Subgroups — C₂³×#10587493 #17898012

Order 4352.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 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	1	yes	maximal, characteristic
2	C₂⁴×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂²×#10587493	C₂	1	28	yes	maximal, direct product factor
4	C₂³×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	—	2	7	no	—
6	C₂²×#63223	C₂²	1	14	yes	—
7	C₂³×#3622	—	2	1	no	—
8	C₂³×#3622	C₄	1	14	yes	—
9	C₂³×#3622	C₄	1	2	yes	—
10	C₂³×#3622	C₂²	1	14	yes	—
11	C₂×#10587493	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₇	—	2	7	no	—
14	C₂²×M₁₆×C₁₇	C₂²	1	14	yes	—
15	C₂³×C₈×C₁₇	—	2	1	no	—
16	C₂⁵×C₁₇	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₇	C₂×C₄	1	7	yes	—
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₇	D₄	1	7	yes	—
20	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₇	Q₈	1	7	yes	—
22	C₂³×C₄×C₁₇	C₂³	1	7	yes	—
23	C₂×M₁₆×C₁₇	—	2	42	no	—
24	C₂×M₁₆×C₁₇	C₂³	1	28	yes	—
25	C₂²×C₈×C₁₇	—	2	14	no	—
26	C₂×#63223	—	2	42	no	—
27	C₂×#63223	C₂³	1	28	yes	—
28	C₂²×#3622	—	2	14	no	—
29	C₂²×#3622	C₂×C₄	1	84	yes	—
30	C₂²×#3622	C₂×C₄	1	28	yes	—
31	C₂²×#3622	C₂³	1	28	yes	—
32	#10587493	C₂³	1	64	yes	direct product factor
33	C₂⁴×C₁₇	—	2	8	no	—
34	C₂⁴×C₁₇	C₄⋊C₄	1	7	yes	—
35	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂⁴×C₁₇	C₂×Q₈	1	7	yes	—
37	C₂²×C₄×C₁₇	C₄⋊C₄	1	42	yes	—
38	C₂²×C₄×C₁₇	C₄⋊C₄	1	7	yes	—
39	C₂²×C₄×C₁₇	C₄⋊C₄	1	7	yes	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	21	yes	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	7	yes	—
42	C₂²×C₄×C₁₇	C₂×D₄	1	7	yes	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	21	yes	—
44	C₂²×C₄×C₁₇	C₂×Q₈	1	21	yes	—
45	C₂²×C₄×C₁₇	C₂⁴	1	7	yes	—
46	M₁₆×C₁₇	—	2	28	no	—
47	M₁₆×C₁₇	C₂⁴	1	8	yes	—
48	C₂×C₈×C₁₇	—	2	28	no	—
49	#63223	—	2	28	no	—
50	#63223	C₂⁴	1	8	yes	—
51	C₂×#3622	—	2	28	no	—
52	C₂×#3622	C₂²×C₄	1	56	yes	—
53	C₂×#3622	C₂²×C₄	1	56	yes	—
54	C₂×#3622	C₂⁴	1	8	yes	—
55	C₂³×#108	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	—	2	28	no	—
57	C₂³×C₁₇	—	2	28	no	—
58	C₂³×C₁₇	#108	1	8	yes	complemented
59	C₂³×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
60	C₂³×C₁₇	C₂×C₄⋊C₄	1	7	yes	—
61	C₂³×C₁₇	C₂²×Q₈	1	7	yes	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	42	yes	—
63	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	21	yes	—
65	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
66	C₂×C₄×C₁₇	C₂³×C₄	1	7	yes	—
67	C₂×C₄×C₁₇	C₂²×D₄	1	7	yes	—
68	C₂×C₄×C₁₇	C₂²×D₄	1	7	yes	—
69	C₂×C₄×C₁₇	C₂²×Q₈	1	7	yes	—
70	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
71	C₈×C₁₇	—	2	8	no	—
72	#3622	—	2	8	no	—
73	#3622	C₂³×C₄	1	16	yes	—
74	C₂³×M₁₆	—	17	1	no	—
75	C₂³×M₁₆	D₁₇	1	1	yes	characteristic
76	C₂⁴×C₈	—	17	1	no	—
77	C₂²×#108	—	17	28	no	—
78	C₂²×C₁₇	—	2	42	no	—
79	C₂²×C₁₇	—	2	14	no	—
80	C₂²×C₁₇	C₂×#108	1	28	yes	complemented
81	C₂²×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
82	C₂²×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
83	C₂²×C₁₇	C₂³×Q₈	1	1	yes	characteristic
84	C₄×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
85	C₄×C₁₇	C₂²×C₄⋊C₄	1	7	yes	—
86	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
87	C₄×C₁₇	C₂³×D₄	1	1	yes	characteristic
88	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
89	C₂²×M₁₆	—	2	7	no	—
90	C₂²×M₁₆	—	17	14	no	—
91	C₂²×M₁₆	—	34	7	no	—
92	C₂²×M₁₆	C₂×D₁₇	1	14	yes	—
93	C₂³×C₈	—	2	1	no	—
94	C₂³×C₈	—	17	2	no	—
95	C₂³×C₈	—	17	14	no	—
96	C₂³×C₈	—	17	14	no	—
97	C₂³×C₈	—	34	1	no	—
98	C₂×#108	—	17	112	no	—
99	C₂×C₁₇	—	2	7	no	—
100	C₂×C₁₇	—	2	1	no	—
101	C₂×C₁₇	C₂²×#108	1	14	yes	complemented
102	C₂×C₁₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
103	C₂⁵	Dic₃₄	1	1	yes	characteristic
104	C₂³×C₄	Dic₃₄	1	7	yes	—
105	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
106	C₂³×C₄	C₄×D₁₇	1	7	yes	—
107	C₂³×C₄	C₂²⋊D₁₇	1	7	yes	—
108	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
109	C₂³×C₄	C₂²×D₁₇	1	7	yes	—
110	C₂×M₁₆	—	2	42	no	—
111	C₂×M₁₆	—	17	28	no	—
112	C₂×M₁₆	—	34	42	no	—
113	C₂×M₁₆	C₂²×D₁₇	1	28	yes	—
114	C₂²×C₈	—	2	14	no	—
115	C₂²×C₈	—	17	84	no	—
116	C₂²×C₈	—	17	28	no	—
117	C₂²×C₈	—	17	28	no	—
118	C₂²×C₈	—	34	14	no	—
119	#108	—	17	64	no	—
120	C₁₇	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
121	C₂⁴	—	2	8	no	—
122	C₂⁴	#63225	1	7	yes	—
123	C₂⁴	#63225	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₃₄	1	7	yes	—
125	C₂²×C₄	#63225	1	42	yes	—
126	C₂²×C₄	#63225	1	7	yes	—
127	C₂²×C₄	#63225	1	7	yes	—
128	C₂²×C₄	C₂×Dic₃₄	1	21	yes	—
129	C₂²×C₄	C₂×C₄×D₁₇	1	21	yes	—
130	C₂²×C₄	C₂×C₄×D₁₇	1	7	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₁₇	1	7	yes	—
132	C₂²×C₄	C₂×C₂²⋊D₁₇	1	21	yes	—
133	C₂²×C₄	C₂³×D₁₇	1	7	yes	—
134	M₁₆	—	2	28	no	—
135	M₁₆	—	17	8	no	—
136	M₁₆	—	34	28	no	—
137	M₁₆	C₂³×D₁₇	1	8	yes	—
138	C₂×C₈	—	2	28	no	—
139	C₂×C₈	—	17	56	no	—
140	C₂×C₈	—	17	56	no	—
141	C₂×C₈	—	17	8	no	—
142	C₂×C₈	—	34	28	no	—
143	C₂³	—	2	28	no	—
144	C₂³	—	2	28	no	—
145	C₂³	#10587493	1	8	yes	direct product factor
146	C₂³	C₂×#63225	1	21	yes	—
147	C₂³	C₂×#63225	1	7	yes	—
148	C₂³	C₂²×Dic₃₄	1	7	yes	—
149	C₂×C₄	C₂×#63225	1	42	yes	—
150	C₂×C₄	C₂×#63225	1	21	yes	—
151	C₂×C₄	C₂×#63225	1	21	yes	—
152	C₂×C₄	C₂²×Dic₃₄	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₁₇	1	7	yes	—
154	C₂×C₄	C₂²×C₄×D₁₇	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₁₇	1	7	yes	—
156	C₂×C₄	C₂²×C₂²⋊D₁₇	1	7	yes	—
157	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
158	C₈	—	2	8	no	—
159	C₈	—	17	16	no	—
160	C₈	—	34	8	no	—
161	C₂²	—	2	14	no	—
162	C₂²	—	2	42	no	—
163	C₂²	C₂×#10587493	1	28	yes	direct product factor
164	C₂²	C₂²×#63225	1	7	yes	—
165	C₂²	C₂²×#63225	1	7	yes	—
166	C₂²	C₂³×Dic₃₄	1	1	yes	characteristic
167	C₄	C₂²×#63225	1	7	yes	—
168	C₄	C₂²×#63225	1	7	yes	—
169	C₄	C₂³×C₄×D₁₇	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₁₇	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#10587493	1	14	yes	direct product factor
174	C₂	C₂³×#63225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂³×M₁₆×C₁₇	C₂
socle	C₂⁴×C₁₇	C₄⋊C₄
prime radical	C₂⁵×C₁₇	Q₈
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂³×M₁₆	D₁₇
hypocenter	C₁₇	C₂³×#108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#10587493	C₂	direct (N × H)
11. C₂×#10587493	C₂²	direct (N × H)
32. #10587493	C₂³	direct (N × H)
58. C₂³×C₁₇	#108	semidirect
80. C₂²×C₁₇	C₂×#108	semidirect
101. C₂×C₁₇	C₂²×#108	semidirect
120. C₁₇	C₂³×#108	semidirect
145. C₂³	#10587493	direct (N × H)
163. C₂²	C₂×#10587493	direct (N × H)
173. C₂	C₂²×#10587493	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
C₂⁴×#3622	2
C₂²×#10587493	2
C₂³×M₁₆×C₁₇	2
C₂³×#108	17

Subgroups — C23×#10587493 #17898012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587493 #17898012