Subgroups — #17981688

Order 4352.

422 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545926	C₂	1	2	yes	maximal
2	C₂×#11763899	C₂	1	2	yes	maximal
3	C₂×#11763898	C₂	1	2	yes	maximal
4	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3633	C₂²	1	4	yes	—
6	(C₂×#3633)⋊C₄	—	2	4	no	—
7	#11763899	—	2	4	no	—
8	#11763898	—	2	4	no	—
9	C₄×#63242	C₄	1	1	yes	characteristic
10	C₂×C₈×#595	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₄	1	1	yes	characteristic
12	C₂×C₈×#595	C₂²	1	1	yes	characteristic
13	C₂×C₈×#595	C₂²	1	1	yes	characteristic
14	C₄²×#595	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
16	C₂²×#3633	—	2	4	no	—
17	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×#595	D₄	1	2	yes	—
19	C₂×C₄×#595	Q₈	1	2	yes	—
20	C₂×C₄×#595	C₂³	1	1	yes	characteristic
21	C₂×#63242	C₂×C₄	1	1	yes	characteristic
22	C₂×#63242	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#3633	—	2	4	no	—
24	C₄⋊#3633	—	2	4	no	—
25	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×#3624	—	2	2	no	—
29	C₈×#595	—	2	2	no	—
30	C₈×#595	—	2	2	no	—
31	C₈×#595	D₄	1	2	yes	—
32	C₈×#595	D₄	1	2	yes	—
33	C₈×#595	Q₈	1	2	yes	—
34	C₈×#595	Q₈	1	2	yes	—
35	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic, complemented
36	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
37	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
38	C₄×#3622	C₈	1	1	yes	characteristic, complemented
39	C₂²×#595	C₄⋊C₄	1	2	yes	—
40	C₂²×#595	C₂×D₄	1	1	yes	characteristic
41	C₂²×#595	C₂×Q₈	1	1	yes	characteristic
42	C₂×#3633	—	4	4	no	—
43	C₂×#3633	—	4	8	no	—
44	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×#3624	—	2	1	no	—
47	C₂×#3624	—	2	1	no	—
48	C₄×#595	—	2	4	no	—
49	C₄×#595	—	2	4	no	—
50	C₄×#595	—	2	1	no	—
51	C₄×#595	—	2	1	no	—
52	C₄×#595	C₂×D₄	1	1	yes	characteristic
53	C₄×#595	C₂×D₄	1	1	yes	characteristic
54	C₄×#595	C₂×Q₈	1	1	yes	characteristic
55	C₄×#595	C₂×Q₈	1	1	yes	characteristic
56	#63242	—	2	1	no	—
57	#63242	—	2	1	no	—
58	#63242	C₄⋊C₄	1	2	yes	—
59	#63242	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₁₇	M₁₆	1	2	yes	—
65	C₂×#3622	C₂×C₈	1	1	yes	characteristic
66	C₂×#3622	C₂×C₈	1	1	yes	characteristic
67	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
68	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
69	C₈×D₁₇	M₁₆	1	2	yes	—
70	C₈×D₁₇	M₁₆	1	2	yes	—
71	#8807	—	17	1	no	maximal, Sylow
72	C₂²×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×#595	—	2	2	no	—
74	C₂×#595	—	2	2	no	—
75	C₂×#595	—	2	2	no	—
76	C₂×#595	#108	1	4	yes	—
77	C₂×#595	#127	1	1	yes	characteristic
78	C₂×#595	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#595	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×#595	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×#595	#135	1	4	yes	—
82	#3633	—	8	4	no	—
83	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
86	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₄×D₁₇	—	2	2	no	—
88	C₄×D₁₇	—	2	2	no	—
89	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
92	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
93	#3624	—	2	2	no	—
94	#3624	—	2	2	no	—
95	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
96	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
97	#3622	C₄⋊C₈	1	2	yes	—
98	#3622	C₄⋊C₈	1	2	yes	—
99	#1831	—	17	2	no	—
100	C₄²×C₈	—	17	1	no	—
101	C₂×#335	—	17	2	no	—
102	C₂×#334	—	17	2	no	—
103	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂×D₁₇	#384	1	1	yes	characteristic
105	C₂×D₁₇	#423	1	1	yes	characteristic
106	C₂×D₁₇	#427	1	2	yes	—
107	C₂×D₁₇	C₂×#108	1	2	yes	—
108	#595	—	2	4	no	—
109	#595	—	2	1	no	—
110	#595	—	2	1	no	—
111	#595	#495	1	2	yes	—
112	#595	#499	1	2	yes	—
113	C₄×C₁₇	—	2	2	no	—
114	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	Dic₁₇	—	2	2	no	—
117	Dic₁₇	#423	1	1	yes	characteristic
118	Dic₁₇	#423	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	17	4	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	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₄³	—	17	1	no	—
127	#336	—	34	4	no	—
128	#335	—	34	4	no	—
129	#334	—	34	4	no	—
130	C₂×C₁₇	#1564	1	2	yes	—
131	C₂×C₁₇	#1842	1	1	yes	characteristic
132	D₁₇	#1568	1	2	yes	—
133	D₁₇	#1846	1	1	yes	characteristic
134	D₁₇	#1848	1	1	yes	characteristic
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	34	4	no	—
140	C₂×C₄²	—	17	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	2	no	—
144	C₂×C₄²	—	17	2	no	—
145	C₄⋊C₈	—	34	4	no	—
146	C₄⋊C₈	—	34	4	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	17	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	1	no	—
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	34	2	no	—
155	C₄×C₈	—	34	2	no	—
156	C₄×C₈	—	34	2	no	—
157	C₄×C₈	#3633	1	1	yes	characteristic, complemented
158	C₁₇	#8807	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	17	2	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	—	17	2	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	2	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	—	34	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	68	4	no	—
177	C₂×C₈	—	68	8	no	—
178	C₂×C₈	C₂×#3633	1	1	yes	characteristic
179	C₂×C₈	C₂×#3633	1	1	yes	characteristic
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	17	1	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	2	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	34	1	no	—
187	C₄²	—	34	1	no	—
188	C₄²	—	34	4	no	—
189	C₄²	—	34	4	no	—
190	C₄²	C₂×#3633	1	1	yes	characteristic
191	C₂³	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	4	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	4	no	—
203	C₂×C₄	—	17	2	no	—
204	C₂×C₄	—	34	2	no	—
205	C₂×C₄	—	34	2	no	—
206	C₂×C₄	—	34	2	no	—
207	C₂×C₄	—	34	2	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	C₄⋊#3633	1	2	yes	—
210	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
211	C₈	—	17	2	no	—
212	C₈	—	17	2	no	—
213	C₈	—	34	2	no	—
214	C₈	—	34	2	no	—
215	C₈	—	136	4	no	—
216	C₈	C₄⋊#3633	1	2	yes	—
217	C₈	C₄⋊#3633	1	2	yes	—
218	C₂²	—	17	1	no	—
219	C₂²	—	17	1	no	—
220	C₂²	—	17	2	no	—
221	C₂²	—	17	2	no	—
222	C₂²	C₂×C₄⋊#3633	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	17	2	no	—
225	C₄	—	17	2	no	—
226	C₄	—	17	1	no	—
227	C₄	—	17	1	no	—
228	C₄	—	34	4	no	—
229	C₄	—	34	1	no	—
230	C₄	—	34	1	no	—
231	C₄	—	34	2	no	—
232	C₄	C₂×C₄⋊#3633	1	1	yes	characteristic
233	C₄	C₂×C₄⋊#3633	1	1	yes	characteristic
234	C₂	—	17	1	no	—
235	C₂	—	17	2	no	—
236	C₂	—	17	1	no	—
237	C₂	#14545908	1	2	yes	—
238	C₂	#14545928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#3633
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×#595	C₂³
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#8807
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₈×C₁₇	C₈	semidirect
36. C₈×Dic₁₇	C₈	semidirect
37. C₈×Dic₁₇	C₈	semidirect
38. C₄×#3622	C₈	semidirect
157. C₄×C₈	#3633	semidirect
158. C₁₇	#8807	semidirect

Maximal subgroups

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

subgroup	index
#14545926	2
C₂×#11763899	2
C₂×#11763898	2
C₄×C₈×#595	2
#8807	17