Subgroups — C₂³×#10587490 #17898013

Order 4352.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63223	C₂	1	2	yes	maximal
2	C₂²×#10587490	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63223	—	2	14	no	—
5	C₂²×#63223	C₂²	1	28	yes	—
6	C₂³×#3622	—	2	2	no	—
7	C₂×#10587490	C₂²	1	112	yes	direct product factor
8	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₇	C₄	1	14	yes	—
10	C₂³×C₈×C₁₇	C₄	1	2	yes	—
11	C₂³×C₈×C₁₇	C₂²	1	14	yes	—
12	C₂⁵×C₁₇	Q₈	1	1	yes	characteristic
13	C₂³×C₄×C₁₇	C₂×C₄	1	7	yes	—
14	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₇	D₄	1	7	yes	—
16	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₇	Q₈	1	7	yes	—
18	C₂³×C₄×C₁₇	C₂³	1	7	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	84	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	28	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	28	yes	—
22	C₂×#63223	—	2	84	no	—
23	C₂×#63223	C₂³	1	56	yes	—
24	C₂²×#3622	—	2	28	no	—
25	#10587490	C₂³	1	64	yes	direct product factor
26	C₂⁴×C₁₇	—	2	8	no	—
27	C₂⁴×C₁₇	C₄⋊C₄	1	7	yes	—
28	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂⁴×C₁₇	C₂×Q₈	1	7	yes	—
30	C₂²×C₄×C₁₇	C₄⋊C₄	1	42	yes	—
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	7	yes	—
32	C₂²×C₄×C₁₇	C₄⋊C₄	1	7	yes	—
33	C₂²×C₄×C₁₇	C₂²×C₄	1	21	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	7	yes	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	7	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	21	yes	—
37	C₂²×C₄×C₁₇	C₂×Q₈	1	21	yes	—
38	C₂²×C₄×C₁₇	C₂⁴	1	7	yes	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	56	yes	—
40	C₂×C₈×C₁₇	C₂²×C₄	1	56	yes	—
41	C₂×C₈×C₁₇	C₂⁴	1	8	yes	—
42	#63223	—	2	56	no	—
43	#63223	C₂⁴	1	16	yes	—
44	C₂×#3622	—	2	56	no	—
45	C₂³×#108	—	17	1	no	maximal, Sylow
46	C₂³×C₁₇	—	2	28	no	—
47	C₂³×C₁₇	—	2	28	no	—
48	C₂³×C₁₇	#108	1	8	yes	complemented
49	C₂³×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
50	C₂³×C₁₇	C₂×C₄⋊C₄	1	7	yes	—
51	C₂³×C₁₇	C₂²×Q₈	1	7	yes	—
52	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	42	yes	—
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
55	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
57	C₂×C₄×C₁₇	C₂²×D₄	1	7	yes	—
58	C₂×C₄×C₁₇	C₂²×D₄	1	7	yes	—
59	C₂×C₄×C₁₇	C₂²×Q₈	1	7	yes	—
60	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
61	C₈×C₁₇	C₂³×C₄	1	16	yes	—
62	#3622	—	2	16	no	—
63	C₂³×M₁₆	—	17	2	no	—
64	C₂⁴×C₈	D₁₇	1	1	yes	characteristic
65	C₂²×#108	—	17	28	no	—
66	C₂²×C₁₇	—	2	42	no	—
67	C₂²×C₁₇	—	2	14	no	—
68	C₂²×C₁₇	C₂×#108	1	28	yes	complemented
69	C₂²×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
70	C₂²×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
71	C₂²×C₁₇	C₂³×Q₈	1	1	yes	characteristic
72	C₄×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
73	C₄×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
74	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
76	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
77	C₂²×M₁₆	—	17	28	no	—
78	C₂²×M₁₆	—	34	14	no	—
79	C₂³×C₈	—	34	2	no	—
80	C₂³×C₈	Dic₁₇	1	2	yes	—
81	C₂³×C₈	Dic₁₇	1	14	yes	—
82	C₂³×C₈	C₂×D₁₇	1	14	yes	—
83	C₂×#108	—	17	112	no	—
84	C₂×C₁₇	—	2	7	no	—
85	C₂×C₁₇	—	2	1	no	—
86	C₂×C₁₇	C₂²×#108	1	14	yes	complemented
87	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
88	C₂⁵	Dic₃₄	1	1	yes	characteristic
89	C₂³×C₄	Dic₃₄	1	7	yes	—
90	C₂³×C₄	D₆₈	1	7	yes	—
91	C₂³×C₄	D₆₈	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
93	C₂³×C₄	C₂×Dic₁₇	1	7	yes	—
94	C₂³×C₄	C₂²×D₁₇	1	7	yes	—
95	C₂×M₁₆	—	17	56	no	—
96	C₂×M₁₆	—	34	84	no	—
97	C₂²×C₈	—	34	28	no	—
98	C₂²×C₈	C₂×Dic₁₇	1	84	yes	—
99	C₂²×C₈	C₂×Dic₁₇	1	28	yes	—
100	C₂²×C₈	C₂²×D₁₇	1	28	yes	—
101	#108	—	17	64	no	—
102	C₁₇	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	—	2	8	no	—
104	C₂⁴	C₄⋊Dic₁₇	1	7	yes	—
105	C₂⁴	C₄⋊Dic₁₇	1	1	yes	characteristic
106	C₂⁴	C₂×Dic₃₄	1	7	yes	—
107	C₂²×C₄	C₄⋊Dic₁₇	1	42	yes	—
108	C₂²×C₄	C₄⋊Dic₁₇	1	7	yes	—
109	C₂²×C₄	C₄⋊Dic₁₇	1	7	yes	—
110	C₂²×C₄	C₂×Dic₃₄	1	21	yes	—
111	C₂²×C₄	C₂×D₆₈	1	7	yes	—
112	C₂²×C₄	C₂×D₆₈	1	21	yes	—
113	C₂²×C₄	C₂²×Dic₁₇	1	21	yes	—
114	C₂²×C₄	C₂²×Dic₁₇	1	7	yes	—
115	C₂²×C₄	C₂³×D₁₇	1	7	yes	—
116	M₁₆	—	17	16	no	—
117	M₁₆	—	34	56	no	—
118	C₂×C₈	—	34	56	no	—
119	C₂×C₈	C₂²×Dic₁₇	1	56	yes	—
120	C₂×C₈	C₂²×Dic₁₇	1	56	yes	—
121	C₂×C₈	C₂³×D₁₇	1	8	yes	—
122	C₂³	—	2	28	no	—
123	C₂³	—	2	28	no	—
124	C₂³	#10587490	1	8	yes	direct product factor
125	C₂³	C₂×C₄⋊Dic₁₇	1	21	yes	—
126	C₂³	C₂×C₄⋊Dic₁₇	1	7	yes	—
127	C₂³	C₂²×Dic₃₄	1	7	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₁₇	1	42	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₁₇	1	21	yes	—
130	C₂×C₄	C₂×C₄⋊Dic₁₇	1	21	yes	—
131	C₂×C₄	C₂²×Dic₃₄	1	7	yes	—
132	C₂×C₄	C₂²×D₆₈	1	7	yes	—
133	C₂×C₄	C₂²×D₆₈	1	7	yes	—
134	C₂×C₄	C₂³×Dic₁₇	1	7	yes	—
135	C₂×C₄	C₂³×Dic₁₇	1	7	yes	—
136	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
137	C₈	—	34	16	no	—
138	C₈	C₂³×Dic₁₇	1	16	yes	—
139	C₂²	—	2	14	no	—
140	C₂²	—	2	42	no	—
141	C₂²	C₂×#10587490	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊Dic₁₇	1	7	yes	—
143	C₂²	C₂²×C₄⋊Dic₁₇	1	7	yes	—
144	C₂²	C₂³×Dic₃₄	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊Dic₁₇	1	7	yes	—
146	C₄	C₂²×C₄⋊Dic₁₇	1	7	yes	—
147	C₄	C₂⁴×Dic₁₇	1	1	yes	characteristic
148	C₄	C₂³×D₆₈	1	1	yes	characteristic
149	C₂	—	2	7	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂²×#10587490	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₆₈
commutator	C₄×C₁₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂⁴×C₈×C₁₇	C₂
socle	C₂⁴×C₁₇	C₄⋊C₄
prime radical	C₂⁵×C₁₇	Q₈
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂⁴×C₈	D₁₇
hypocenter	C₁₇	C₂³×#108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#10587490	C₂	direct (N × H)
7. C₂×#10587490	C₂²	direct (N × H)
25. #10587490	C₂³	direct (N × H)
48. C₂³×C₁₇	#108	semidirect
68. C₂²×C₁₇	C₂×#108	semidirect
86. C₂×C₁₇	C₂²×#108	semidirect
102. C₁₇	C₂³×#108	semidirect
124. C₂³	#10587490	direct (N × H)
141. C₂²	C₂×#10587490	direct (N × H)
151. C₂	C₂²×#10587490	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×#63223	2
C₂²×#10587490	2
C₂⁴×C₈×C₁₇	2
C₂³×#108	17

Subgroups — C23×#10587490 #17898013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587490 #17898013