Subgroups — #17961608

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763730	C₂	1	2	yes	maximal
2	#14543823	C₂	1	2	yes	maximal
3	#14543822	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	C₂²	1	4	yes	—
6	#11763730	—	2	4	no	—
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂³	1	1	yes	characteristic
25	C₂×#63216	—	2	4	no	—
26	C₂×#63216	—	2	4	no	—
27	C₂×#63216	—	2	4	no	—
28	C₄×C₈×C₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	—	2	2	no	—
31	C₈×Dic₁₇	—	2	2	no	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	C₄×#3622	—	2	2	no	—
35	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
48	C₄²×C₁₇	—	2	1	no	—
49	C₄²×C₁₇	—	2	1	no	—
50	C₄×Dic₁₇	—	2	2	no	—
51	C₄×Dic₁₇	—	2	1	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	—	2	2	no	—
54	C₄×Dic₁₇	—	2	2	no	—
55	C₄×Dic₁₇	C₁₆	1	4	yes	complemented
56	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₁₇	M₁₆	1	2	yes	—
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	M₁₆	1	2	yes	—
61	C₂×#3622	C₂×D₄	1	1	yes	characteristic
62	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
63	C₂×#3622	#41	1	2	yes	—
64	C₂×#3622	#41	1	2	yes	—
65	#63216	—	4	4	no	—
66	#12879	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
73	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	—	2	1	no	—
82	C₂×Dic₁₇	—	2	1	no	—
83	C₂×Dic₁₇	—	2	1	no	—
84	C₂×Dic₁₇	—	2	1	no	—
85	C₂×Dic₁₇	—	2	1	no	—
86	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
88	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
89	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
90	C₂×Dic₁₇	M₃₂	1	2	yes	—
91	C₂×Dic₁₇	M₃₂	1	2	yes	—
92	C₈×C₁₇	—	2	2	no	—
93	C₈×C₁₇	—	2	2	no	—
94	#3622	—	2	2	no	—
95	#3622	#124	1	2	yes	—
96	#3622	#125	1	2	yes	—
97	C₂×C₄⋊C₁₆	—	17	2	no	—
98	C₄²×C₈	—	17	1	no	—
99	#1374	—	17	2	no	—
100	#1374	—	17	2	no	—
101	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₇	#431	1	1	yes	characteristic
103	C₂²×C₁₇	#431	1	1	yes	characteristic
104	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
106	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
107	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	#382	1	1	yes	characteristic
111	C₄×C₁₇	#423	1	1	yes	characteristic
112	C₄×C₁₇	#432	1	1	yes	characteristic
113	C₄×C₁₇	#432	1	1	yes	characteristic
114	Dic₁₇	—	2	2	no	—
115	Dic₁₇	—	2	2	no	—
116	Dic₁₇	—	2	2	no	—
117	Dic₁₇	C₄⋊C₁₆	1	4	yes	—
118	C₂²×C₁₆	—	17	4	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	#595	1	1	yes	characteristic
125	C₄³	—	17	1	no	—
126	C₄⋊C₁₆	—	34	4	no	—
127	C₂×C₁₇	#1831	1	1	yes	characteristic
128	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
129	C₂×C₁₇	#2150	1	1	yes	characteristic
130	C₂×C₁₇	#2162	1	1	yes	characteristic
131	C₂×C₁₇	#2162	1	1	yes	characteristic
132	C₂×C₁₇	#2163	1	1	yes	characteristic
133	C₂×C₁₇	#2163	1	1	yes	characteristic
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	C₂×#595	1	1	yes	characteristic
137	C₂²×C₈	C₂×#595	1	1	yes	characteristic
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	—	17	1	no	—
140	C₂×C₄²	—	17	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	C₂×#595	1	1	yes	characteristic
145	C₂×C₁₆	—	34	4	no	—
146	C₂×C₁₆	—	34	4	no	—
147	C₂×C₁₆	—	34	4	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	34	2	no	—
152	C₄×C₈	—	34	2	no	—
153	C₄×C₈	—	34	2	no	—
154	C₄×C₈	—	34	2	no	—
155	C₁₇	#12879	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	#63242	1	1	yes	characteristic
161	C₂²×C₄	#63242	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	2	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	1	no	—
174	C₂×C₈	C₄⋊#595	1	2	yes	—
175	C₂×C₈	#63247	1	2	yes	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	17	2	no	—
180	C₄²	—	17	4	no	—
181	C₄²	—	34	2	no	—
182	C₄²	—	34	2	no	—
183	C₄²	—	34	1	no	—
184	C₄²	—	34	1	no	—
185	C₄²	—	34	2	no	—
186	C₁₆	—	68	4	no	—
187	C₂³	C₂×#63242	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	17	2	no	—
193	C₂×C₄	—	17	2	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	34	1	no	—
199	C₂×C₄	—	34	1	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	1	no	—
203	C₂×C₄	—	34	1	no	—
204	C₂×C₄	#10587529	1	2	yes	—
205	C₂×C₄	#10587530	1	2	yes	—
206	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
207	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
209	C₂×C₄	#10587677	1	1	yes	characteristic
210	C₂×C₄	#10587677	1	1	yes	characteristic
211	C₂×C₄	C₂×#63247	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	17	2	no	—
215	C₈	—	17	2	no	—
216	C₈	—	34	2	no	—
217	C₂²	#11763810	1	1	yes	characteristic
218	C₂²	#11763826	1	1	yes	characteristic
219	C₂²	#11763827	1	1	yes	characteristic
220	C₂²	C₂×#10587529	1	1	yes	characteristic
221	C₂²	#11763865	1	1	yes	characteristic
222	C₂²	#11763865	1	1	yes	characteristic
223	C₂²	C₂×#10587530	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	17	4	no	—
227	C₄	—	34	2	no	—
228	C₄	—	34	2	no	—
229	C₄	—	34	2	no	—
230	C₄	#11763753	1	1	yes	characteristic
231	C₄	#11763812	1	1	yes	characteristic
232	C₄	#11763829	1	1	yes	characteristic
233	C₄	#11763832	1	1	yes	characteristic
234	C₂	#14544700	1	1	yes	characteristic
235	C₂	#14545114	1	1	yes	characteristic
236	C₂	#14545117	1	1	yes	characteristic
237	C₂	#14545142	1	1	yes	characteristic
238	C₂	#14545143	1	1	yes	characteristic
239	C₂	#14545146	1	1	yes	characteristic
240	C₂	#14545147	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12879
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
55. C₄×Dic₁₇	C₁₆	semidirect
155. 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×#11763730	2
#14543823	2
#14543822	2
C₄×C₈×Dic₁₇	2
#12879	17