Subgroups — #17981627

Order 4352.

414 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	#14545908	C₂	1	2	yes	maximal
2	#14545893	C₂	1	2	yes	maximal
3	#14545789	C₂	1	2	yes	maximal
4	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
5	#11763922	C₂²	1	4	yes	—
6	#11763899	C₄	1	4	yes	complemented
7	#11763898	C₄	1	4	yes	complemented
8	#11763893	—	2	2	no	—
9	C₈×#3633	—	2	2	no	—
10	C₄×#63242	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₄	1	1	yes	characteristic
12	C₂×C₈×#595	C₄	1	1	yes	characteristic
13	C₂×C₈×#595	C₂²	1	1	yes	characteristic
14	C₂×C₈×#595	C₂²	1	1	yes	characteristic
15	C₄²×#595	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
17	C₂×C₄×#595	—	2	2	no	—
18	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×#595	C₂³	1	1	yes	characteristic
20	C₂×#63242	C₂×C₄	1	1	yes	characteristic
21	C₂×#63242	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊#3633	—	2	4	no	—
23	C₄⋊#3633	C₂×C₄	1	8	yes	—
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	1	no	—
30	C₈×#595	—	2	1	no	—
31	C₈×#595	C₈	1	2	yes	complemented
32	C₈×#595	C₈	1	2	yes	complemented
33	C₈×#595	C₂×C₄	1	2	yes	—
34	C₈×#595	C₂×C₄	1	2	yes	—
35	C₈×#595	D₄	1	1	yes	characteristic
36	C₈×#595	D₄	1	1	yes	characteristic
37	C₈×#595	Q₈	1	1	yes	characteristic
38	C₈×#595	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic, complemented
40	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
41	C₈×Dic₁₇	C₈	1	1	yes	characteristic, complemented
42	C₄×#3622	C₈	1	1	yes	characteristic, complemented
43	C₂²×#595	—	2	1	no	—
44	C₂²×#595	#41	1	2	yes	—
45	C₂×#3633	—	2	8	no	—
46	C₂×C₄×D₁₇	—	2	1	no	—
47	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×#3624	—	2	1	no	—
49	C₂×#3624	M₁₆	1	2	yes	—
50	C₄×#595	—	2	8	no	—
51	C₄×#595	—	2	1	no	—
52	C₄×#595	C₂×C₈	1	2	yes	—
53	C₄×#595	C₂²×C₄	1	2	yes	—
54	C₄×#595	C₂×D₄	1	1	yes	characteristic
55	C₄×#595	C₂×Q₈	1	1	yes	characteristic
56	#63242	—	2	1	no	—
57	#63242	C₄²	1	2	yes	—
58	#63242	C₄⋊C₄	1	1	yes	characteristic
59	#63242	C₄⋊C₄	1	1	yes	characteristic
60	#63242	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₁₇	—	2	1	no	—
65	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
66	C₂×#3622	C₂×C₈	1	1	yes	characteristic
67	C₂×#3622	C₂×C₈	1	1	yes	characteristic
68	C₈×D₁₇	C₄²	1	1	yes	characteristic
69	C₈×D₁₇	C₄²	1	1	yes	characteristic
70	C₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
71	C₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
72	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
73	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
74	C₈×D₁₇	M₁₆	1	1	yes	characteristic
75	C₈×D₁₇	M₁₆	1	1	yes	characteristic
76	#8266	—	17	1	no	maximal, Sylow
77	C₂²×D₁₇	#117	1	1	yes	characteristic
78	C₂×#595	—	2	4	no	—
79	C₂×#595	—	2	2	no	—
80	C₂×#595	—	2	4	no	—
81	C₂×#595	C₄×D₄	1	2	yes	—
82	C₂×#595	C₄×Q₈	1	2	yes	—
83	#3633	—	4	8	no	—
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×Dic₁₇	—	2	1	no	—
87	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
88	C₄×D₁₇	—	2	4	no	—
89	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₄×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
92	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
93	#3624	—	2	2	no	—
94	#3624	C₄⋊C₈	1	2	yes	—
95	#3624	C₄⋊C₈	1	2	yes	—
96	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
97	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
98	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
99	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
100	#3622	C₄×C₈	1	1	yes	characteristic
101	#3622	C₄×C₈	1	1	yes	characteristic
102	#3622	C₄⋊C₈	1	1	yes	characteristic
103	#3622	C₄⋊C₈	1	1	yes	characteristic
104	C₄²×C₈	—	17	1	no	—
105	#1564	—	17	2	no	—
106	#1450	—	17	2	no	—
107	#1276	—	17	2	no	—
108	C₂²×C₁₇	#431	1	1	yes	characteristic
109	C₂×D₁₇	—	2	2	no	—
110	C₂×D₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₁₇	#431	1	1	yes	characteristic
112	#595	—	2	4	no	—
113	#595	—	2	2	no	—
114	#595	#443	1	4	yes	—
115	C₄×C₁₇	—	2	2	no	—
116	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
117	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
118	Dic₁₇	—	2	2	no	—
119	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
120	Dic₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	#431	—	17	4	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	C₂×C₄×C₈	—	17	1	no	—
127	C₂×C₄×C₈	—	17	1	no	—
128	C₄³	—	17	1	no	—
129	#335	—	17	4	no	—
130	#334	—	17	4	no	—
131	#322	—	34	2	no	—
132	C₈²	—	34	2	no	—
133	C₂×C₁₇	—	2	1	no	—
134	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
135	D₁₇	—	2	1	no	—
136	D₁₇	#1772	1	2	yes	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	34	2	no	—
145	C₄⋊C₈	—	17	8	no	—
146	C₂²⋊C₈	—	34	4	no	—
147	C₄×C₈	—	17	1	no	—
148	C₄×C₈	—	17	1	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	2	no	—
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	—	34	4	no	—
159	C₄×C₈	—	34	1	no	—
160	C₄×C₈	—	34	1	no	—
161	C₄×C₈	—	34	2	no	—
162	C₄×C₈	#3633	1	1	yes	characteristic, complemented
163	C₁₇	#8266	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	2	no	—
166	C₂²×C₄	—	34	1	no	—
167	C₂²×C₄	—	34	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	17	1	no	—
176	C₂×C₈	—	17	1	no	—
177	C₂×C₈	—	17	1	no	—
178	C₂×C₈	—	17	1	no	—
179	C₂×C₈	—	17	1	no	—
180	C₂×C₈	—	17	1	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	2	no	—
183	C₂×C₈	—	34	1	no	—
184	C₂×C₈	—	34	1	no	—
185	C₂×C₈	—	34	8	no	—
186	C₂×C₈	C₂×#3633	1	1	yes	characteristic
187	C₂×C₈	C₂×#3633	1	1	yes	characteristic
188	C₄²	—	17	2	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	1	no	—
191	C₄²	—	17	2	no	—
192	C₄²	—	17	1	no	—
193	C₄²	—	34	1	no	—
194	C₄²	—	34	1	no	—
195	C₄²	—	34	8	no	—
196	C₄²	C₂×#3633	1	1	yes	characteristic
197	C₂³	—	17	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	17	2	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	17	1	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	2	no	—
206	C₂×C₄	—	34	4	no	—
207	C₂×C₄	—	34	4	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	—	34	4	no	—
210	C₂×C₄	—	34	1	no	—
211	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
212	C₈	—	17	2	no	—
213	C₈	—	17	2	no	—
214	C₈	—	17	1	no	—
215	C₈	—	17	1	no	—
216	C₈	—	17	1	no	—
217	C₈	—	17	1	no	—
218	C₈	—	34	2	no	—
219	C₈	—	68	8	no	—
220	C₈	C₄×#3633	1	1	yes	characteristic
221	C₈	C₄×#3633	1	1	yes	characteristic
222	C₈	C₄⋊#3633	1	1	yes	characteristic
223	C₈	C₄⋊#3633	1	1	yes	characteristic
224	C₂²	—	17	1	no	—
225	C₂²	—	17	1	no	—
226	C₂²	—	34	2	no	—
227	C₂²	#11763922	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	17	4	no	—
230	C₄	—	17	1	no	—
231	C₄	—	17	1	no	—
232	C₄	—	34	2	no	—
233	C₄	—	34	2	no	—
234	C₄	—	34	4	no	—
235	C₄	C₂×C₄×#3633	1	1	yes	characteristic
236	C₄	C₂×C₄⋊#3633	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	17	2	no	—
239	C₂	—	34	1	no	—
240	C₂	C₄×C₄⋊#3633	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄⋊#3633
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂×C₁₇	C₄×C₄⋊C₈
prime radical	C₂²×D₁₇	#117
cosocle	C₂×C₄×#595	C₂³
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#8266
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11763899	C₄	semidirect
7. #11763898	C₄	semidirect
31. C₈×#595	C₈	semidirect
32. C₈×#595	C₈	semidirect
39. C₄×C₈×C₁₇	C₈	semidirect
40. C₈×Dic₁₇	C₈	semidirect
41. C₈×Dic₁₇	C₈	semidirect
42. C₄×#3622	C₈	semidirect
162. C₄×C₈	#3633	semidirect
163. C₁₇	#8266	semidirect

Maximal subgroups

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

subgroup	index
#14545908	2
#14545893	2
#14545789	2
C₄×C₈×#595	2
#8266	17