Subgroups — #17963998

Order 4352.

890 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#3624	C₂	1	2	yes	maximal
2	#14543987	C₂	1	3	yes	maximal, complemented
3	M₁₆×#3624	C₂	1	6	yes	maximal, complemented
4	C₂×#11763650	C₂	1	3	yes	maximal, complemented
5	#131×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3624	C₄	1	4	yes	—
7	D₄×#3624	C₄	1	12	yes	—
8	#11763825	C₂²	1	6	yes	—
9	C₂×C₄×#3624	C₂²	1	6	yes	—
10	#11763650	C₂²	1	1	yes	characteristic
11	#11763650	C₂²	1	6	yes	—
12	#11763650	C₂²	1	3	yes	—
13	C₈×#3624	C₂²	1	6	yes	—
14	C₂×#10587634	C₄	1	1	yes	characteristic
15	#41×Dic₁₇	C₂²	1	1	yes	characteristic
16	#11762767	C₄	1	3	yes	—
17	#11762764	C₄	1	3	yes	—
18	M₁₆×Dic₁₇	C₂²	1	3	yes	—
19	C₂×C₈×Dic₁₇	C₂²	1	3	yes	—
20	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
21	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3624	C₂×C₄	1	12	yes	—
23	C₁₇×#131	C₈	1	8	yes	complemented
24	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
26	#10587634	C₈	1	8	yes	complemented
27	Q₈×Dic₁₇	C₂×C₄	1	2	yes	—
28	D₄×Dic₁₇	C₂×C₄	1	6	yes	—
29	#10587612	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₁₇	C₂³	1	3	yes	—
31	C₂×#63223	C₂×C₄	1	3	yes	—
32	C₂²×#3622	C₂×C₄	1	3	yes	—
33	C₂²⋊#3624	C₂×C₄	1	12	yes	—
34	C₄⋊#3624	C₂×C₄	1	12	yes	—
35	C₄×#3624	C₂×C₄	1	12	yes	—
36	C₄×#3624	C₂³	1	2	yes	—
37	C₄×#3624	C₂³	1	6	yes	—
38	#10587486	C₂×C₄	1	6	yes	—
39	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
40	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
41	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
42	C₈×Dic₁₇	C₂³	1	3	yes	—
43	#41×C₁₇	C₂×C₈	1	4	yes	—
44	#41×C₁₇	C₂×C₈	1	4	yes	—
45	C₂×Q₈×C₁₇	C₄²	1	1	yes	characteristic
46	C₂×D₄×C₁₇	C₄²	1	3	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₇	C₂²×C₄	1	6	yes	—
49	C₂×#3624	—	4	12	no	—
50	C₂×#3624	C₂²×C₄	1	4	yes	—
51	C₂×#3624	C₂²×C₄	1	12	yes	—
52	M₁₆×C₁₇	C₂×C₈	1	12	yes	complemented
53	C₂×C₈×C₁₇	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
56	C₂²⋊Dic₁₇	C₄²	1	6	yes	—
57	C₄⋊Dic₁₇	C₄²	1	6	yes	—
58	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
59	C₄×Dic₁₇	C₂²×C₄	1	6	yes	—
60	C₄×Dic₁₇	C₂⁴	1	1	yes	characteristic
61	#63223	C₂×C₈	1	12	yes	complemented
62	C₂×#3622	C₂×C₈	1	12	yes	complemented
63	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
64	C₂×#3622	C₂²×C₄	1	3	yes	—
65	#17316	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×C₄²	1	3	yes	—
67	Q₈×C₁₇	C₄×C₈	1	4	yes	—
68	D₄×C₁₇	C₄×C₈	1	12	yes	—
69	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
71	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
72	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₁₇	—	2	12	no	—
74	C₂×Dic₁₇	C₂×C₄²	1	6	yes	—
75	C₂×Dic₁₇	C₂³×C₄	1	2	yes	—
76	#3624	—	2	4	no	—
77	#3624	—	2	12	no	—
78	C₈×C₁₇	C₂²×C₈	1	2	yes	—
79	C₈×C₁₇	C₂²×C₈	1	6	yes	—
80	#3622	C₂²×C₈	1	2	yes	—
81	#3622	C₂²×C₈	1	6	yes	—
82	C₈×#41	—	17	2	no	—
83	C₄×#131	—	17	1	no	—
84	#1450	—	17	3	no	—
85	C₈×M₁₆	—	17	6	no	—
86	C₂×#322	—	17	3	no	—
87	C₂²×C₁₇	—	2	6	no	—
88	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
89	C₂²×C₁₇	C₂²×C₄²	1	1	yes	characteristic
90	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
91	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
93	Dic₁₇	—	2	6	no	—
94	Dic₁₇	#568	1	4	yes	—
95	C₂×#131	—	17	1	no	—
96	C₂×#131	#595	1	1	yes	characteristic
97	C₄×#41	—	17	1	no	—
98	C₈×Q₈	—	17	4	no	—
99	C₈×D₄	—	17	12	no	—
100	#431	—	17	6	no	—
101	#405	—	17	3	no	—
102	#405	—	17	3	no	—
103	C₄×M₁₆	—	17	3	no	—
104	C₂×C₄×C₈	—	17	6	no	—
105	C₂×C₄×C₈	—	17	3	no	—
106	#322	—	17	1	no	—
107	#322	—	17	6	no	—
108	#322	—	17	3	no	—
109	C₈²	—	17	6	no	—
110	C₂×C₁₇	—	2	6	no	—
111	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
112	C₂×C₁₇	#2875	1	2	yes	—
113	C₂×#41	C₂×#595	1	1	yes	characteristic
114	#131	—	17	8	no	—
115	#131	#3624	1	8	yes	complemented
116	C₂×M₁₆	—	17	3	no	—
117	C₂×M₁₆	C₂×#595	1	3	yes	—
118	C₂²×C₈	—	17	3	no	—
119	C₂²×C₈	—	17	12	no	—
120	C₂²×C₈	C₂×#595	1	3	yes	—
121	C₄×Q₈	—	17	2	no	—
122	C₄×D₄	—	17	6	no	—
123	#117	—	17	3	no	—
124	C₂×C₄²	—	17	3	no	—
125	C₄⋊C₈	—	17	12	no	—
126	C₂²⋊C₈	—	17	12	no	—
127	#97	—	17	6	no	—
128	C₄×C₈	—	17	6	no	—
129	C₄×C₈	—	17	2	no	—
130	C₄×C₈	—	17	12	no	—
131	C₄×C₈	—	17	3	no	—
132	C₄×C₈	—	17	1	no	—
133	C₄×C₈	—	17	3	no	—
134	C₄×C₈	—	17	3	no	—
135	C₁₇	#17316	1	1	yes	characteristic, complemented, Sylow
136	#41	#63242	1	4	yes	—
137	#41	C₂×#3624	1	4	yes	—
138	C₂×Q₈	C₄×#595	1	1	yes	characteristic
139	C₂×D₄	C₄×#595	1	3	yes	—
140	C₂²×C₄	—	17	6	no	—
141	C₂²×C₄	C₂²×#595	1	3	yes	—
142	M₁₆	—	17	12	no	—
143	M₁₆	C₂×#3624	1	12	yes	complemented
144	C₂×C₈	—	17	12	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	3	no	—
147	C₂×C₈	—	17	4	no	—
148	C₂×C₈	—	17	12	no	—
149	C₂×C₈	—	68	12	no	—
150	C₂×C₈	C₂×#3624	1	12	yes	complemented
151	C₂×C₈	C₂²×#595	1	1	yes	characteristic
152	C₂×C₈	C₂²×#595	1	3	yes	—
153	C₄⋊C₄	—	17	6	no	—
154	C₂²⋊C₄	—	17	6	no	—
155	C₄²	—	17	6	no	—
156	C₄²	—	17	1	no	—
157	C₄²	—	17	3	no	—
158	C₂³	C₂×C₄×#595	1	3	yes	—
159	Q₈	C₄×#3624	1	4	yes	—
160	D₄	C₄×#3624	1	12	yes	—
161	C₂×C₄	—	17	2	no	—
162	C₂×C₄	—	17	6	no	—
163	C₂×C₄	—	34	12	no	—
164	C₂×C₄	C₂×#63242	1	6	yes	—
165	C₂×C₄	C₂×C₄×#595	1	3	yes	—
166	C₂×C₄	C₂²×#3624	1	6	yes	—
167	C₂×C₄	C₂³×#595	1	1	yes	characteristic
168	C₈	—	17	2	no	—
169	C₈	—	17	6	no	—
170	C₈	—	34	4	no	—
171	C₈	—	34	12	no	—
172	C₈	C₂²×#3624	1	2	yes	—
173	C₈	C₂²×#3624	1	6	yes	—
174	C₂²	—	2	6	no	—
175	C₂²	C₂×C₄×#3624	1	6	yes	—
176	C₂²	C₂²×C₄×#595	1	1	yes	characteristic
177	C₄	—	17	4	no	—
178	C₄	—	34	6	no	—
179	C₄	C₂×C₄×#3624	1	6	yes	—
180	C₄	C₂³×#3624	1	1	yes	characteristic
181	C₄	C₂²×#63242	1	1	yes	characteristic
182	C₂	—	2	6	no	—
183	C₂	C₂²×C₄×#3624	1	1	yes	characteristic
184	C₂	#14545354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Fitting	C₂×C₁₇×#131	C₄
socle	C₂²×C₁₇	C₂²×C₄²
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₄×Dic₁₇	C₂⁴
hypercenter	C₂×#131	#595
hypocenter	C₁₇	#17316
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #14543987	C₂	semidirect
3. M₁₆×#3624	C₂	semidirect
4. C₂×#11763650	C₂	semidirect
23. C₁₇×#131	C₈	semidirect
26. #10587634	C₈	semidirect
52. M₁₆×C₁₇	C₂×C₈	semidirect
53. C₂×C₈×C₁₇	C₂×C₈	semidirect
61. #63223	C₂×C₈	semidirect
62. C₂×#3622	C₂×C₈	semidirect
115. #131	#3624	semidirect
135. C₁₇	#17316	semidirect
143. M₁₆	C₂×#3624	semidirect
150. C₂×C₈	C₂×#3624	semidirect

Maximal subgroups

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

subgroup	index
#41×#3624	2
#14543987	2
M₁₆×#3624	2
C₂×#11763650	2
#131×Dic₁₇	2
#17316	17