×C₄	—	17	1	no	—
127	C₂³×C₄	—	17	6	no	—
128	C₂³×C₄	—	17	6	no	—
129	C₂³×C₄	—	34	4	no	—
130	C₂³×C₄	—	34	4	no	—
131	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
132	C₂²×C₈	—	17	6	no	—
133	C₂²×C₈	—	17	12	no	—
134	C₂²×C₈	—	17	4	no	—
135	C₂²×C₈	—	34	4	no	—
136	C₂²×C₈	—	34	12	no	—
137	C₂²×C₈	—	34	12	no	—
138	C₂²×C₈	—	34	24	no	—
139	C₂²×C₈	C₄×D₁₇	1	3	yes	—
140	C₂²×C₈	D₆₈	1	4	yes	—
141	C₂²×C₈	C₂²⋊D₁₇	1	4	yes	—
142	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
143	C₂×C₁₆	—	2	12	no	—
144	C₂×C₁₆	—	17	4	no	—
145	C₂×C₁₆	—	34	12	no	—
146	C₂×C₁₆	C₂²×D₁₇	1	4	yes	complemented
147	C₁₇	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
148	C₂⁴	—	17	8	no	—
149	C₂⁴	—	17	6	no	—
150	C₂⁴	—	34	4	no	—
151	C₂⁴	—	34	4	no	—
152	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
153	C₂²×C₄	—	17	6	no	—
154	C₂²×C₄	—	17	4	no	—
155	C₂²×C₄	—	17	12	no	—
156	C₂²×C₄	—	34	4	no	—
157	C₂²×C₄	—	34	24	no	—
158	C₂²×C₄	—	34	12	no	—
159	C₂²×C₄	—	34	12	no	—
160	C₂²×C₄	C₈×D₁₇	1	3	yes	—
161	C₂²×C₄	#63218	1	4	yes	—
162	C₂²×C₄	#63227	1	4	yes	—
163	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
164	C₂×C₈	—	17	4	no	—
165	C₂×C₈	—	34	12	no	—
166	C₂×C₈	—	34	4	no	—
167	C₂×C₈	—	34	4	no	—
168	C₂×C₈	—	34	24	no	—
169	C₂×C₈	#63227	1	12	yes	—
170	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
171	C₂×C₈	C₂×D₆₈	1	6	yes	—
172	C₂×C₈	C₂×C₂²⋊D₁₇	1	6	yes	—
173	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₁₆	—	34	4	no	—
176	C₂³	—	17	24	no	—
177	C₂³	—	17	4	no	—
178	C₂³	—	34	12	no	—
179	C₂³	—	34	12	no	—
180	C₂³	—	34	32	no	—
181	C₂³	C₁₆×D₁₇	1	4	yes	—
182	C₂³	#10587469	1	4	yes	—
183	C₂³	#10587491	1	4	yes	—
184	C₂³	C₂×C₈×D₁₇	1	3	yes	—
185	C₂×C₄	—	17	4	no	—
186	C₂×C₄	—	34	12	no	—
187	C₂×C₄	—	34	4	no	—
188	C₂×C₄	—	34	24	no	—
189	C₂×C₄	—	34	4	no	—
190	C₂×C₄	#10587491	1	12	yes	—
191	C₂×C₄	C₂×C₈×D₁₇	1	3	yes	—
192	C₂×C₄	C₂×#63218	1	6	yes	—
193	C₂×C₄	C₂×#63227	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
195	C₈	—	34	4	no	—
196	C₈	C₂×#63227	1	6	yes	—
197	C₈	C₂²×D₆₈	1	1	yes	characteristic
198	C₈	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
199	C₂²	—	17	8	no	—
200	C₂²	—	34	4	no	—
201	C₂²	—	34	48	no	—
202	C₂²	—	34	4	no	—
203	C₂²	#11762588	1	16	yes	direct product factor
204	C₂²	C₂×#10587491	1	6	yes	—
205	C₂²	C₂×C₁₆×D₁₇	1	6	yes	—
206	C₂²	C₂×#10587469	1	6	yes	—
207	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
208	C₄	—	34	4	no	—
209	C₄	C₂×#10587491	1	6	yes	—
210	C₄	C₂²×#63227	1	1	yes	characteristic
211	C₄	C₂²×#63218	1	1	yes	characteristic
212	C₂	—	34	8	no	—
213	C₂	C₂×#11762588	1	12	yes	direct product factor
214	C₂	C₂²×#10587491	1	1	yes	characteristic
215	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic
216	C₂	C₂²×#10587469	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂⁴×C₁₇	C₂×C₈
prime radical	C₂⁴×D₁₇	C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#63214	C₂	semidirect
3. C₂×#11762588	C₂	direct (N × H)
4. C₂³×C₁₆×C₁₇	C₂	semidirect
12. C₂²×#63214	C₂²	semidirect
13. #11762588	C₂²	direct (N × H)
16. C₂²×C₁₆×C₁₇	C₂²	semidirect
36. C₂×C₁₆×C₁₇	C₂³	semidirect
38. C₂×#63214	C₂³	semidirect
42. C₂³×D₁₇	C₁₆	semidirect
75. C₂²×D₁₇	C₂×C₁₆	semidirect
93. C₂³×C₁₆	D₁₇	semidirect
95. C₂²×C₁₇	C₂²⋊C₁₆	semidirect
103. C₂×D₁₇	C₂²×C₁₆	semidirect
118. C₂²×C₁₆	C₂×D₁₇	semidirect
120. C₂×C₁₇	C₂×C₂²⋊C₁₆	semidirect
146. C₂×C₁₆	C₂²×D₁₇	semidirect
147. C₁₇	C₂²×C₂²⋊C₁₆	semidirect
203. C₂²	#11762588	direct (N × H)
213. C₂	C₂×#11762588	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×C₈×D₁₇	2
C₂³×#63214	2
C₂×#11762588	2
C₂³×C₁₆×C₁₇	2
C₂²×C₂²⋊C₁₆	17

Subgroups — C22×#11762588 #17186988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762588 #17186988