Subgroups — #17953767

Order 4352.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544717	C₂	1	1	yes	maximal, characteristic
2	#14544701	C₂	1	1	yes	maximal, characteristic
3	#14543614	C₂	1	2	yes	maximal
4	#14543614	C₂	1	2	yes	maximal
5	#97×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587544	—	2	2	no	—
7	C₂×C₄⋊#3624	C₂²	1	2	yes	—
8	C₂×C₄⋊#3624	C₂²	1	2	yes	—
9	#11763675	—	2	2	no	—
10	C₂×#10587474	C₄	1	1	yes	characteristic
11	C₂×#10587486	C₄	1	1	yes	characteristic
12	C₂×#10587486	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
17	C₂²×#3624	—	2	2	no	—
18	C₂²×#3624	—	2	2	no	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
24	C₂×C₄×Dic₁₇	Q₈	1	2	yes	—
25	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
28	#10587544	—	4	4	no	—
29	C₄⋊#3624	—	2	2	no	—
30	C₄⋊#3624	—	2	2	no	—
31	C₄⋊#3624	—	2	2	no	—
32	C₄⋊#3624	—	2	2	no	—
33	C₁₇×#97	—	2	2	no	—
34	#10587486	—	2	2	no	—
35	#10587486	C₈	1	4	yes	complemented
36	C₈×Dic₁₇	—	2	1	no	—
37	C₈×Dic₁₇	—	2	1	no	—
38	C₈×Dic₁₇	—	2	1	no	—
39	C₈×Dic₁₇	—	2	1	no	—
40	#10587474	—	2	2	no	—
41	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×#3624	—	4	2	no	—
47	C₂×#3624	—	4	2	no	—
48	C₂×#3624	—	4	2	no	—
49	C₂×#3624	—	4	4	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×C₈×C₁₇	—	2	1	no	—
52	C₂×C₈×C₁₇	—	2	1	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	—	2	1	no	—
55	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
57	C₄²×C₁₇	—	2	1	no	—
58	C₄²×C₁₇	—	2	1	no	—
59	C₄×Dic₁₇	—	2	2	no	—
60	C₄×Dic₁₇	—	2	2	no	—
61	C₄×Dic₁₇	—	4	2	no	—
62	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
63	C₄×Dic₁₇	M₁₆	1	2	yes	—
64	C₄×Dic₁₇	#41	1	1	yes	characteristic
65	C₄×Dic₁₇	#41	1	1	yes	characteristic
66	C₄×Dic₁₇	#41	1	1	yes	characteristic
67	C₄×Dic₁₇	#41	1	1	yes	characteristic
68	C₂×#3622	—	2	1	no	—
69	C₂×#3622	—	2	1	no	—
70	C₂×#3622	—	2	1	no	—
71	C₂×#3622	—	2	1	no	—
72	C₂×#3622	C₂×C₈	1	2	yes	—
73	C₂×#3622	M₁₆	1	2	yes	—
74	#8786	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₁₇	—	2	2	no	—
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	—	2	2	no	—
88	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
89	C₂×Dic₁₇	#124	1	1	yes	characteristic
90	C₂×Dic₁₇	#124	1	1	yes	characteristic
91	C₂×Dic₁₇	#125	1	1	yes	characteristic
92	C₂×Dic₁₇	#125	1	1	yes	characteristic
93	C₂×Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×Dic₁₇	#137	1	2	yes	—
95	#3624	—	8	2	no	—
96	#3624	—	8	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	2	no	—
99	#3622	—	2	2	no	—
100	#3622	—	2	2	no	—
101	#1842	—	17	1	no	—
102	#1831	—	17	1	no	—
103	C₄×#97	—	17	1	no	—
104	#1277	—	17	2	no	—
105	#1277	—	17	2	no	—
106	C₂²×C₁₇	#382	1	1	yes	characteristic
107	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₁₇	#423	1	1	yes	characteristic
109	C₂²×C₁₇	#428	1	2	yes	—
110	C₂²×C₁₇	#430	1	2	yes	—
111	C₄×C₁₇	—	4	2	no	—
112	C₄×C₁₇	#431	1	1	yes	characteristic
113	C₄×C₁₇	#431	1	1	yes	characteristic
114	C₄×C₁₇	#432	1	1	yes	characteristic
115	C₄×C₁₇	#432	1	1	yes	characteristic
116	Dic₁₇	—	2	4	no	—
117	Dic₁₇	—	2	1	no	—
118	Dic₁₇	—	2	1	no	—
119	Dic₁₇	#489	1	2	yes	—
120	Dic₁₇	#490	1	1	yes	characteristic
121	Dic₁₇	#491	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	17	2	no	—
123	C₂×C₄⋊C₈	—	17	2	no	—
124	C₂×C₄⋊C₈	—	34	2	no	—
125	C₂×#97	—	17	1	no	—
126	C₂×#97	—	17	1	no	—
127	C₂×#97	—	17	1	no	—
128	C₂×#97	#595	1	1	yes	characteristic
129	C₂×C₄×C₈	—	17	1	no	—
130	C₂×C₄×C₈	—	17	1	no	—
131	C₄³	—	17	1	no	—
132	#336	—	34	2	no	—
133	C₂×C₁₇	#1565	1	2	yes	—
134	C₂×C₁₇	#1569	1	2	yes	—
135	C₂×C₁₇	#1831	1	1	yes	characteristic
136	C₂×C₁₇	#1836	1	1	yes	characteristic
137	C₂×C₁₇	#1838	1	1	yes	characteristic
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	34	2	no	—
141	C₂²×C₈	—	34	2	no	—
142	C₂²×C₈	C₂×#595	1	1	yes	characteristic
143	C₂²×C₈	C₂×#595	1	1	yes	characteristic
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	—	17	2	no	—
147	C₂×C₄²	—	17	2	no	—
148	C₂×C₄²	C₂×#595	1	1	yes	characteristic
149	C₄⋊C₈	—	34	2	no	—
150	C₄⋊C₈	—	34	2	no	—
151	C₄⋊C₈	—	34	2	no	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄⋊C₈	—	68	4	no	—
154	#97	—	2	2	no	—
155	#97	—	17	4	no	—
156	#97	—	34	2	no	—
157	#97	—	34	2	no	—
158	C₄×C₈	—	34	1	no	—
159	C₄×C₈	—	34	1	no	—
160	C₄×C₈	—	34	1	no	—
161	C₄×C₈	—	34	1	no	—
162	C₁₇	#8786	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	2	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	C₄⋊#595	1	2	yes	—
167	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	34	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	68	2	no	—
179	C₂×C₈	—	68	2	no	—
180	C₂×C₈	—	68	4	no	—
181	C₂×C₈	—	68	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	#63242	1	2	yes	—
184	C₂×C₈	#63247	1	2	yes	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	17	2	no	—
188	C₄²	—	17	2	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	1	no	—
191	C₄²	—	17	1	no	—
192	C₄²	—	17	1	no	—
193	C₄²	—	34	2	no	—
194	C₄²	—	34	2	no	—
195	C₄²	—	68	2	no	—
196	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	2	no	—
203	C₂×C₄	—	17	2	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	1	no	—
206	C₂×C₄	—	17	4	no	—
207	C₂×C₄	—	34	2	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	—	34	2	no	—
210	C₂×C₄	C₂×#63242	1	1	yes	characteristic
211	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
212	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	#10587677	1	1	yes	characteristic
214	C₂×C₄	#10587677	1	1	yes	characteristic
215	C₂×C₄	C₂×#63247	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	34	2	no	—
219	C₈	—	34	2	no	—
220	C₈	—	136	2	no	—
221	C₈	—	136	2	no	—
222	C₂²	#11763753	1	1	yes	characteristic
223	C₂²	#11763810	1	1	yes	characteristic
224	C₂²	#11763812	1	1	yes	characteristic
225	C₂²	#11763820	1	2	yes	—
226	C₂²	#11763824	1	2	yes	—
227	C₄	—	4	2	no	—
228	C₄	—	17	1	no	—
229	C₄	—	17	2	no	—
230	C₄	—	17	1	no	—
231	C₄	—	34	1	no	—
232	C₄	—	34	1	no	—
233	C₄	—	34	4	no	—
234	C₄	#11763826	1	1	yes	characteristic
235	C₄	#11763827	1	1	yes	characteristic
236	C₄	#11763829	1	1	yes	characteristic
237	C₄	#11763832	1	1	yes	characteristic
238	C₂	#14544336	1	2	yes	—
239	C₂	#14544348	1	2	yes	—
240	C₂	#14544700	1	1	yes	characteristic
241	C₂	#14544707	1	1	yes	characteristic
242	C₂	#14544710	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#97	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#97	#595
hypocenter	C₁₇	#8786
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
35. #10587486	C₈	semidirect
162. C₁₇	#8786	semidirect

Maximal subgroups

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

subgroup	index
#14544717	2
#14544701	2
#14543614	2
#97×Dic₁₇	2
#8786	17