Subgroups — #16944103

Order 4352.

386 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	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic
3	#14526754	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	—	2	1	no	—
6	#11762862	—	2	1	no	—
7	#11762585	C₂²	1	2	yes	—
8	C₄×#63214	—	2	1	no	—
9	#11762562	—	2	1	no	—
10	#11762560	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
13	C₄×Q₁₆×C₁₇	—	2	1	no	—
14	C₄×Q₁₆×C₁₇	—	2	1	no	—
15	C₄×Q₁₆×C₁₇	C₄	1	2	yes	—
16	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
17	C₂×C₁₇×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₇×#103	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	1	no	—
21	C₂×Q₁₆×C₁₇	—	2	1	no	—
22	C₂×Q₁₆×C₁₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₇	C₈	1	4	yes	complemented
24	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₇	—	2	2	no	—
28	C₄×Q₈×C₁₇	—	2	2	no	—
29	C₄×Q₈×C₁₇	—	4	1	no	—
30	C₄×Q₈×C₁₇	—	4	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
33	C₂×C₄²×C₁₇	—	4	1	no	—
34	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
35	C₁₇×#107	—	2	1	no	—
36	C₁₇×#107	C₂×C₄	1	2	yes	—
37	C₁₇×#103	—	2	2	no	—
38	C₁₇×#103	—	2	2	no	—
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²⋊#3622	—	4	1	no	—
44	#10587483	—	2	4	no	—
45	C₂×#63214	—	2	2	no	—
46	C₄⋊#3622	—	2	2	no	—
47	C₄×#3622	—	4	1	no	—
48	C₂×Q₈×C₁₇	—	2	4	no	—
49	C₂×Q₈×C₁₇	—	2	2	no	—
50	C₂×Q₈×C₁₇	—	4	1	no	—
51	C₂×Q₈×C₁₇	—	4	2	no	—
52	C₂²×C₄×C₁₇	—	4	1	no	—
53	C₂²×C₄×C₁₇	—	4	1	no	—
54	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
57	Q₁₆×C₁₇	—	2	4	no	—
58	Q₁₆×C₁₇	—	2	2	no	—
59	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₇	—	2	2	no	—
68	C₄⋊C₄×C₁₇	—	4	1	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	—	4	2	no	—
71	C₄²×C₁₇	—	4	2	no	—
72	C₄²×C₁₇	—	4	2	no	—
73	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₇	D₈	1	1	yes	characteristic
75	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
76	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×#3622	—	4	2	no	—
78	#63214	—	2	4	no	—
79	#11600	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	#102	1	1	yes	characteristic
81	Q₈×C₁₇	—	2	4	no	—
82	Q₈×C₁₇	—	4	4	no	—
83	Q₈×C₁₇	—	4	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	4	1	no	—
86	C₂×C₄×C₁₇	—	4	4	no	—
87	C₂×C₄×C₁₇	—	4	2	no	—
88	C₂×C₄×C₁₇	—	4	2	no	—
89	C₂×C₄×C₁₇	—	8	2	no	—
90	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#104	1	2	yes	—
95	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₁₇	—	2	1	no	—
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
101	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
103	#3622	—	8	2	no	—
104	C₂×C₄×Q₁₆	D₁₇	1	1	yes	characteristic
105	C₄×M₃₂	—	17	1	no	—
106	#1564	—	17	1	no	—
107	#1334	—	17	4	no	—
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	D₄⋊C₈	1	2	yes	—
111	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	4	4	no	—
116	C₄×C₁₇	—	8	2	no	—
117	C₄×C₁₇	D₄⋊C₈	1	2	yes	—
118	C₄×C₁₇	#362	1	2	yes	—
119	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
121	C₂²×Q₁₆	Dic₁₇	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	2	1	no	—
123	C₂×M₃₂	—	34	1	no	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	Dic₁₇	1	2	yes	—
127	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
128	#431	—	34	1	no	—
129	C₂×#107	Dic₁₇	1	1	yes	characteristic
130	C₂×#103	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
132	#346	—	17	2	no	—
133	C₄×C₁₆	—	34	1	no	—
134	#335	—	34	1	no	—
135	#334	—	17	2	no	—
136	C₂×C₁₇	—	2	2	no	—
137	C₂×C₁₇	C₂×D₄⋊C₈	1	1	yes	characteristic
138	C₂×C₁₇	#2145	1	1	yes	characteristic
139	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	#3622	1	4	yes	complemented
144	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
145	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
155	M₃₂	—	34	4	no	—
156	C₂×C₁₆	—	34	2	no	—
157	#107	—	2	1	no	—
158	#107	C₂×Dic₁₇	1	2	yes	—
159	C₄⋊C₈	—	34	2	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₂²⋊C₈	—	68	1	no	—
163	C₄×C₈	—	68	1	no	—
164	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
168	C₁₇	#11600	1	1	yes	characteristic, complemented, Sylow
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	#63228	1	1	yes	characteristic
176	C₂²×C₄	#63229	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
178	Q₁₆	—	2	4	no	—
179	Q₁₆	—	2	2	no	—
180	Q₁₆	C₂×#3622	1	4	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	C₂×#3622	1	2	yes	—
184	C₂×C₈	#63223	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	#63228	1	1	yes	characteristic
196	C₄²	#63229	1	1	yes	characteristic
197	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
199	C₁₆	—	34	4	no	—
200	C₂³	#10587503	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂²⋊#3622	1	2	yes	—
211	C₂×C₄	#10587503	1	1	yes	characteristic
212	C₂×C₄	#10587503	1	1	yes	characteristic
213	C₂×C₄	#10587503	1	1	yes	characteristic
214	C₂×C₄	#10587508	1	2	yes	—
215	C₂×C₄	C₂×#63228	1	1	yes	characteristic
216	C₂×C₄	C₂×#63229	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	136	2	no	—
221	C₈	C₂²⋊#3622	1	2	yes	—
222	C₈	C₂²×#3622	1	1	yes	characteristic
223	C₈	C₂×#63223	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	D₄⋊#3622	1	2	yes	—
227	C₂²	C₂×#10587503	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	8	2	no	—
233	C₄	D₄⋊#3622	1	2	yes	—
234	C₄	#11762625	1	2	yes	—
235	C₄	C₂×C₂²⋊#3622	1	1	yes	characteristic
236	C₄	C₂×#10587508	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×D₄⋊#3622	1	1	yes	characteristic
239	C₂	#14530520	1	1	yes	characteristic
240	C₂	C₂×#11762625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587503
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₄×Q₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₇
hypocenter	C₁₇	#11600
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #14526754	C₂	semidirect
23. C₂×Q₁₆×C₁₇	C₈	semidirect
143. C₂×Q₁₆	#3622	semidirect
168. C₁₇	#11600	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587483	2
#14527532	2
#14526754	2
C₂×C₄×Q₁₆×C₁₇	2
#11600	17