Subgroups — #17961607

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63216	C₂	1	2	yes	maximal
2	#14543822	C₂	1	4	yes	maximal
3	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	C₂²	1	4	yes	—
5	C₄⋊#63216	—	2	4	no	—
6	C₂×C₄×#3622	C₄	1	2	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	2	yes	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₄³×C₁₇	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₈	1	2	yes	—
12	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	C₂×C₄	1	2	yes	—
18	C₂²×#3622	C₂³	1	1	yes	characteristic
19	C₂×#63216	—	2	4	no	—
20	C₂×#63216	—	2	8	no	—
21	C₄×#3622	—	2	4	no	—
22	C₄×#3622	—	2	4	no	—
23	C₄×#3622	—	2	2	no	—
24	C₄×#3622	D₄	1	2	yes	—
25	C₄×#3622	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₁₇	—	2	4	no	—
32	C₄²×C₁₇	—	2	2	no	—
33	C₄²×C₁₇	—	2	2	no	—
34	C₄²×C₁₇	—	2	2	no	—
35	C₄²×C₁₇	C₁₆	1	4	yes	complemented
36	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₄²×C₁₇	M₁₆	1	2	yes	—
38	C₂×#3622	—	2	2	no	—
39	C₂×#3622	—	2	2	no	—
40	C₂×#3622	—	2	2	no	—
41	C₂×#3622	C₄⋊C₄	1	2	yes	—
42	C₂×#3622	M₁₆	1	4	yes	—
43	C₂×#3622	C₂×D₄	1	1	yes	characteristic
44	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
45	C₂×#3622	#41	1	4	yes	—
46	#63216	—	4	4	no	—
47	#12879	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
57	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
59	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#117	1	2	yes	—
62	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
64	#3622	—	2	2	no	—
65	#3622	—	2	4	no	—
66	#3622	#124	1	2	yes	—
67	#3622	#125	1	2	yes	—
68	C₂×C₄⋊C₁₆	—	17	2	no	—
69	C₄²×C₈	—	17	1	no	—
70	#1374	—	17	4	no	—
71	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₁₇	#431	1	2	yes	—
73	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
74	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
75	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	—	2	4	no	—
78	C₄×C₁₇	—	2	4	no	—
79	C₄×C₁₇	C₄⋊C₁₆	1	4	yes	—
80	C₄×C₁₇	#382	1	1	yes	characteristic
81	C₄×C₁₇	#423	1	1	yes	characteristic
82	C₄×C₁₇	#432	1	2	yes	—
83	C₂²×C₁₆	—	17	4	no	—
84	C₂×C₄×C₈	—	17	1	no	—
85	C₂×C₄×C₈	—	17	2	no	—
86	C₂×C₄×C₈	—	17	1	no	—
87	C₂×C₄×C₈	—	17	2	no	—
88	C₄³	#595	1	1	yes	characteristic
89	C₄⋊C₁₆	—	34	4	no	—
90	C₂×C₁₇	#1831	1	1	yes	characteristic
91	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
92	C₂×C₁₇	#2150	1	1	yes	characteristic
93	C₂×C₁₇	#2162	1	2	yes	—
94	C₂×C₁₇	#2163	1	2	yes	—
95	C₂²×C₈	—	17	1	no	—
96	C₂²×C₈	—	17	2	no	—
97	C₂²×C₈	—	17	1	no	—
98	C₂×C₄²	#3624	1	2	yes	—
99	C₂×C₄²	#3624	1	1	yes	characteristic
100	C₂×C₄²	#3624	1	1	yes	characteristic
101	C₂×C₄²	C₂×#595	1	2	yes	—
102	C₂×C₄²	C₂×#595	1	1	yes	characteristic
103	C₂×C₁₆	—	34	4	no	—
104	C₂×C₁₆	—	34	8	no	—
105	C₄×C₈	—	17	2	no	—
106	C₄×C₈	—	17	2	no	—
107	C₄×C₈	—	34	4	no	—
108	C₄×C₈	—	34	4	no	—
109	C₄×C₈	—	34	2	no	—
110	C₁₇	#12879	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₄	C₂×#3624	1	2	yes	—
112	C₂²×C₄	C₂×#3624	1	2	yes	—
113	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
114	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
115	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
116	C₂×C₈	—	17	1	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	17	4	no	—
119	C₂×C₈	—	17	4	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	—	34	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	4	no	—
127	C₄²	—	2	2	no	—
128	C₄²	#63216	1	4	yes	complemented
129	C₄²	#63243	1	2	yes	—
130	C₄²	C₄⋊#595	1	2	yes	—
131	C₁₆	—	68	4	no	—
132	C₂³	C₂²×#3624	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×#63216	1	4	yes	—
139	C₂×C₄	C₂×#63216	1	2	yes	—
140	C₂×C₄	#10587538	1	4	yes	—
141	C₂×C₄	#10587538	1	2	yes	—
142	C₂×C₄	C₄⋊#3624	1	2	yes	—
143	C₂×C₄	C₄⋊#3624	1	2	yes	—
144	C₂×C₄	C₂×#63243	1	1	yes	characteristic
145	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
147	C₂×C₄	#10587677	1	2	yes	—
148	C₈	—	17	2	no	—
149	C₈	—	17	2	no	—
150	C₈	—	34	2	no	—
151	C₈	—	34	4	no	—
152	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
153	C₂²	#11763825	1	2	yes	—
154	C₂²	C₂²×#63216	1	1	yes	characteristic
155	C₂²	C₂×#10587538	1	1	yes	characteristic
156	C₂²	C₂×#10587538	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊#63216	1	4	yes	—
161	C₄	#11763753	1	1	yes	characteristic
162	C₄	#11763814	1	1	yes	characteristic
163	C₄	#11763831	1	2	yes	—
164	C₂	#14544699	1	1	yes	characteristic
165	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic
166	C₂	#14545116	1	1	yes	characteristic
167	C₂	#14545141	1	2	yes	—
168	C₂	#14545145	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₄³×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₄³	#595
hypocenter	C₁₇	#12879
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
35. C₄²×C₁₇	C₁₆	semidirect
110. C₁₇	#12879	semidirect
128. C₄²	#63216	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63216	2
#14543822	2
C₄²×#3622	2
#12879	17