Subgroups — #16933038

Order 4352.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529559	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
4	#14526680	C₂	1	2	yes	maximal
5	C₁₇×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587493	—	2	2	no	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	#11762573	—	2	2	no	—
10	#11762562	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762560	C₄	1	2	yes	—
13	#11762560	C₄	1	2	yes	—
14	C₁₇×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#322	—	2	1	no	—
18	C₁₇×#322	—	2	1	no	—
19	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	—	2	2	no	—
25	C₂²×#3622	—	2	2	no	—
26	C₁₇×C₄⋊C₈	—	2	1	no	—
27	C₁₇×C₂²⋊C₈	Q₈	1	2	yes	—
28	C₁₇×#97	—	2	1	no	—
29	C₄×C₈×C₁₇	—	2	1	no	—
30	C₄×C₈×C₁₇	—	2	1	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587493	—	4	4	no	—
36	C₄⋊#3622	—	2	2	no	—
37	C₄⋊#3622	—	2	2	no	—
38	C₄⋊#3622	—	2	2	no	—
39	C₄⋊#3622	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₁₇	—	2	2	no	—
44	M₁₆×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	2	1	no	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	—	4	1	no	—
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₇	#41	1	1	yes	characteristic
57	C₄²×C₁₇	#41	1	1	yes	characteristic
58	#63223	—	4	2	no	—
59	#63223	—	4	2	no	—
60	C₂×#3622	—	2	4	no	—
61	C₂×#3622	—	4	2	no	—
62	C₂×#3622	—	4	2	no	—
63	C₂×#3622	—	4	2	no	—
64	#9626	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	4	1	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₁₇	#137	1	2	yes	—
77	C₈×C₁₇	—	2	1	no	—
78	C₈×C₁₇	—	2	1	no	—
79	C₈×C₁₇	—	2	1	no	—
80	C₈×C₁₇	—	4	2	no	—
81	C₈×C₁₇	—	4	1	no	—
82	C₈×C₁₇	—	4	1	no	—
83	C₈×C₁₇	#108	1	2	yes	complemented
84	#3622	—	4	4	no	—
85	#3622	—	8	2	no	—
86	#1950	—	17	2	no	—
87	#1567	—	17	1	no	—
88	C₂×#334	—	17	1	no	—
89	#1451	D₁₇	1	1	yes	characteristic
90	#1301	—	17	2	no	—
91	C₂²×C₁₇	—	2	1	no	—
92	C₂²×C₁₇	—	2	1	no	—
93	C₂²×C₁₇	#389	1	1	yes	characteristic
94	C₂²×C₁₇	#501	1	2	yes	—
95	C₄×C₁₇	—	2	1	no	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
99	C₄×C₁₇	#430	1	1	yes	characteristic
100	C₄×C₁₇	#440	1	2	yes	—
101	C₄×C₁₇	#469	1	1	yes	characteristic
102	C₄×C₁₇	#470	1	1	yes	characteristic
103	#431	C₂×D₁₇	1	1	yes	characteristic
104	C₂×#108	—	34	2	no	—
105	#423	—	17	2	no	—
106	C₂×C₄⋊C₈	—	17	2	no	—
107	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
109	#341	—	34	2	no	—
110	#335	—	34	1	no	—
111	#335	—	34	1	no	—
112	#334	—	17	2	no	—
113	#334	—	17	2	no	—
114	#322	—	2	1	no	—
115	#322	—	2	1	no	—
116	C₂×C₁₇	—	2	2	no	—
117	C₂×C₁₇	#1944	1	1	yes	characteristic
118	C₂×C₁₇	#1950	1	1	yes	characteristic
119	C₂×C₁₇	#1953	1	1	yes	characteristic
120	C₂×M₁₆	—	34	2	no	—
121	C₂×M₁₆	D₆₈	1	1	yes	characteristic
122	C₂×M₁₆	D₆₈	1	1	yes	characteristic
123	C₂²×C₈	—	34	2	no	—
124	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
125	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
127	#108	—	68	4	no	—
128	C₄⋊C₈	—	2	1	no	—
129	C₄⋊C₈	—	17	4	no	—
130	C₄⋊C₈	—	34	2	no	—
131	C₄⋊C₈	—	34	2	no	—
132	C₄⋊C₈	—	34	2	no	—
133	C₂²⋊C₈	Dic₃₄	1	2	yes	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
138	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
141	C₁₇	#9626	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	68	2	no	—
148	M₁₆	—	68	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	34	4	no	—
155	C₂×C₈	—	68	2	no	—
156	C₂×C₈	—	68	2	no	—
157	C₂×C₈	—	68	2	no	—
158	C₂×C₈	#63225	1	1	yes	characteristic
159	C₂×C₈	#63225	1	1	yes	characteristic
160	C₂×C₈	#63225	1	1	yes	characteristic
161	C₂×C₈	#63225	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
163	C₄²	#63252	1	1	yes	characteristic
164	C₄²	#63252	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	C₂³	#10587555	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
171	C₂×C₄	#10587556	1	1	yes	characteristic
172	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
173	C₂×C₄	#10587594	1	2	yes	—
174	C₂×C₄	#10587595	1	2	yes	—
175	C₂×C₄	C₂×#63225	1	1	yes	characteristic
176	C₂×C₄	#10587610	1	1	yes	characteristic
177	C₂×C₄	#10587616	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	1	no	—
184	C₈	—	68	4	no	—
185	C₈	—	136	2	no	—
186	C₈	#10587493	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#11762707	1	1	yes	characteristic
190	C₂²	#11763218	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#10587493	1	1	yes	characteristic
195	C₄	#11762857	1	1	yes	characteristic
196	C₄	#11762911	1	2	yes	—
197	C₄	#11763062	1	1	yes	characteristic
198	C₄	#11763067	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#14529527	1	1	yes	characteristic
201	C₂	#14529559	1	1	yes	characteristic
202	C₂	#14529574	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1451	C₂
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1451	D₁₇
hypocenter	C₁₇	#9626
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1451

Complements

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

N	H ≅ G/N	type
83. C₈×C₁₇	#108	semidirect
141. C₁₇	#9626	semidirect
186. C₈	#10587493	semidirect

Maximal subgroups

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

subgroup	index
#14529559	2
#14527543	2
C₂×#11762560	2
#14526680	2
C₁₇×#1451	2
#9626	17