Subgroups — #17962839

Order 4352.

202 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	C₄⋊#10587467	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#10587467	C₂	1	2	yes	maximal
3	C₄×#10587467	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587467	C₂²	1	4	yes	—
6	C₂²×#63214	C₄	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₄×#63214	C₄	1	1	yes	characteristic
9	C₄×#63214	C₄	1	1	yes	characteristic
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
16	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
17	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
18	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#63214	—	2	2	no	—
21	C₂×#63214	—	2	2	no	—
22	C₂×#63214	C₂×C₄	1	1	yes	characteristic
23	C₂×#63214	C₂×C₄	1	1	yes	characteristic
24	C₂×#63214	C₂×C₄	1	1	yes	characteristic
25	C₂×#63214	C₂³	1	1	yes	characteristic
26	#10587467	—	2	4	no	—
27	C₂²×C₄×C₁₇	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₁₇	—	2	2	no	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₁₆	1	1	yes	characteristic
38	C₄²×C₁₇	C₁₆	1	1	yes	characteristic
39	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
41	#63214	—	2	1	no	—
42	#63214	—	2	1	no	—
43	#63214	M₁₆	1	2	yes	—
44	#63214	#41	1	2	yes	—
45	#13639	—	17	1	no	maximal, Sylow
46	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	C₃₂	1	4	yes	complemented
51	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₇	—	2	1	no	—
58	C₈×C₁₇	—	2	1	no	—
59	C₈×C₁₇	M₃₂	1	2	yes	—
60	C₈×C₁₇	#117	1	1	yes	characteristic
61	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₆	—	17	1	no	—
63	C₄⋊C₃₂	—	17	2	no	—
64	C₂²⋊C₃₂	—	17	2	no	—
65	C₄×C₃₂	—	17	2	no	—
66	C₂²×C₁₇	—	2	1	no	—
67	C₂²×C₁₇	—	2	1	no	—
68	C₂²×C₁₇	C₂×C₃₂	1	2	yes	—
69	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	C₂×C₃₂	1	2	yes	—
73	C₄×C₁₇	C₂×C₃₂	1	2	yes	—
74	C₄×C₁₇	M₆₄	1	2	yes	—
75	C₄×C₁₇	#431	1	1	yes	characteristic
76	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
77	C₂²×C₁₆	—	17	1	no	—
78	C₂²×C₁₆	—	17	1	no	—
79	C₂×C₄×C₈	#595	1	1	yes	characteristic
80	C₂×C₃₂	—	17	4	no	—
81	C₄×C₁₆	—	17	1	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₄×C₁₆	—	17	1	no	—
84	C₄×C₁₆	—	17	1	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	#2162	1	1	yes	characteristic
87	C₂×C₁₇	C₂²×C₃₂	1	1	yes	characteristic
88	C₂×C₁₇	C₂×M₆₄	1	1	yes	characteristic
89	C₂²×C₈	#3624	1	1	yes	characteristic
90	C₂²×C₈	C₂×#595	1	1	yes	characteristic
91	C₂×C₄²	#3624	1	1	yes	characteristic
92	C₂×C₁₆	—	17	1	no	—
93	C₂×C₁₆	—	17	1	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	34	2	no	—
97	C₂×C₁₆	—	34	2	no	—
98	C₄×C₈	#3624	1	1	yes	characteristic
99	C₄×C₈	#3624	1	1	yes	characteristic
100	C₄×C₈	C₂×#595	1	1	yes	characteristic
101	C₄×C₈	C₂×#595	1	1	yes	characteristic
102	C₃₂	—	34	4	no	—
103	C₁₇	#13639	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	#63216	1	1	yes	characteristic
105	C₂²×C₄	#63216	1	1	yes	characteristic
106	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₂×#3624	1	1	yes	characteristic
111	C₂×C₈	C₂×#3624	1	1	yes	characteristic
112	C₂×C₈	C₂×#3624	1	1	yes	characteristic
113	C₂×C₈	C₂²×#595	1	1	yes	characteristic
114	C₄²	#63216	1	1	yes	characteristic
115	C₄²	#63216	1	1	yes	characteristic
116	C₄²	C₂×#3624	1	1	yes	characteristic
117	C₄²	C₂×#3624	1	1	yes	characteristic
118	C₁₆	—	17	2	no	—
119	C₁₆	—	17	2	no	—
120	C₁₆	—	34	1	no	—
121	C₁₆	—	34	1	no	—
122	C₂³	C₂×#63216	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	#10587467	1	4	yes	complemented
127	C₂×C₄	C₂×#63216	1	1	yes	characteristic
128	C₂×C₄	C₂×#63216	1	1	yes	characteristic
129	C₂×C₄	C₂×#63216	1	1	yes	characteristic
130	C₂×C₄	C₂×#63216	1	1	yes	characteristic
131	C₂×C₄	C₂×#63216	1	1	yes	characteristic
132	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	#10587538	1	2	yes	—
136	C₈	D₁₇⋊M₁₆	1	1	yes	characteristic
137	C₈	#10587677	1	1	yes	characteristic
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#10587467	1	2	yes	—
141	C₂²	C₂²×#63216	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	C₂×#10587467	1	2	yes	—
145	C₄	C₂×#10587467	1	2	yes	—
146	C₄	#11763741	1	2	yes	—
147	C₄	#11763825	1	1	yes	characteristic
148	C₄	C₂×#10587538	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#14545141	1	1	yes	characteristic
151	C₂	C₂²×#10587467	1	1	yes	characteristic
152	C₂	C₂×#11763741	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#595
commutator	C₂×C₁₇	C₂²×C₃₂
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×#63214	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#13639
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
1. C₄⋊#10587467	C₂	semidirect
3. C₄×#10587467	C₂	semidirect
50. C₂×C₄×C₁₇	C₃₂	semidirect
103. C₁₇	#13639	semidirect
126. C₂×C₄	#10587467	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#10587467	2
C₂²⋊#10587467	2
C₄×#10587467	2
C₂×C₄×#63214	2
#13639	17