Subgroups — C₂²×#11762639 #17187376

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587490	C₂	1	2	yes	maximal
2	C₂×#11762639	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63223	—	2	2	no	—
5	C₂×#10587490	—	2	6	no	—
6	C₂×#10587490	C₂²	1	12	yes	—
7	#11762639	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₇	C₄	1	6	yes	—
10	C₂²×C₁₆×C₁₇	C₄	1	2	yes	—
11	C₂²×C₁₆×C₁₇	C₂²	1	6	yes	—
12	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	D₄	1	3	yes	—
16	C₂²×C₈×C₁₇	Q₈	1	3	yes	—
17	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
19	C₂×#63223	—	2	12	no	—
20	C₂×#63223	—	4	6	no	—
21	C₂²×#3622	—	4	2	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	—
24	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
25	#10587490	—	2	12	no	—
26	#10587490	C₂³	1	8	yes	—
27	C₂⁴×C₁₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
29	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	D₈	1	3	yes	—
31	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	Q₁₆	1	3	yes	—
33	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	6	yes	—
35	C₂×C₈×C₁₇	C₄⋊C₄	1	3	yes	—
36	C₂×C₈×C₁₇	C₄⋊C₄	1	3	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
38	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
39	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
40	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
41	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
43	#63223	—	2	8	no	—
44	#63223	—	4	12	no	—
45	C₂×#3622	—	4	12	no	—
46	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
47	C₂²×#368	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	—	2	4	no	—
49	C₂³×C₁₇	#107	1	3	yes	—
50	C₂³×C₁₇	#107	1	1	yes	characteristic
51	C₂³×C₁₇	C₂×Q₁₆	1	3	yes	—
52	C₂×C₄×C₁₇	#107	1	6	yes	—
53	C₂×C₄×C₁₇	#107	1	3	yes	—
54	C₂×C₄×C₁₇	#107	1	3	yes	—
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
59	C₂×C₄×C₁₇	C₂×Q₁₆	1	3	yes	—
60	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
61	C₈×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
63	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
64	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
65	#3622	—	4	8	no	—
66	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
67	C₂²×#108	—	17	2	no	—
68	C₂×#368	—	17	12	no	—
69	C₂²×C₁₇	—	2	6	no	—
70	C₂²×C₁₇	—	2	6	no	—
71	C₂²×C₁₇	#368	1	4	yes	complemented
72	C₂²×C₁₇	C₂×#107	1	3	yes	—
73	C₂²×C₁₇	C₂×#107	1	3	yes	—
74	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
75	C₄×C₁₇	C₂×#107	1	3	yes	—
76	C₄×C₁₇	C₂×#107	1	3	yes	—
77	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
79	C₂²×M₁₆	—	34	2	no	—
80	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
81	C₂²×C₁₆	Dic₁₇	1	2	yes	—
82	C₂²×C₁₆	Dic₁₇	1	6	yes	—
83	C₂²×C₁₆	C₂×D₁₇	1	6	yes	—
84	C₂×#108	—	17	12	no	—
85	C₂×#108	—	34	6	no	—
86	#368	—	17	16	no	—
87	C₂×C₁₇	—	2	1	no	—
88	C₂×C₁₇	—	2	3	no	—
89	C₂×C₁₇	C₂×#368	1	6	yes	complemented
90	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
91	C₂³×C₄	D₆₈	1	1	yes	characteristic
92	C₂×M₁₆	—	34	12	no	—
93	C₂×M₁₆	—	68	6	no	—
94	C₂²×C₈	—	68	2	no	—
95	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
96	C₂²×C₈	Dic₃₄	1	3	yes	—
97	C₂²×C₈	D₆₈	1	3	yes	—
98	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
100	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
101	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	—
102	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	—
103	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
104	#108	—	17	8	no	—
105	#108	—	34	12	no	—
106	C₁₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	Dic₆₈	1	1	yes	characteristic
108	C₂²×C₄	D₁₃₆	1	3	yes	—
109	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
110	C₂²×C₄	Dic₆₈	1	3	yes	—
111	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₁₇	1	3	yes	—
113	C₂²×C₄	C₂×D₆₈	1	3	yes	—
114	M₁₆	—	34	8	no	—
115	M₁₆	—	68	12	no	—
116	C₂×C₈	—	68	12	no	—
117	C₂×C₈	C₄⋊Dic₁₇	1	6	yes	—
118	C₂×C₈	C₄⋊Dic₁₇	1	3	yes	—
119	C₂×C₈	C₄⋊Dic₁₇	1	3	yes	—
120	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
121	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
122	C₂×C₈	C₂×D₆₈	1	3	yes	—
123	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
124	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
125	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₁₇	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#10587489	1	3	yes	—
129	C₂³	#10587489	1	1	yes	characteristic
130	C₂³	C₂×Dic₆₈	1	3	yes	—
131	C₂×C₄	#10587489	1	6	yes	—
132	C₂×C₄	#10587489	1	3	yes	—
133	C₂×C₄	#10587489	1	3	yes	—
134	C₂×C₄	C₂×D₁₃₆	1	3	yes	—
135	C₂×C₄	C₂×D₁₃₆	1	3	yes	—
136	C₂×C₄	C₂×Dic₆₈	1	3	yes	—
137	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
138	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
139	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
140	C₈	—	68	8	no	—
141	C₈	C₂×C₄⋊Dic₁₇	1	3	yes	—
142	C₈	C₂×C₄⋊Dic₁₇	1	3	yes	—
143	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
144	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#11762639	1	4	yes	direct product factor
148	C₂²	C₂×#10587489	1	3	yes	—
149	C₂²	C₂×#10587489	1	3	yes	—
150	C₂²	C₂²×Dic₆₈	1	1	yes	characteristic
151	C₄	C₂×#10587489	1	3	yes	—
152	C₄	C₂×#10587489	1	3	yes	—
153	C₄	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
154	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	3	no	—
157	C₂	C₂×#11762639	1	6	yes	direct product factor
158	C₂	C₂²×#10587489	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	#107
prime radical	C₂⁴×C₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₂²×#368
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11762639	C₂	direct (N × H)
7. #11762639	C₂²	direct (N × H)
71. C₂²×C₁₇	#368	semidirect
89. C₂×C₁₇	C₂×#368	semidirect
106. C₁₇	C₂²×#368	semidirect
147. C₂²	#11762639	direct (N × H)
157. C₂	C₂×#11762639	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×#10587490	2
C₂×#11762639	2
C₂³×C₁₆×C₁₇	2
C₂²×#368	17

Subgroups — C22×#11762639 #17187376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762639 #17187376