Subgroups — C₂×#14530557 #17187336

Order 4352.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587490	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762847	C₂	1	1	yes	maximal, characteristic
3	#14530557	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762632	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63223	—	2	1	no	—
7	C₂×#10587612	—	2	1	no	—
8	C₂×#10587490	—	2	1	no	—
9	C₂×#10587490	—	2	2	no	—
10	C₂×#10587490	C₂²	1	2	yes	—
11	C₂×#10587490	C₂²	1	4	yes	—
12	#11762847	—	2	2	no	—
13	#11762847	C₂²	1	4	yes	—
14	C₂×#10587489	—	2	1	no	—
15	C₂×#10587488	C₂²	1	2	yes	—
16	#11762632	C₂²	1	16	yes	complemented
17	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×M₃₂	C₄	1	2	yes	complemented
19	C₂×C₁₇×M₃₂	C₄	1	4	yes	complemented
20	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
21	C₂×C₁₇×M₃₂	C₂²	1	4	yes	—
22	C₂²×C₁₆×C₁₇	C₄	1	2	yes	complemented
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	D₄	1	2	yes	—
29	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
31	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
34	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
35	#10587612	—	2	4	no	—
36	#10587612	—	4	2	no	—
37	C₂×C₄⋊Dic₁₇	—	2	2	no	—
38	C₂×C₄×Dic₁₇	—	4	1	no	—
39	C₂×#63223	—	2	2	no	—
40	C₂×#63223	—	2	4	no	—
41	C₂×#63223	—	4	1	no	—
42	C₂×#63223	—	4	2	no	—
43	C₂²×#3622	—	4	1	no	—
44	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
45	C₁₇×M₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₇×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
49	#10587490	—	2	2	no	—
50	#10587490	—	2	4	no	—
51	#10587490	C₂³	1	4	yes	—
52	#10587489	—	2	4	no	—
53	#10587488	—	2	2	no	—
54	#10587488	C₂³	1	4	yes	—
55	C₂⁴×C₁₇	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
66	C₂²×Dic₁₇	—	4	2	no	—
67	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	4	4	no	—
86	C₄⋊Dic₁₇	—	2	4	no	—
87	C₄⋊Dic₁₇	—	4	2	no	—
88	C₄×Dic₁₇	—	4	4	no	—
89	#63223	—	2	4	no	—
90	#63223	—	4	2	no	—
91	#63223	—	4	4	no	—
92	C₂×#3622	—	4	4	no	—
93	C₂×#3622	—	4	2	no	—
94	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
95	C₂×#2155	—	17	1	no	maximal, Sylow
96	C₂³×C₁₇	—	4	2	no	—
97	C₂³×C₁₇	#106	1	1	yes	characteristic
98	C₂³×C₁₇	#106	1	1	yes	characteristic
99	C₂³×C₁₇	#106	1	2	yes	—
100	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂³×C₁₇	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₁₇	#106	1	2	yes	—
103	C₂×C₄×C₁₇	#106	1	2	yes	—
104	C₂×C₄×C₁₇	#106	1	2	yes	—
105	C₂×C₄×C₁₇	#106	1	2	yes	—
106	C₂×C₄×C₁₇	#106	1	2	yes	—
107	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₁₇	—	4	4	no	—
121	C₂×Dic₁₇	—	8	4	no	—
122	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
128	#3622	—	4	4	no	—
129	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
130	C₂²×#108	—	17	1	no	—
131	C₂×#427	—	17	1	no	—
132	#2155	—	17	8	no	—
133	C₂×#365	—	17	4	no	—
134	C₂²×C₁₇	—	2	2	no	—
135	C₂²×C₁₇	—	2	2	no	—
136	C₂²×C₁₇	—	2	2	no	—
137	C₂²×C₁₇	—	4	2	no	—
138	C₂²×C₁₇	—	4	2	no	—
139	C₂²×C₁₇	C₂×#106	1	2	yes	—
140	C₂²×C₁₇	C₂×#106	1	2	yes	—
141	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
142	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
143	C₂²×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
144	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
145	C₄×C₁₇	C₂×#106	1	2	yes	—
146	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
147	C₄×C₁₇	C₂×#106	1	2	yes	—
148	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
150	Dic₁₇	—	8	4	no	—
151	C₂²×M₁₆	—	34	1	no	—
152	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
153	C₂×#117	—	34	1	no	—
154	C₂×M₃₂	Dic₁₇	1	2	yes	complemented
155	C₂×M₃₂	Dic₁₇	1	4	yes	complemented
156	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
157	C₂×M₃₂	C₂×D₁₇	1	4	yes	—
158	C₂²×C₁₆	Dic₁₇	1	2	yes	complemented
159	C₂×#108	—	17	2	no	—
160	C₂×#108	—	17	4	no	—
161	C₂×#108	—	34	1	no	—
162	C₂×#108	—	34	2	no	—
163	#427	—	17	4	no	—
164	#427	—	34	2	no	—
165	C₂×#107	—	34	1	no	—
166	C₂×#106	—	17	2	no	—
167	#365	—	17	16	no	—
168	C₂×C₁₇	—	2	1	no	—
169	C₂×C₁₇	—	2	1	no	—
170	C₂×C₁₇	—	2	2	no	—
171	C₂×C₁₇	—	2	2	no	—
172	C₂×C₁₇	#2155	1	2	yes	complemented
173	C₂×C₁₇	C₂²×#106	1	1	yes	characteristic
174	C₂³×C₄	D₆₈	1	1	yes	characteristic
175	C₂×M₁₆	—	34	2	no	—
176	C₂×M₁₆	—	34	4	no	—
177	C₂×M₁₆	—	68	1	no	—
178	C₂×M₁₆	—	68	2	no	—
179	C₂²×C₈	—	68	1	no	—
180	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
181	C₂²×C₈	Dic₃₄	1	2	yes	—
182	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
183	C₂²×C₈	D₆₈	1	1	yes	characteristic
184	C₂²×C₈	D₆₈	1	2	yes	—
185	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
190	#117	—	34	4	no	—
191	#117	—	68	2	no	—
192	C₂×C₄⋊C₄	—	34	2	no	—
193	C₂×C₂²⋊C₄	—	68	1	no	—
194	C₂×C₄²	—	68	1	no	—
195	M₃₂	C₂×Dic₁₇	1	4	yes	—
196	M₃₂	C₂×Dic₁₇	1	8	yes	complemented
197	M₃₂	C₂²×D₁₇	1	4	yes	—
198	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
199	C₂×C₁₆	C₂×Dic₁₇	1	8	yes	complemented
200	#108	—	17	4	no	—
201	#108	—	34	2	no	—
202	#108	—	34	4	no	—
203	#107	—	34	4	no	—
204	#106	—	17	4	no	—
205	#106	—	34	2	no	—
206	C₁₇	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
207	C₂⁴	#63219	1	1	yes	characteristic
208	C₂²×C₄	—	68	2	no	—
209	C₂²×C₄	#63219	1	2	yes	—
210	C₂²×C₄	#63219	1	1	yes	characteristic
211	C₂²×C₄	#63219	1	1	yes	characteristic
212	C₂²×C₄	#63219	1	2	yes	—
213	C₂²×C₄	#63219	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
216	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₆₈	1	2	yes	—
219	M₁₆	—	34	4	no	—
220	M₁₆	—	68	2	no	—
221	M₁₆	—	68	4	no	—
222	C₂×C₈	—	68	2	no	—
223	C₂×C₈	—	68	4	no	—
224	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
225	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
226	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
227	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
228	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
229	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
231	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
233	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
234	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
235	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
236	C₂×C₈	C₂×D₆₈	1	2	yes	—
237	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
239	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
241	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
242	C₄⋊C₄	—	34	4	no	—
243	C₄⋊C₄	—	68	2	no	—
244	C₂²⋊C₄	—	68	4	no	—
245	C₄²	—	68	4	no	—
246	C₁₆	C₂²×Dic₁₇	1	8	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#10587488	1	1	yes	characteristic
249	C₂³	#10587488	1	1	yes	characteristic
250	C₂³	#10587488	1	2	yes	—
251	C₂³	C₂×#63219	1	1	yes	characteristic
252	C₂³	C₂×#63219	1	2	yes	—
253	C₂×C₄	—	68	4	no	—
254	C₂×C₄	—	136	4	no	—
255	C₂×C₄	#10587488	1	2	yes	—
256	C₂×C₄	#10587488	1	2	yes	—
257	C₂×C₄	#10587488	1	1	yes	characteristic
258	C₂×C₄	#10587488	1	2	yes	—
259	C₂×C₄	#10587488	1	2	yes	—
260	C₂×C₄	#10587488	1	2	yes	—
261	C₂×C₄	#10587488	1	1	yes	characteristic
262	C₂×C₄	C₂×#63219	1	1	yes	characteristic
263	C₂×C₄	C₂×#63219	1	2	yes	—
264	C₂×C₄	C₂×#63219	1	1	yes	characteristic
265	C₂×C₄	C₂×#63219	1	2	yes	—
266	C₂×C₄	C₂×#63219	1	2	yes	—
267	C₂×C₄	C₂×#63219	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
270	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
272	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
273	C₈	—	68	4	no	—
274	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
275	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
276	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
277	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
278	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
279	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×#10587488	1	2	yes	—
286	C₂²	C₂×#10587488	1	2	yes	—
287	C₂²	C₂×#10587488	1	1	yes	characteristic
288	C₂²	C₂×#10587488	1	1	yes	characteristic
289	C₂²	C₂²×#63219	1	1	yes	characteristic
290	C₄	—	136	4	no	—
291	C₄	C₂×#10587488	1	1	yes	characteristic
292	C₄	C₂×#10587488	1	2	yes	—
293	C₄	C₂×#10587488	1	1	yes	characteristic
294	C₄	C₂×#10587488	1	2	yes	—
295	C₄	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
296	C₄	C₂²×#63219	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#14530557	1	2	yes	direct product factor
302	C₂	C₂²×#10587488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587488
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂⁴×C₁₇	SD₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	C₂×#2155
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14530557	C₂	direct (N × H)
4. C₂×#11762632	C₂	semidirect
16. #11762632	C₂²	semidirect
18. C₂×C₁₇×M₃₂	C₄	semidirect
19. C₂×C₁₇×M₃₂	C₄	semidirect
22. C₂²×C₁₆×C₁₇	C₄	semidirect
45. C₁₇×M₃₂	C₂×C₄	semidirect
48. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
154. C₂×M₃₂	Dic₁₇	semidirect
155. C₂×M₃₂	Dic₁₇	semidirect
158. C₂²×C₁₆	Dic₁₇	semidirect
172. C₂×C₁₇	#2155	semidirect
196. M₃₂	C₂×Dic₁₇	semidirect
199. C₂×C₁₆	C₂×Dic₁₇	semidirect
206. C₁₇	C₂×#2155	semidirect
301. C₂	#14530557	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×#10587490	2
C₂×#11762847	2
#14530557	2
C₂×#11762632	2
C₂²×C₁₇×M₃₂	2
C₂×#2155	17

Subgroups — C2×#14530557 #17187336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530557 #17187336