Subgroups — #16950670

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14526861	C₂	1	4	yes	maximal
4	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
9	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
10	C₄⋊#63214	—	2	2	no	—
11	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
18	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
21	C₂²×#3622	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₁₇	—	2	2	no	—
23	C₂×C₁₆×C₁₇	—	2	4	no	—
24	C₄×C₈×C₁₇	D₄	1	2	yes	—
25	C₄×C₈×C₁₇	Q₈	1	2	yes	—
26	C₈×Dic₁₇	—	2	4	no	—
27	C₈×Dic₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₂×#63214	—	2	2	no	—
30	C₂×#63214	—	2	4	no	—
31	C₄×#3622	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
36	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₇	—	2	2	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	#41	1	2	yes	—
42	C₂×C₈×C₁₇	#41	1	2	yes	—
43	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₇	M₁₆	1	2	yes	—
45	C₄×Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	—	2	2	no	—
47	C₄×Dic₁₇	—	2	2	no	—
48	C₄×Dic₁₇	—	2	4	no	—
49	C₄×Dic₁₇	C₁₆	1	4	yes	complemented
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	M₁₆	1	4	yes	—
53	C₁₆×C₁₇	—	4	2	no	—
54	#63214	—	4	2	no	—
55	#12879	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
62	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#117	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₁₇	—	2	2	no	—
67	C₂×Dic₁₇	—	2	2	no	—
68	C₂×Dic₁₇	—	2	2	no	—
69	C₂×Dic₁₇	—	2	2	no	—
70	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
71	C₂×Dic₁₇	M₃₂	1	4	yes	—
72	C₈×C₁₇	—	2	2	no	—
73	C₈×C₁₇	#124	1	1	yes	characteristic
74	C₈×C₁₇	#124	1	1	yes	characteristic
75	C₈×C₁₇	#125	1	2	yes	—
76	#3622	—	2	4	no	—
77	C₂×C₄⋊C₁₆	—	17	1	no	—
78	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic
79	C₄²×C₈	—	17	1	no	—
80	#1374	—	17	4	no	—
81	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂²×C₁₇	#431	1	2	yes	—
83	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
84	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
85	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
86	C₄×C₁₇	—	2	2	no	—
87	C₄×C₁₇	C₄⋊C₁₆	1	4	yes	—
88	C₄×C₁₇	#382	1	1	yes	characteristic
89	C₄×C₁₇	#423	1	1	yes	characteristic
90	C₄×C₁₇	#432	1	2	yes	—
91	Dic₁₇	—	2	4	no	—
92	Dic₁₇	—	2	4	no	—
93	C₂²×C₁₆	—	17	2	no	—
94	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
95	C₂×C₄×C₈	—	17	2	no	—
96	C₂×C₄×C₈	—	17	2	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
99	C₄³	—	17	1	no	—
100	C₄⋊C₁₆	—	2	2	no	—
101	C₄⋊C₁₆	—	34	2	no	—
102	C₂×C₁₇	#1831	1	1	yes	characteristic
103	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
104	C₂×C₁₇	#2150	1	1	yes	characteristic
105	C₂×C₁₇	#2162	1	2	yes	—
106	C₂×C₁₇	#2163	1	2	yes	—
107	C₂²×C₈	—	17	2	no	—
108	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
110	C₂×C₄²	—	17	1	no	—
111	C₂×C₄²	—	17	2	no	—
112	C₂×C₄²	—	17	2	no	—
113	C₂×C₄²	—	17	1	no	—
114	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	34	4	no	—
118	C₂×C₁₆	—	34	2	no	—
119	C₄×C₈	—	34	4	no	—
120	C₄×C₈	—	34	2	no	—
121	C₄×C₈	—	34	2	no	—
122	C₄×C₈	—	34	2	no	—
123	C₄×C₈	Dic₃₄	1	2	yes	—
124	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
125	C₁₇	#12879	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₄	—	17	2	no	—
127	C₂²×C₄	—	17	2	no	—
128	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	17	4	no	—
133	C₂×C₈	—	34	2	no	—
134	C₂×C₈	—	34	2	no	—
135	C₂×C₈	#63225	1	2	yes	—
136	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
137	C₂×C₈	#63254	1	2	yes	—
138	C₂×C₈	#63256	1	2	yes	—
139	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄²	—	17	4	no	—
141	C₄²	—	34	4	no	—
142	C₄²	—	34	2	no	—
143	C₄²	—	34	2	no	—
144	C₄²	—	34	2	no	—
145	C₄²	#63218	1	2	yes	—
146	C₄²	#63225	1	2	yes	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	68	2	no	—
149	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	17	4	no	—
152	C₂×C₄	—	17	4	no	—
153	C₂×C₄	—	34	2	no	—
154	C₂×C₄	—	34	2	no	—
155	C₂×C₄	—	34	2	no	—
156	C₂×C₄	—	34	2	no	—
157	C₂×C₄	C₁₆×D₁₇	1	2	yes	—
158	C₂×C₄	#10587469	1	2	yes	—
159	C₂×C₄	#10587485	1	2	yes	—
160	C₂×C₄	#10587485	1	2	yes	—
161	C₂×C₄	#10587578	1	2	yes	—
162	C₂×C₄	C₂×#63218	1	1	yes	characteristic
163	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
164	C₂×C₄	C₂×#63225	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	34	4	no	—
167	C₈	#10587576	1	2	yes	—
168	C₈	#10587622	1	1	yes	characteristic
169	C₈	#10587633	1	1	yes	characteristic
170	C₂²	C₂×#10587485	1	1	yes	characteristic
171	C₂²	#11762861	1	2	yes	—
172	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
173	C₂²	M₃₂×D₁₇	1	2	yes	—
174	C₂²	C₂×#10587469	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	34	4	no	—
177	C₄	—	34	4	no	—
178	C₄	#11762627	1	4	yes	—
179	C₄	#11762674	1	1	yes	characteristic
180	C₄	#11762832	1	1	yes	characteristic
181	C₄	#11762865	1	2	yes	—
182	C₂	#14528900	1	1	yes	characteristic
183	C₂	C₂×#11762627	1	1	yes	characteristic
184	C₂	#14530539	1	1	yes	characteristic
185	C₂	#14530582	1	2	yes	—
186	C₂	#14530586	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	#12879
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
49. C₄×Dic₁₇	C₁₆	semidirect
125. C₁₇	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63214	2
C₄×C₈×Dic₁₇	2
#14526861	2
C₂×C₁₇×C₄⋊C₁₆	2
#12879	17