Subgroups — #17958057

Order 4352.

306 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	#14545141	C₂	1	2	yes	maximal
2	#14543697	C₂	1	2	yes	maximal, complemented
3	C₈×#63216	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#63216	C₄	1	4	yes	—
6	C₂²⋊#63216	C₄	1	4	yes	—
7	C₄×#63216	C₂²	1	4	yes	—
8	C₂×C₄×#3622	C₄	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₈×#3622	C₄	1	1	yes	characteristic
11	C₈×#3622	C₄	1	1	yes	characteristic
12	C₈×#3622	C₂²	1	1	yes	characteristic
13	C₈×#3622	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₈	1	2	yes	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	C₈	1	2	yes	—
18	C₂²×#3622	C₂×C₄	1	2	yes	—
19	C₂×#63216	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₁₇	C₈	1	2	yes	—
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×#3622	C₈	1	2	yes	—
24	C₄×#3622	C₂×C₄	1	2	yes	—
25	C₄×#3622	C₂×C₄	1	1	yes	characteristic
26	C₄×#3622	C₂×C₄	1	1	yes	characteristic
27	C₄×#3622	C₂×C₄	1	1	yes	characteristic
28	C₄×#3622	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₇	C₁₆	1	8	yes	complemented
33	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
35	C₄²×C₁₇	C₄²	1	1	yes	characteristic
36	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×#3622	—	2	4	no	—
40	C₂×#3622	—	2	4	no	—
41	C₂×#3622	C₄²	1	2	yes	—
42	C₂×#3622	C₂×C₈	1	2	yes	—
43	C₂×#3622	C₂×C₈	1	2	yes	—
44	C₂×#3622	C₂²×C₄	1	2	yes	—
45	#63216	—	2	8	no	—
46	#11410	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
54	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₇	C₂×C₁₆	1	4	yes	—
58	C₈×C₁₇	C₂×C₁₆	1	4	yes	—
59	#3622	—	2	2	no	—
60	#3622	—	2	2	no	—
61	#3622	M₃₂	1	4	yes	—
62	#3622	#131	1	4	yes	—
63	#2162	—	17	2	no	—
64	C₂×C₈²	—	17	1	no	—
65	#1312	—	17	2	no	—
66	C₈×C₁₆	—	17	2	no	—
67	C₂²×C₁₇	—	2	1	no	—
68	C₂²×C₁₇	—	2	1	no	—
69	C₂²×C₁₇	C₄×C₁₆	1	2	yes	—
70	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
74	C₄×C₁₇	#405	1	2	yes	—
75	C₄×C₁₇	C₂²×C₁₆	1	2	yes	—
76	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
77	C₂×C₄×C₈	—	17	1	no	—
78	C₂×C₄×C₈	—	17	1	no	—
79	C₂×C₄×C₈	#595	1	1	yes	characteristic
80	C₄⋊C₁₆	—	17	4	no	—
81	C₂²⋊C₁₆	—	17	4	no	—
82	C₄×C₁₆	—	17	4	no	—
83	C₈²	—	17	1	no	—
84	C₈²	—	17	1	no	—
85	C₈²	—	17	1	no	—
86	C₈²	—	17	1	no	—
87	C₂×C₁₇	—	2	2	no	—
88	C₂×C₁₇	#1450	1	1	yes	characteristic
89	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
90	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
91	C₂²×C₈	—	17	2	no	—
92	C₂²×C₈	—	17	2	no	—
93	C₂²×C₈	#3624	1	2	yes	—
94	C₂×C₄²	C₂×#595	1	1	yes	characteristic
95	C₂×C₁₆	—	17	8	no	—
96	C₄×C₈	—	17	2	no	—
97	C₄×C₈	—	17	2	no	—
98	C₄×C₈	—	17	1	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	#3624	1	2	yes	—
103	C₄×C₈	C₂×#595	1	1	yes	characteristic
104	C₄×C₈	C₂×#595	1	1	yes	characteristic
105	C₁₇	#11410	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₄	#63242	1	1	yes	characteristic
107	C₂²×C₄	C₄×#595	1	1	yes	characteristic
108	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
109	C₂×C₈	—	17	2	no	—
110	C₂×C₈	—	17	2	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	34	4	no	—
114	C₂×C₈	—	34	4	no	—
115	C₂×C₈	#63216	1	8	yes	complemented
116	C₂×C₈	C₂×#3624	1	2	yes	—
117	C₂×C₈	C₂×#3624	1	2	yes	—
118	C₄²	#63242	1	1	yes	characteristic
119	C₄²	C₄×#595	1	1	yes	characteristic
120	C₄²	C₂×#3624	1	1	yes	characteristic
121	C₄²	C₂²×#595	1	1	yes	characteristic
122	C₁₆	—	34	8	no	—
123	C₂³	C₄×#3624	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₂×#63216	1	4	yes	—
127	C₂×C₄	C₄×#3624	1	1	yes	characteristic
128	C₂×C₄	C₄×#3624	1	1	yes	characteristic
129	C₂×C₄	C₄×#3624	1	1	yes	characteristic
130	C₂×C₄	C₂×#63242	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
132	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
133	C₈	—	17	4	no	—
134	C₈	—	17	4	no	—
135	C₈	—	34	2	no	—
136	C₈	—	34	2	no	—
137	C₈	C₂×#63216	1	4	yes	—
138	C₈	C₂×#63216	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×#63216	1	2	yes	—
142	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×#63216	1	2	yes	—
146	C₄	#11763772	1	2	yes	—
147	C₄	C₂²×#63216	1	2	yes	—
148	C₄	#11763865	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#14543987	1	1	yes	characteristic
151	C₂	C₂×C₄×#63216	1	1	yes	characteristic
152	C₂	#14544990	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#595
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂³×C₁₇	C₄×C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#11410
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14543697	C₂	semidirect
3. C₈×#63216	C₂	semidirect
32. C₂×C₈×C₁₇	C₁₆	semidirect
105. C₁₇	#11410	semidirect
115. C₂×C₈	#63216	semidirect

Maximal subgroups

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

subgroup	index
#14545141	2
#14543697	2
C₈×#63216	2
C₂×C₈×#3622	2
#11410	17