Subgroups — C₂×#14543844 #17960508

Order 4352.

574 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587536	C₂	1	2	yes	maximal
2	#14543844	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63243	—	2	2	no	—
5	C₂×#10587536	—	2	4	no	—
6	C₂×#10587536	—	2	2	no	—
7	C₂×#10587536	C₄	1	8	yes	—
8	C₂×#10587536	C₂²	1	4	yes	—
9	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₁₇	—	2	4	no	—
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
13	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
16	C₂²×#3624	—	4	2	no	—
17	C₂×#63243	—	2	8	no	—
18	C₂×#63243	—	2	4	no	—
19	C₂×#63243	—	4	2	no	—
20	C₂×#63243	—	4	4	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	Q₈	1	1	yes	characteristic
23	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	D₄	1	4	yes	—
26	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	#10587536	—	2	8	no	—
30	#10587536	—	2	4	no	—
31	#10587536	C₂×C₄	1	8	yes	—
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	C₂×C₄	1	2	yes	—
36	C₁₆×D₁₇	—	2	4	no	—
37	C₁₆×D₁₇	—	2	4	no	—
38	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
41	C₂×C₄×D₁₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
44	C₂×C₄×D₁₇	SD₁₆	1	4	yes	—
45	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×#3624	—	4	8	no	—
48	C₂×#3624	—	4	4	no	—
49	#63243	—	2	8	no	—
50	#63243	—	4	8	no	—
51	#63243	—	4	4	no	—
52	C₂×C₈×C₁₇	C₄²	1	2	yes	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#3622	C₄⋊C₄	1	4	yes	—
58	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₁₇	C₄²	1	2	yes	—
60	C₈×D₁₇	C₂²⋊C₄	1	4	yes	—
61	C₈×D₁₇	C₂²⋊C₄	1	2	yes	—
62	C₈×D₁₇	C₄⋊C₄	1	4	yes	—
63	C₈×D₁₇	C₂²×C₄	1	2	yes	—
64	C₈×D₁₇	C₂×D₄	1	2	yes	—
65	C₁₆×C₁₇	—	2	2	no	—
66	#63214	—	2	2	no	—
67	C₂×#1381	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#107	1	1	yes	characteristic
69	C₂²×D₁₇	—	2	2	no	—
70	C₂²×D₁₇	—	2	2	no	—
71	C₂²×D₁₇	#103	1	4	yes	—
72	C₂²×D₁₇	#107	1	1	yes	characteristic
73	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#95	1	2	yes	—
75	C₂×C₄×C₁₇	#106	1	2	yes	—
76	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₁₇	#102	1	4	yes	—
79	C₂×Dic₁₇	#107	1	1	yes	characteristic
80	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
81	C₄×D₁₇	#95	1	2	yes	—
82	C₄×D₁₇	#102	1	4	yes	—
83	C₄×D₁₇	#103	1	4	yes	—
84	C₄×D₁₇	#106	1	2	yes	—
85	C₄×D₁₇	C₂×C₂²⋊C₄	1	2	yes	—
86	C₄×D₁₇	C₂×SD₁₆	1	2	yes	—
87	#3624	—	4	8	no	—
88	C₈×C₁₇	#95	1	2	yes	—
89	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
90	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	#3622	#95	1	2	yes	—
92	#3622	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₁₆	—	17	1	no	—
94	C₂²×#108	—	17	2	no	—
95	#1381	—	17	4	no	—
96	C₂²×C₁₇	#340	1	2	yes	—
97	C₂²×C₁₇	#368	1	2	yes	—
98	C₂²×C₁₇	#368	1	2	yes	—
99	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
100	C₂×D₁₇	—	2	8	no	—
101	C₂×D₁₇	—	2	1	no	—
102	C₂×D₁₇	—	2	1	no	—
103	C₂×D₁₇	—	2	1	no	—
104	C₂×D₁₇	—	2	1	no	—
105	C₂×D₁₇	#340	1	2	yes	—
106	C₂×D₁₇	C₂×#103	1	2	yes	—
107	C₄×C₁₇	#340	1	2	yes	—
108	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
109	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
110	Dic₁₇	#340	1	2	yes	—
111	Dic₁₇	C₂×#102	1	2	yes	—
112	C₂²×M₁₆	—	34	2	no	—
113	C₂³×C₈	—	17	1	no	—
114	C₂²×C₁₆	—	17	1	no	—
115	C₂²×C₁₆	—	17	2	no	—
116	C₂²×C₁₆	—	17	2	no	—
117	C₂²×C₁₆	—	34	4	no	—
118	C₂²×C₁₆	#595	1	1	yes	characteristic
119	C₂×#108	—	17	4	no	—
120	C₂×#108	—	17	8	no	—
121	C₂×#108	—	34	2	no	—
122	C₂×#108	—	34	4	no	—
123	C₂×C₁₇	#1381	1	4	yes	complemented
124	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
125	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
126	C₂×C₁₇	C₂×#368	1	1	yes	characteristic
127	D₁₇	—	2	2	no	—
128	D₁₇	—	2	2	no	—
129	C₂³×C₄	—	17	1	no	—
130	C₂×M₁₆	—	34	4	no	—
131	C₂×M₁₆	—	34	8	no	—
132	C₂×M₁₆	—	68	4	no	—
133	C₂×M₁₆	—	68	2	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	4	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	4	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	68	2	no	—
142	C₂²×C₈	C₂×#595	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	17	2	no	—
145	C₂×C₁₆	—	34	4	no	—
146	C₂×C₁₆	—	34	4	no	—
147	C₂×C₁₆	—	34	2	no	—
148	C₂×C₁₆	C₂×#595	1	2	yes	—
149	#108	—	17	8	no	—
150	#108	—	34	4	no	—
151	#108	—	34	8	no	—
152	C₁₇	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
153	C₂⁴	—	17	1	no	—
154	C₂²×C₄	—	17	1	no	—
155	C₂²×C₄	—	17	1	no	—
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	4	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	4	no	—
161	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
162	M₁₆	—	34	8	no	—
163	M₁₆	—	68	4	no	—
164	M₁₆	—	68	8	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	17	4	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	—	17	4	no	—
173	C₂×C₈	—	17	4	no	—
174	C₂×C₈	—	68	4	no	—
175	C₂×C₈	—	68	8	no	—
176	C₂×C₈	C₄×#595	1	2	yes	—
177	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊#595	1	2	yes	—
179	C₂×C₈	C₂²×#595	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	34	2	no	—
182	C₂³	—	17	1	no	—
183	C₂³	—	17	1	no	—
184	C₂³	—	17	4	no	—
185	C₂³	—	34	2	no	—
186	C₂³	—	34	2	no	—
187	C₂³	#10587534	1	1	yes	characteristic
188	C₂×C₄	—	17	1	no	—
189	C₂×C₄	—	17	1	no	—
190	C₂×C₄	—	17	4	no	—
191	C₂×C₄	—	17	2	no	—
192	C₂×C₄	—	17	2	no	—
193	C₂×C₄	—	17	2	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	4	no	—
196	C₂×C₄	—	17	4	no	—
197	C₂×C₄	#10587533	1	2	yes	—
198	C₂×C₄	#10587534	1	1	yes	characteristic
199	C₂×C₄	(C₂×#595)⋊C₄	1	2	yes	—
200	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
201	C₈	—	17	2	no	—
202	C₈	—	17	2	no	—
203	C₈	—	68	8	no	—
204	C₈	(C₂×#595)⋊C₄	1	2	yes	—
205	C₈	C₂×C₄×#595	1	1	yes	characteristic
206	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
207	C₂²	—	17	2	no	—
208	C₂²	—	17	2	no	—
209	C₂²	—	34	8	no	—
210	C₂²	—	34	1	no	—
211	C₂²	—	34	1	no	—
212	C₂²	—	34	1	no	—
213	C₂²	—	34	1	no	—
214	C₂²	#11763683	1	2	yes	—
215	C₂²	#11763737	1	2	yes	—
216	C₂²	#11763738	1	2	yes	—
217	C₂²	C₂×#10587534	1	1	yes	characteristic
218	C₄	—	17	2	no	—
219	C₄	—	17	2	no	—
220	C₄	#11763683	1	2	yes	—
221	C₄	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic
222	C₄	C₂×#10587533	1	1	yes	characteristic
223	C₂	—	34	2	no	—
224	C₂	—	34	2	no	—
225	C₂	#14543844	1	4	yes	direct product factor
226	C₂	C₂×#11763683	1	1	yes	characteristic
227	C₂	C₂×#11763737	1	1	yes	characteristic
228	C₂	C₂×#11763738	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587534
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂³×C₁₇	#107
prime radical	C₂³×D₁₇	Q₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂×#1381
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14543844	C₂	direct (N × H)
123. C₂×C₁₇	#1381	semidirect
152. C₁₇	C₂×#1381	semidirect
225. C₂	#14543844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10587536	2
#14543844	2
C₂²×C₁₆×D₁₇	2
C₂×#1381	17

Subgroups — C2×#14543844 #17960508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543844 #17960508