Subgroups — #17978141

Order 4352.

834 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763867	C₂	1	2	yes	maximal
2	#14544995	C₂	1	6	yes	maximal, complemented
3	#14544990	C₂	1	3	yes	maximal, complemented
4	C₂×#11763695	C₂	1	3	yes	maximal, complemented
5	#131×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763867	C₄	1	16	yes	complemented
7	C₂×#10587538	C₂²	1	6	yes	—
8	C₂²×#63216	C₂²	1	6	yes	—
9	#11763698	C₂²	1	6	yes	—
10	#11763695	C₂²	1	1	yes	characteristic
11	#11763695	C₂²	1	3	yes	—
12	#11763695	C₂²	1	6	yes	—
13	C₂×#10587634	C₂²	1	1	yes	characteristic
14	#41×Dic₁₇	C₄	1	1	yes	characteristic
15	#11762767	C₂²	1	3	yes	—
16	#11762764	C₄	1	3	yes	—
17	M₁₆×Dic₁₇	C₄	1	3	yes	—
18	C₂×C₈×Dic₁₇	C₂²	1	3	yes	—
19	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
20	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₁₇×#131	C₈	1	8	yes	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
24	#10587634	C₂×C₄	1	8	yes	—
25	Q₈×Dic₁₇	C₈	1	2	yes	—
26	D₄×Dic₁₇	C₈	1	6	yes	—
27	#10587612	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
29	C₂×#63223	C₂×C₄	1	3	yes	—
30	C₂²×#3622	C₂³	1	3	yes	—
31	#10587538	C₂×C₄	1	24	yes	complemented
32	C₂×#63216	C₂×C₄	1	24	yes	complemented
33	C₂×#63216	C₂³	1	2	yes	—
34	C₂×#63216	C₂³	1	6	yes	—
35	#10587486	C₂×C₄	1	6	yes	—
36	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
37	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
38	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂³	1	3	yes	—
40	#41×C₁₇	C₄²	1	4	yes	—
41	#41×C₁₇	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₁₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₁₇	C₂×C₈	1	6	yes	—
46	M₁₆×C₁₇	C₂×C₈	1	12	yes	—
47	C₂×C₈×C₁₇	C₂×C₈	1	12	yes	—
48	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊Dic₁₇	C₂×C₈	1	6	yes	—
51	C₄⋊Dic₁₇	C₂×C₈	1	6	yes	—
52	C₄×Dic₁₇	C₂×C₈	1	6	yes	—
53	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
55	#63223	C₄²	1	12	yes	—
56	C₂×#3622	C₂²×C₄	1	3	yes	—
57	C₂×#3622	C₂²×C₄	1	12	yes	—
58	C₂×#3622	C₂⁴	1	1	yes	characteristic
59	#63216	C₂²×C₄	1	4	yes	—
60	#63216	C₂²×C₄	1	12	yes	—
61	C₄×#504	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
63	Q₈×C₁₇	C₄×C₈	1	4	yes	—
64	D₄×C₁₇	C₄×C₈	1	12	yes	—
65	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
69	C₂×Dic₁₇	—	2	12	no	—
70	C₂×Dic₁₇	C₂²×C₈	1	6	yes	—
71	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
72	C₈×C₁₇	C₂²×C₈	1	2	yes	—
73	C₈×C₁₇	C₂²×C₈	1	6	yes	—
74	#3622	C₂×C₄²	1	6	yes	—
75	#3622	C₂³×C₄	1	2	yes	—
76	C₂×#504	—	17	2	no	—
77	C₄×#131	—	17	1	no	—
78	#2108	—	17	6	no	—
79	C₄×M₃₂	—	17	3	no	—
80	C₂×C₄×C₁₆	—	17	3	no	—
81	C₂²×C₁₇	—	2	6	no	—
82	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
83	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
85	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
87	Dic₁₇	—	2	6	no	—
88	Dic₁₇	#504	1	4	yes	complemented
89	C₂×#131	—	17	1	no	—
90	C₂×#131	#595	1	1	yes	characteristic
91	C₄×#41	—	17	1	no	—
92	#504	—	17	16	no	—
93	C₂×M₃₂	—	17	6	no	—
94	C₂²×C₁₆	—	17	6	no	—
95	#405	—	17	3	no	—
96	#405	—	17	3	no	—
97	C₄×M₁₆	—	17	3	no	—
98	C₂×C₄×C₈	—	17	3	no	—
99	#346	—	17	6	no	—
100	C₄×C₁₆	—	17	1	no	—
101	C₄×C₁₆	—	17	3	no	—
102	C₄×C₁₆	—	17	6	no	—
103	C₂×C₁₇	—	2	6	no	—
104	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#504	1	2	yes	—
106	C₂×#41	C₂×#595	1	1	yes	characteristic
107	#131	—	17	8	no	—
108	#131	#3624	1	8	yes	—
109	C₂×M₁₆	—	17	3	no	—
110	C₂×M₁₆	C₂×#595	1	3	yes	—
111	C₂²×C₈	—	17	3	no	—
112	C₂²×C₈	C₂×#595	1	3	yes	—
113	C₄×Q₈	—	17	2	no	—
114	C₄×D₄	—	17	6	no	—
115	#117	—	17	3	no	—
116	C₂×C₄²	—	17	3	no	—
117	M₃₂	—	17	24	no	—
118	C₂×C₁₆	—	17	2	no	—
119	C₂×C₁₆	—	17	6	no	—
120	C₂×C₁₆	—	17	24	no	—
121	#97	—	17	6	no	—
122	C₄×C₈	—	17	3	no	—
123	C₄×C₈	—	17	3	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	3	no	—
126	C₁₇	C₄×#504	1	1	yes	characteristic, complemented, Sylow
127	#41	C₄×#595	1	4	yes	—
128	#41	C₂×#3624	1	4	yes	—
129	C₂×Q₈	#63242	1	1	yes	characteristic
130	C₂×D₄	#63242	1	3	yes	—
131	C₂²×C₄	—	17	6	no	—
132	C₂²×C₄	C₂²×#595	1	3	yes	—
133	M₁₆	—	17	12	no	—
134	M₁₆	C₂×#3624	1	12	yes	—
135	C₂×C₈	—	17	12	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	3	no	—
138	C₂×C₈	C₂×#3624	1	12	yes	—
139	C₂×C₈	C₂²×#595	1	3	yes	—
140	C₂×C₈	C₂²×#595	1	1	yes	characteristic
141	C₄⋊C₄	—	17	6	no	—
142	C₂²⋊C₄	—	17	6	no	—
143	C₄²	—	17	6	no	—
144	C₄²	—	17	1	no	—
145	C₄²	—	17	3	no	—
146	C₁₆	—	17	4	no	—
147	C₁₆	—	17	12	no	—
148	C₂³	C₂×#63242	1	3	yes	—
149	Q₈	C₄×#3624	1	4	yes	—
150	D₄	C₄×#3624	1	12	yes	—
151	C₂×C₄	—	17	2	no	—
152	C₂×C₄	—	17	6	no	—
153	C₂×C₄	—	34	12	no	—
154	C₂×C₄	C₂×#63242	1	3	yes	—
155	C₂×C₄	C₂×C₄×#595	1	6	yes	—
156	C₂×C₄	C₂²×#3624	1	6	yes	—
157	C₂×C₄	C₂³×#595	1	1	yes	characteristic
158	C₈	—	17	2	no	—
159	C₈	—	17	6	no	—
160	C₈	C₂²×#3624	1	2	yes	—
161	C₈	C₂²×#3624	1	6	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#3624	1	6	yes	—
164	C₂²	C₂²×#63242	1	1	yes	characteristic
165	C₄	—	17	4	no	—
166	C₄	—	34	6	no	—
167	C₄	C₂×C₄×#3624	1	6	yes	—
168	C₄	C₂²×C₄×#595	1	1	yes	characteristic
169	C₄	C₂³×#3624	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#3624	1	1	yes	characteristic
172	C₂	#14545749	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂×C₁₇×#131	C₄
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×#131	#595
hypocenter	C₁₇	C₄×#504
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #14544995	C₂	semidirect
3. #14544990	C₂	semidirect
4. C₂×#11763695	C₂	semidirect
6. #11763867	C₄	semidirect
31. #10587538	C₂×C₄	semidirect
32. C₂×#63216	C₂×C₄	semidirect
88. Dic₁₇	#504	semidirect
126. C₁₇	C₄×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763867	2
#14544995	2
#14544990	2
C₂×#11763695	2
#131×Dic₁₇	2
C₄×#504	17