Subgroups — C₂×#14530388 #17186981

Order 4352.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762764	C₂	1	1	yes	maximal, characteristic
3	#14530388	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762586	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63223	C₄	1	1	yes	characteristic
7	C₂×#10587612	C₄	1	1	yes	characteristic
8	C₂×#10587483	C₄	1	2	yes	complemented
9	C₂×#10587483	C₄	1	4	yes	complemented
10	C₂×#10587483	C₂²	1	2	yes	—
11	C₂×#10587483	C₂²	1	4	yes	—
12	C₂²×#63214	C₄	1	2	yes	complemented
13	#11762764	C₄	1	4	yes	—
14	#11762764	C₂²	1	4	yes	—
15	C₂×#10587486	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
17	#11762586	C₂²	1	16	yes	complemented
18	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×M₃₂	C₄	1	2	yes	complemented
20	C₂×C₁₇×M₃₂	C₄	1	4	yes	complemented
21	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
22	C₂×C₁₇×M₃₂	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₁₇	C₄	1	2	yes	complemented
24	C₂³×C₄×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	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₁₇	—	2	1	no	—
36	#10587612	—	2	2	no	—
37	#10587612	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₇	—	2	1	no	—
39	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
40	C₂×#63223	—	2	1	no	—
41	C₂×#63223	—	2	1	no	—
42	C₂×#63223	—	2	2	no	—
43	C₂×#63223	C₂×C₄	1	4	yes	—
44	C₂²×#3622	C₂×C₄	1	2	yes	—
45	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
46	C₁₇×M₃₂	C₂×C₄	1	8	yes	complemented
47	C₁₇×M₃₂	C₂³	1	4	yes	—
48	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
49	C₂×C₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
50	#10587486	C₂×C₄	1	4	yes	—
51	#10587486	C₂³	1	4	yes	—
52	C₈×Dic₁₇	C₂×C₄	1	8	yes	—
53	#10587483	C₂×C₄	1	4	yes	—
54	#10587483	C₂×C₄	1	8	yes	complemented
55	#10587483	C₂³	1	4	yes	—
56	C₂×#63214	C₂×C₄	1	4	yes	—
57	C₂×#63214	C₂×C₄	1	8	yes	complemented
58	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
69	C₂²×Dic₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₄²	1	2	yes	—
72	C₂×C₈×C₁₇	C₄²	1	2	yes	—
73	C₂×C₈×C₁₇	C₄²	1	2	yes	—
74	C₂×C₈×C₁₇	C₄²	1	2	yes	—
75	C₂×C₈×C₁₇	C₄²	1	2	yes	—
76	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	2	4	no	—
89	C₄⋊Dic₁₇	—	2	4	no	—
90	C₄×Dic₁₇	—	2	2	no	—
91	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
92	#63223	—	2	4	no	—
93	#63223	—	2	4	no	—
94	C₂×#3622	—	2	2	no	—
95	C₂×#3622	—	2	2	no	—
96	C₂×#3622	C₂²×C₄	1	4	yes	—
97	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
98	#63214	C₂²×C₄	1	8	yes	—
99	C₂×#2110	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	—	4	2	no	—
101	C₂³×C₁₇	#97	1	1	yes	characteristic
102	C₂³×C₁₇	#97	1	1	yes	characteristic
103	C₂³×C₁₇	#97	1	2	yes	—
104	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂³×C₁₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₇	#97	1	2	yes	—
107	C₂×C₄×C₁₇	#97	1	2	yes	—
108	C₂×C₄×C₁₇	#97	1	2	yes	—
109	C₂×C₄×C₁₇	#97	1	2	yes	—
110	C₂×C₄×C₁₇	#97	1	2	yes	—
111	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₁₇	—	2	4	no	—
125	C₂×Dic₁₇	—	4	4	no	—
126	C₈×C₁₇	C₂×C₄²	1	2	yes	—
127	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
128	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₁₇	C₂×C₄²	1	2	yes	—
130	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
132	#3622	—	2	4	no	—
133	C₂²×M₃₂	—	17	1	no	—
134	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
135	C₂×#405	—	17	1	no	—
136	#2110	—	17	8	no	—
137	C₂×#347	—	17	4	no	—
138	C₂²×C₁₇	—	2	2	no	—
139	C₂²×C₁₇	—	2	2	no	—
140	C₂²×C₁₇	—	2	2	no	—
141	C₂²×C₁₇	—	4	2	no	—
142	C₂²×C₁₇	—	4	2	no	—
143	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
144	C₂²×C₁₇	C₂×#97	1	2	yes	—
145	C₂²×C₁₇	C₂×#97	1	2	yes	—
146	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
147	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
148	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
149	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
150	C₄×C₁₇	C₂×#97	1	2	yes	—
151	C₄×C₁₇	C₂×#97	1	2	yes	—
152	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
154	Dic₁₇	—	4	4	no	—
155	C₂²×M₁₆	—	17	1	no	—
156	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₂×#117	—	17	1	no	—
158	C₂×M₃₂	—	17	2	no	—
159	C₂×M₃₂	—	17	4	no	—
160	C₂×M₃₂	—	17	2	no	—
161	C₂×M₃₂	—	17	4	no	—
162	C₂×M₃₂	Dic₁₇	1	2	yes	complemented
163	C₂×M₃₂	Dic₁₇	1	4	yes	complemented
164	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
165	C₂×M₃₂	C₂×D₁₇	1	4	yes	—
166	C₂²×C₁₆	—	17	2	no	—
167	C₂²×C₁₆	Dic₁₇	1	2	yes	complemented
168	#405	—	17	4	no	—
169	#405	—	17	4	no	—
170	C₂×#97	—	17	2	no	—
171	C₂×C₄×C₈	—	17	2	no	—
172	#347	—	17	16	no	—
173	C₂×C₁₇	—	2	1	no	—
174	C₂×C₁₇	—	2	1	no	—
175	C₂×C₁₇	—	2	2	no	—
176	C₂×C₁₇	—	2	2	no	—
177	C₂×C₁₇	#2110	1	2	yes	complemented
178	C₂×C₁₇	C₂²×#97	1	1	yes	characteristic
179	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
180	C₂×M₁₆	—	17	4	no	—
181	C₂×M₁₆	—	34	1	no	—
182	C₂×M₁₆	—	34	1	no	—
183	C₂×M₁₆	—	34	2	no	—
184	C₂²×C₈	—	17	2	no	—
185	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₁₇	1	2	yes	—
187	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₁₇	1	2	yes	—
189	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
192	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
195	#117	—	17	4	no	—
196	#117	—	34	2	no	—
197	C₂×C₄⋊C₄	—	34	1	no	—
198	C₂×C₂²⋊C₄	—	34	1	no	—
199	C₂×C₄²	—	17	2	no	—
200	M₃₂	—	17	4	no	—
201	M₃₂	—	17	8	no	—
202	M₃₂	—	17	4	no	—
203	M₃₂	C₂×Dic₁₇	1	4	yes	—
204	M₃₂	C₂×Dic₁₇	1	8	yes	complemented
205	M₃₂	C₂²×D₁₇	1	4	yes	—
206	C₂×C₁₆	—	17	4	no	—
207	C₂×C₁₆	—	17	8	no	—
208	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
209	C₂×C₁₆	C₂×Dic₁₇	1	8	yes	complemented
210	#97	—	17	4	no	—
211	#97	—	17	4	no	—
212	C₄×C₈	—	17	8	no	—
213	C₁₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
214	C₂⁴	#63218	1	1	yes	characteristic
215	C₂²×C₄	—	34	2	no	—
216	C₂²×C₄	#63218	1	1	yes	characteristic
217	C₂²×C₄	#63218	1	2	yes	—
218	C₂²×C₄	#63218	1	1	yes	characteristic
219	C₂²×C₄	#63218	1	1	yes	characteristic
220	C₂²×C₄	#63218	1	2	yes	—
221	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
222	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
223	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₁₇	1	2	yes	—
226	M₁₆	—	34	4	no	—
227	M₁₆	—	34	4	no	—
228	C₂×C₈	—	17	4	no	—
229	C₂×C₈	—	34	2	no	—
230	C₂×C₈	—	34	2	no	—
231	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
232	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
233	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
234	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
235	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
236	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
237	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
246	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
248	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
249	C₄⋊C₄	—	34	4	no	—
250	C₂²⋊C₄	—	34	4	no	—
251	C₄²	—	17	4	no	—
252	C₄²	—	34	2	no	—
253	C₁₆	—	17	8	no	—
254	C₁₆	C₂²×Dic₁₇	1	8	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#10587486	1	1	yes	characteristic
257	C₂³	#10587486	1	1	yes	characteristic
258	C₂³	#10587486	1	2	yes	—
259	C₂³	C₂×#63218	1	1	yes	characteristic
260	C₂³	C₂×#63218	1	2	yes	—
261	C₂×C₄	—	34	4	no	—
262	C₂×C₄	—	68	4	no	—
263	C₂×C₄	#10587486	1	2	yes	—
264	C₂×C₄	#10587486	1	2	yes	—
265	C₂×C₄	#10587486	1	2	yes	—
266	C₂×C₄	#10587486	1	2	yes	—
267	C₂×C₄	#10587486	1	2	yes	—
268	C₂×C₄	#10587486	1	1	yes	characteristic
269	C₂×C₄	#10587486	1	1	yes	characteristic
270	C₂×C₄	C₂×#63218	1	1	yes	characteristic
271	C₂×C₄	C₂×#63218	1	2	yes	—
272	C₂×C₄	C₂×#63218	1	1	yes	characteristic
273	C₂×C₄	C₂×#63218	1	1	yes	characteristic
274	C₂×C₄	C₂×#63218	1	2	yes	—
275	C₂×C₄	C₂×#63218	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₇	1	2	yes	—
278	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
281	C₈	—	34	4	no	—
282	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
283	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
284	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
285	C₈	C₂×C₄×Dic₁₇	1	2	yes	—
286	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
287	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#10587486	1	1	yes	characteristic
294	C₂²	C₂×#10587486	1	2	yes	—
295	C₂²	C₂×#10587486	1	2	yes	—
296	C₂²	C₂×#10587486	1	1	yes	characteristic
297	C₂²	C₂²×#63218	1	1	yes	characteristic
298	C₄	—	68	4	no	—
299	C₄	C₂×#10587486	1	1	yes	characteristic
300	C₄	C₂×#10587486	1	1	yes	characteristic
301	C₄	C₂×#10587486	1	2	yes	—
302	C₄	C₂×#10587486	1	2	yes	—
303	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
304	C₄	C₂²×#63218	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#14530388	1	2	yes	direct product factor
310	C₂	C₂²×#10587486	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	C₂×#2110
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14530388	C₂	direct (N × H)
4. C₂×#11762586	C₂	semidirect
8. C₂×#10587483	C₄	semidirect
9. C₂×#10587483	C₄	semidirect
12. C₂²×#63214	C₄	semidirect
17. #11762586	C₂²	semidirect
19. C₂×C₁₇×M₃₂	C₄	semidirect
20. C₂×C₁₇×M₃₂	C₄	semidirect
23. C₂²×C₁₆×C₁₇	C₄	semidirect
46. C₁₇×M₃₂	C₂×C₄	semidirect
49. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
54. #10587483	C₂×C₄	semidirect
57. C₂×#63214	C₂×C₄	semidirect
162. C₂×M₃₂	Dic₁₇	semidirect
163. C₂×M₃₂	Dic₁₇	semidirect
167. C₂²×C₁₆	Dic₁₇	semidirect
177. C₂×C₁₇	#2110	semidirect
204. M₃₂	C₂×Dic₁₇	semidirect
209. C₂×C₁₆	C₂×Dic₁₇	semidirect
213. C₁₇	C₂×#2110	semidirect
309. C₂	#14530388	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×#10587483	2
C₂×#11762764	2
#14530388	2
C₂×#11762586	2
C₂²×C₁₇×M₃₂	2
C₂×#2110	17

Subgroups — C2×#14530388 #17186981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530388 #17186981