Subgroups — #16945754

Order 4352.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530583	C₂	1	1	yes	maximal, characteristic
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14526839	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
5	#11762833	—	2	1	no	—
6	C₄⋊#63214	C₂²	1	2	yes	—
7	C₂²⋊#63214	—	2	1	no	—
8	C₄×#63214	—	2	1	no	—
9	#11762562	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
12	C₁₇×#500	—	2	1	no	—
13	C₁₇×#500	—	2	1	no	—
14	C₁₇×#500	C₄	1	2	yes	—
15	C₁₇×#500	C₂²	1	2	yes	—
16	C₂×C₁₇×#107	—	2	1	no	—
17	C₂×C₁₇×#107	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	—	4	1	no	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	4	1	no	—
25	C₁₇×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×#63223	—	4	1	no	—
30	C₁₇×#107	—	2	2	no	—
31	C₁₇×#107	—	2	2	no	—
32	C₁₇×#107	—	2	2	no	—
33	C₁₇×#107	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×#63214	—	2	2	no	—
39	C₄⋊#3622	—	2	2	no	—
40	C₄⋊#3622	—	4	1	no	—
41	C₂×Q₈×C₁₇	—	4	2	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂×Q₈×C₁₇	—	8	2	no	—
44	C₂×Q₈×C₁₇	—	8	2	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₇	—	2	4	no	—
56	C₄⋊C₄×C₁₇	—	4	2	no	—
57	C₄⋊C₄×C₁₇	—	4	2	no	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₇	D₈	1	1	yes	characteristic
61	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
63	#63223	—	8	2	no	—
64	C₂×#3622	—	4	2	no	—
65	#63214	—	4	2	no	—
66	#11971	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	#102	1	1	yes	characteristic
68	Q₈×C₁₇	—	8	4	no	—
69	Q₈×C₁₇	—	8	2	no	—
70	Q₈×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	4	4	no	—
75	C₂×C₄×C₁₇	—	8	4	no	—
76	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
80	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
81	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₁₇	—	2	1	no	—
85	C₈×C₁₇	—	2	1	no	—
86	C₈×C₁₇	#101	1	2	yes	—
87	C₈×C₁₇	#137	1	2	yes	—
88	#3622	—	8	2	no	—
89	C₂×#500	D₁₇	1	1	yes	characteristic
90	#2162	—	17	1	no	—
91	#1567	—	17	1	no	—
92	#1364	—	17	4	no	—
93	C₂²×C₁₇	—	2	1	no	—
94	C₂²×C₁₇	—	2	1	no	—
95	C₂²×C₁₇	#357	1	2	yes	—
96	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	—	2	1	no	—
99	C₄×C₁₇	—	8	4	no	—
100	C₄×C₁₇	#357	1	2	yes	—
101	C₄×C₁₇	#362	1	2	yes	—
102	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
103	C₄×C₁₇	#419	1	1	yes	characteristic
104	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
105	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	#500	—	2	1	no	—
108	#500	—	2	1	no	—
109	#500	Dic₁₇	1	2	yes	—
110	#500	C₂×D₁₇	1	2	yes	—
111	C₂×#107	—	2	1	no	—
112	C₂×#107	Dic₁₇	1	2	yes	—
113	#423	—	34	1	no	—
114	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	17	2	no	—
116	C₂²⋊C₁₆	—	34	1	no	—
117	C₄×C₁₆	—	34	1	no	—
118	#335	—	17	2	no	—
119	#335	—	34	1	no	—
120	C₂×C₁₇	—	2	2	no	—
121	C₂×C₁₇	#1546	1	1	yes	characteristic
122	C₂×C₁₇	C₂×#357	1	1	yes	characteristic
123	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
124	C₂²×Q₈	—	4	1	no	—
125	C₂×M₁₆	—	68	1	no	—
126	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	4	1	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₂×C₁₆	—	34	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	2	no	—
139	#107	C₂×Dic₁₇	1	4	yes	—
140	C₄⋊C₈	—	34	2	no	—
141	C₄⋊C₈	—	68	1	no	—
142	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
146	C₁₇	#11971	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	2	no	—
150	C₂×Q₈	—	8	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	#63228	1	1	yes	characteristic
153	C₂²×C₄	#63229	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
155	M₁₆	—	136	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	68	2	no	—
159	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	#63228	1	1	yes	characteristic
168	C₄²	#63229	1	1	yes	characteristic
169	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
171	C₁₆	—	68	2	no	—
172	C₂³	#10587503	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	#10587497	1	2	yes	—
182	C₂×C₄	#10587498	1	2	yes	—
183	C₂×C₄	#10587503	1	1	yes	characteristic
184	C₂×C₄	#10587503	1	1	yes	characteristic
185	C₂×C₄	#10587503	1	1	yes	characteristic
186	C₂×C₄	C₂×#63228	1	1	yes	characteristic
187	C₂×C₄	C₂×#63229	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	136	2	no	—
192	C₈	#10587507	1	2	yes	—
193	C₈	#10587628	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#11762610	1	2	yes	—
197	C₂²	C₂×#10587503	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#11762610	1	2	yes	—
202	C₄	#11762625	1	2	yes	—
203	C₄	C₂×#10587507	1	1	yes	characteristic
204	C₄	#11762818	1	1	yes	characteristic
205	C₄	C₂×#10587497	1	1	yes	characteristic
206	C₄	C₂×#10587498	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	#14527456	1	1	yes	characteristic
209	C₂	C₂×#11762610	1	1	yes	characteristic
210	C₂	C₂×#11762625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587503
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#11971
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #14526839	C₂	semidirect
146. C₁₇	#11971	semidirect

Maximal subgroups

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

subgroup	index
#14530583	2
#14527543	2
#14526839	2
C₂×C₁₇×#500	2
#11971	17