Subgroups — #17186965

Order 4352.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 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	2	yes	maximal
2	#14530381	C₂	1	8	yes	maximal, complemented
3	C₄×#10587483	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₄	1	16	yes	complemented
6	C₂×#10587483	C₂²	1	4	yes	—
7	C₂×#10587483	C₂²	1	4	yes	—
8	C₂²×#63214	C₂²	1	4	yes	—
9	#11762585	C₂²	1	8	yes	complemented
10	C₄×#63214	C₂²	1	8	yes	complemented
11	C₂²×M₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
14	C₁₇×#405	C₄	1	4	yes	—
15	C₁₇×#405	C₂²	1	4	yes	—
16	C₂×C₁₇×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₈	1	8	yes	—
20	C₂×M₁₆×C₁₇	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₁₇×#117	C₈	1	4	yes	—
28	C₁₇×#117	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
32	C₁₇×#97	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₇	C₂³	1	4	yes	—
35	#10587483	C₂×C₄	1	16	yes	—
36	#10587483	C₂×C₄	1	16	yes	complemented
37	C₂×#63214	C₂×C₄	1	16	yes	complemented
38	C₂×#63214	C₂³	1	8	yes	—
39	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₄²	1	4	yes	—
41	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
47	M₁₆×C₁₇	C₂×C₈	1	16	yes	—
48	C₂×C₈×C₁₇	C₄²	1	4	yes	—
49	C₂×C₈×C₁₇	C₄²	1	8	yes	—
50	C₂×C₈×C₁₇	C₂×C₈	1	8	yes	—
51	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
59	C₂²⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
60	C₄²×C₁₇	C₂×C₈	1	4	yes	—
61	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
62	#63214	C₂²×C₄	1	16	yes	—
63	#33324	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	—	4	2	no	—
65	C₂³×C₁₇	C₄×C₈	1	4	yes	—
66	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
67	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	4	4	no	—
69	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₁₇	C₄×C₈	1	8	yes	—
71	C₂×C₄×C₁₇	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
73	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₁₇	C₂×C₄²	1	4	yes	—
80	C₈×C₁₇	C₂×C₄²	1	2	yes	—
81	C₈×C₁₇	C₂²×C₈	1	8	yes	—
82	C₈×C₁₇	C₂³×C₄	1	2	yes	—
83	C₂²×M₃₂	—	17	2	no	—
84	C₂×#405	D₁₇	1	1	yes	characteristic
85	#2108	—	17	8	no	—
86	C₄×M₃₂	—	17	4	no	—
87	C₂²×C₁₇	—	2	2	no	—
88	C₂²×C₁₇	—	2	4	no	—
89	C₂²×C₁₇	—	4	4	no	—
90	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
91	C₂²×C₁₇	C₂×C₄×C₈	1	4	yes	—
92	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₁₇	—	2	8	no	—
94	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
95	C₄×C₁₇	C₂×C₄×C₈	1	4	yes	—
96	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
98	C₂²×M₁₆	Dic₁₇	1	1	yes	characteristic
99	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
100	C₂×#117	Dic₁₇	1	1	yes	characteristic
101	C₂×M₃₂	—	17	16	no	—
102	C₂×M₃₂	—	17	4	no	—
103	C₂×M₃₂	—	17	4	no	—
104	C₂²×C₁₆	—	17	4	no	—
105	#405	Dic₁₇	1	4	yes	—
106	#405	C₂×D₁₇	1	4	yes	—
107	C₂×#97	Dic₁₇	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
109	#346	—	17	8	no	—
110	C₄×C₁₆	—	17	8	no	—
111	C₂×C₁₇	—	2	2	no	—
112	C₂×C₁₇	—	2	4	no	—
113	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
114	C₂×C₁₇	#3407	1	2	yes	—
115	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
116	C₂×M₁₆	#3622	1	8	yes	—
117	C₂×M₁₆	C₂×Dic₁₇	1	4	yes	—
118	C₂²×C₈	C₄×D₁₇	1	8	yes	—
119	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
120	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
122	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
124	#117	#3622	1	4	yes	—
125	#117	C₂×Dic₁₇	1	4	yes	—
126	C₂×C₄⋊C₄	#3622	1	2	yes	—
127	C₂×C₂²⋊C₄	#3622	1	2	yes	—
128	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
129	M₃₂	—	17	16	no	—
130	M₃₂	—	17	16	no	—
131	C₂×C₁₆	—	17	16	no	—
132	C₂×C₁₆	—	17	8	no	—
133	#97	C₂×Dic₁₇	1	8	yes	—
134	C₄×C₈	C₂×Dic₁₇	1	4	yes	—
135	C₄×C₈	C₂²×D₁₇	1	4	yes	—
136	C₁₇	#33324	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	C₂×#3622	1	1	yes	characteristic
138	C₂²×C₄	C₂×#3622	1	4	yes	—
139	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
140	C₂²×C₄	C₂×#3622	1	2	yes	—
141	C₂²×C₄	C₂×#3622	1	4	yes	—
142	C₂²×C₄	C₄×Dic₁₇	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
145	M₁₆	C₂×#3622	1	16	yes	—
146	C₂×C₈	C₂×#3622	1	8	yes	—
147	C₂×C₈	C₄×Dic₁₇	1	4	yes	—
148	C₂×C₈	C₄×Dic₁₇	1	8	yes	—
149	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
150	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
151	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
152	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
153	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
154	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
155	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
156	C₄⋊C₄	C₂×#3622	1	8	yes	—
157	C₂²⋊C₄	C₂×#3622	1	8	yes	—
158	C₄²	C₂×#3622	1	4	yes	—
159	C₄²	C₂²×Dic₁₇	1	4	yes	—
160	C₁₆	—	17	16	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#3622	1	4	yes	—
163	C₂³	C₂²×#3622	1	2	yes	—
164	C₂³	C₂²×#3622	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#3622	1	4	yes	—
167	C₂×C₄	C₄×#3622	1	8	yes	—
168	C₂×C₄	C₂²×#3622	1	8	yes	—
169	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
170	C₂×C₄	C₂²×#3622	1	2	yes	—
171	C₂×C₄	C₂²×#3622	1	2	yes	—
172	C₂×C₄	C₂²×#3622	1	4	yes	—
173	C₂×C₄	C₂×C₄×Dic₁₇	1	4	yes	—
174	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
175	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
176	C₈	C₂²×#3622	1	8	yes	—
177	C₈	C₂×C₄×Dic₁₇	1	4	yes	—
178	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
179	C₈	C₂²×C₄×D₁₇	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#3622	1	2	yes	—
184	C₂²	C₂×C₄×#3622	1	4	yes	—
185	C₂²	C₂³×#3622	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#3622	1	2	yes	—
188	C₄	C₂×C₄×#3622	1	4	yes	—
189	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
190	C₄	C₂³×#3622	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#3622	1	1	yes	characteristic
194	C₂	#14541887	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Fitting	C₂×C₁₇×#405	C₂
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#405	D₁₇
hypocenter	C₁₇	#33324
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #14530381	C₂	semidirect
3. C₄×#10587483	C₂	semidirect
5. C₂×#10587483	C₄	semidirect
9. #11762585	C₂²	semidirect
10. C₄×#63214	C₂²	semidirect
36. #10587483	C₂×C₄	semidirect
37. C₂×#63214	C₂×C₄	semidirect
136. C₁₇	#33324	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#10587483	2
#14530381	2
C₄×#10587483	2
C₂×C₁₇×#405	2
#33324	17