Subgroups — C₂×#14544992 #17978127

Order 4352.

854 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763865	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763772	C₂	1	1	yes	maximal, characteristic
3	#14544992	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11763696	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×#595	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂×D₁₇⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×#10587677	C₄	1	1	yes	characteristic
9	C₂×#10587530	C₄	1	2	yes	complemented
10	#11763865	C₄	1	4	yes	complemented
11	#11763865	C₂²	1	4	yes	—
12	C₂×#10587538	C₄	1	2	yes	complemented
13	C₂×#10587529	C₂²	1	2	yes	—
14	#11763772	C₄	1	4	yes	—
15	#11763772	C₂²	1	4	yes	—
16	C₂×#10587532	C₄	1	2	yes	—
17	C₂×C₈×#595	C₂²	1	2	yes	—
18	#11763696	C₂²	1	8	yes	complemented
19	C₁₆×#595	C₂²	1	8	yes	complemented
20	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
21	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
22	C₂×C₁₆×D₁₇	C₄	1	4	yes	complemented
23	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
24	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
25	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
28	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
29	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊#595	C₈	1	2	yes	—
31	C₂×#63247	C₈	1	2	yes	—
32	#10587677	C₈	1	4	yes	—
33	#10587677	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊#595	C₈	1	2	yes	—
35	C₂×C₄×#595	C₂×C₄	1	2	yes	—
36	D₁₇⋊M₁₆	C₈	1	4	yes	—
37	D₁₇⋊M₁₆	C₂×C₄	1	4	yes	—
38	C₂×#63243	C₈	1	2	yes	—
39	C₂×#63242	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₁₇	C₂³	1	2	yes	—
50	#10587538	C₂×C₄	1	8	yes	complemented
51	C₂×#63216	C₂×C₄	1	4	yes	—
52	#10587532	C₂×C₄	1	8	yes	—
53	C₈×#595	C₂×C₄	1	4	yes	—
54	C₈×#595	C₂³	1	4	yes	—
55	#10587530	C₂×C₄	1	8	yes	complemented
56	#10587529	C₂×C₄	1	4	yes	—
57	#10587529	C₂³	1	4	yes	—
58	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
61	C₂×#63214	C₂×C₄	1	1	yes	characteristic
62	C₂×#63214	C₂×C₄	1	1	yes	characteristic
63	C₂×#63214	C₂×C₄	1	4	yes	complemented
64	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
65	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
66	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
67	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
68	C₁₆×D₁₇	C₂³	1	2	yes	—
69	C₁₆×D₁₇	C₂³	1	2	yes	—
70	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×#595	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
73	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₇	C₄²	1	2	yes	—
75	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
82	C₂²⋊#595	C₂×C₈	1	8	yes	—
83	#63247	C₂×C₈	1	8	yes	—
84	C₂×#3624	C₂×C₈	1	4	yes	—
85	C₄⋊#595	C₂×C₈	1	8	yes	—
86	C₄×#595	C₂×C₈	1	4	yes	—
87	C₄×#595	C₂²×C₄	1	4	yes	—
88	#63243	C₂×C₈	1	8	yes	—
89	#63242	C₂×C₈	1	4	yes	—
90	#63242	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₇	C₄²	1	2	yes	—
92	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
93	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
95	C₂×#3622	C₄²	1	1	yes	characteristic
96	C₂×#3622	C₄²	1	2	yes	—
97	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
98	C₂×#3622	C₂²×C₄	1	2	yes	—
99	C₈×D₁₇	C₄²	1	2	yes	—
100	C₈×D₁₇	C₄²	1	2	yes	—
101	C₈×D₁₇	C₄²	1	2	yes	—
102	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₁₇	C₂²×C₄	1	2	yes	—
104	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₁₇	C₂²×C₄	1	2	yes	—
106	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
107	C₈×D₁₇	C₂²×C₄	1	2	yes	—
108	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
109	#63216	C₂²×C₄	1	8	yes	—
110	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
111	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
112	#63214	C₂²×C₄	1	2	yes	—
113	#63214	C₂²×C₄	1	2	yes	—
114	C₂×#2108	—	17	1	no	maximal, Sylow
115	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
116	C₂²×D₁₇	—	2	4	no	—
117	C₂²×D₁₇	C₄×C₈	1	1	yes	characteristic
118	C₂²×D₁₇	C₄×C₈	1	2	yes	—
119	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
120	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×#595	—	2	8	no	—
122	C₂×#595	C₂²×C₈	1	8	yes	—
123	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
124	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
127	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
128	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
131	C₄×D₁₇	C₄×C₈	1	2	yes	—
132	C₄×D₁₇	C₄×C₈	1	2	yes	—
133	C₄×D₁₇	C₄×C₈	1	2	yes	—
134	C₄×D₁₇	C₂×C₄²	1	2	yes	—
135	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
136	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₁₇	C₂²×C₈	1	2	yes	—
138	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
139	C₄×D₁₇	C₂²×C₈	1	2	yes	—
140	C₄×D₁₇	C₂³×C₄	1	1	yes	characteristic
141	#3624	C₂²×C₈	1	8	yes	—
142	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
143	C₈×C₁₇	C₂×C₄²	1	2	yes	—
144	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
145	#3622	C₂×C₄²	1	1	yes	characteristic
146	#3622	C₂×C₄²	1	2	yes	—
147	#3622	C₂³×C₄	1	1	yes	characteristic
148	C₂²×M₃₂	—	17	1	no	—
149	C₂³×C₁₆	—	17	1	no	—
150	C₂×#405	—	17	1	no	—
151	#2108	—	17	8	no	—
152	C₂×#346	—	17	2	no	—
153	C₂×C₄×C₁₆	—	17	2	no	—
154	C₂²×C₁₇	—	2	2	no	—
155	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
156	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂×D₁₇	—	2	2	no	—
158	C₂×D₁₇	—	2	4	no	—
159	C₂×D₁₇	—	2	4	no	—
160	C₂×D₁₇	C₂×C₄×C₈	1	2	yes	—
161	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂×D₁₇	#504	1	4	yes	—
163	C₂×D₁₇	C₂³×C₈	1	1	yes	characteristic
164	#595	—	2	8	no	—
165	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
166	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
167	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
168	Dic₁₇	C₂×C₄×C₈	1	1	yes	characteristic
169	Dic₁₇	C₂×C₄×C₈	1	2	yes	—
170	Dic₁₇	C₂³×C₈	1	1	yes	characteristic
171	C₂²×M₁₆	—	17	1	no	—
172	C₂³×C₈	—	17	1	no	—
173	C₂×#117	—	17	1	no	—
174	C₂×M₃₂	—	17	4	no	—
175	C₂×M₃₂	—	17	2	no	—
176	C₂×M₃₂	—	17	4	no	—
177	C₂×M₃₂	—	17	2	no	—
178	C₂²×C₁₆	—	17	2	no	—
179	C₂²×C₁₆	—	17	1	no	—
180	C₂²×C₁₆	—	17	1	no	—
181	C₂²×C₁₆	—	17	1	no	—
182	C₂²×C₁₆	—	17	1	no	—
183	C₂²×C₁₆	—	17	1	no	—
184	C₂²×C₁₆	—	17	4	no	—
185	C₂²×C₁₆	—	17	4	no	—
186	C₂²×C₁₆	#595	1	1	yes	characteristic, complemented
187	#405	—	17	4	no	—
188	#405	—	17	4	no	—
189	C₂×#97	—	17	2	no	—
190	C₂×C₄×C₈	—	17	2	no	—
191	#346	—	17	8	no	—
192	C₄×C₁₆	—	17	8	no	—
193	C₂×C₁₇	—	2	2	no	—
194	C₂×C₁₇	#2108	1	2	yes	complemented
195	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
196	D₁₇	—	2	2	no	—
197	D₁₇	#2108	1	2	yes	complemented
198	D₁₇	C₂×#504	1	2	yes	—
199	C₂³×C₄	—	17	1	no	—
200	C₂×M₁₆	—	17	2	no	—
201	C₂×M₁₆	—	17	4	no	—
202	C₂×M₁₆	—	17	4	no	—
203	C₂×M₁₆	—	17	2	no	—
204	C₂²×C₈	—	17	2	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₂²×C₈	—	17	2	no	—
208	C₂²×C₈	—	17	1	no	—
209	C₂²×C₈	—	17	2	no	—
210	C₂²×C₈	—	17	1	no	—
211	C₂²×C₈	—	17	2	no	—
212	C₂²×C₈	—	17	1	no	—
213	C₂²×C₈	—	17	2	no	—
214	C₂²×C₈	C₂×#595	1	1	yes	characteristic
215	#117	—	17	4	no	—
216	#117	—	17	4	no	—
217	C₂×C₄⋊C₄	—	17	2	no	—
218	C₂×C₂²⋊C₄	—	17	2	no	—
219	C₂×C₄²	—	17	2	no	—
220	M₃₂	—	17	8	no	—
221	M₃₂	—	17	8	no	—
222	C₂×C₁₆	—	17	4	no	—
223	C₂×C₁₆	—	17	4	no	—
224	C₂×C₁₆	—	17	4	no	—
225	C₂×C₁₆	—	17	1	no	—
226	C₂×C₁₆	—	17	1	no	—
227	C₂×C₁₆	—	17	4	no	—
228	C₂×C₁₆	—	17	2	no	—
229	C₂×C₁₆	—	17	2	no	—
230	C₂×C₁₆	—	17	4	no	—
231	C₂×C₁₆	—	17	4	no	—
232	C₂×C₁₆	—	17	2	no	—
233	C₂×C₁₆	—	17	2	no	—
234	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
235	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#595	1	4	yes	complemented
237	#97	—	17	8	no	—
238	C₄×C₈	—	17	4	no	—
239	C₄×C₈	—	17	4	no	—
240	C₁₇	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	17	1	no	—
242	C₂²×C₄	—	17	1	no	—
243	C₂²×C₄	—	17	2	no	—
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	2	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	17	2	no	—
249	C₂²×C₄	—	17	1	no	—
250	C₂²×C₄	—	17	2	no	—
251	C₂²×C₄	—	17	4	no	—
252	C₂²×C₄	C₄×#595	1	1	yes	characteristic
253	M₁₆	—	17	8	no	—
254	M₁₆	—	17	8	no	—
255	C₂×C₈	—	17	4	no	—
256	C₂×C₈	—	17	4	no	—
257	C₂×C₈	—	17	4	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	2	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	2	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	2	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	—	17	2	no	—
266	C₂×C₈	—	17	2	no	—
267	C₂×C₈	—	17	2	no	—
268	C₂×C₈	—	17	2	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	17	1	no	—
271	C₂×C₈	—	17	2	no	—
272	C₂×C₈	C₄×#595	1	1	yes	characteristic
273	C₂×C₈	C₄×#595	1	2	yes	—
274	C₂×C₈	C₂²×#595	1	1	yes	characteristic
275	C₂×C₈	C₂²×#595	1	2	yes	—
276	C₄⋊C₄	—	17	8	no	—
277	C₂²⋊C₄	—	17	8	no	—
278	C₄²	—	17	4	no	—
279	C₄²	—	17	4	no	—
280	C₁₆	—	17	8	no	—
281	C₁₆	—	17	2	no	—
282	C₁₆	—	17	2	no	—
283	C₁₆	C₂²×#595	1	2	yes	—
284	C₁₆	C₂²×#595	1	2	yes	—
285	C₂³	—	17	1	no	—
286	C₂³	—	17	2	no	—
287	C₂³	—	17	2	no	—
288	C₂³	—	17	1	no	—
289	C₂³	—	34	4	no	—
290	C₂³	C₈×#595	1	1	yes	characteristic
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	17	2	no	—
293	C₂×C₄	—	17	2	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	2	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	2	no	—
299	C₂×C₄	—	17	2	no	—
300	C₂×C₄	—	17	2	no	—
301	C₂×C₄	—	17	2	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	2	no	—
305	C₂×C₄	—	17	8	no	—
306	C₂×C₄	—	34	8	no	—
307	C₂×C₄	C₈×#595	1	2	yes	—
308	C₂×C₄	C₈×#595	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×#595	1	2	yes	—
310	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
311	C₈	—	17	8	no	—
312	C₈	—	17	1	no	—
313	C₈	—	17	1	no	—
314	C₈	—	17	2	no	—
315	C₈	C₂×C₄×#595	1	1	yes	characteristic
316	C₈	C₂×C₄×#595	1	2	yes	—
317	C₈	C₂³×#595	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	17	4	no	—
320	C₂²	—	17	1	no	—
321	C₂²	—	17	1	no	—
322	C₂²	—	17	2	no	—
323	C₂²	—	34	4	no	—
324	C₂²	—	34	4	no	—
325	C₂²	—	34	2	no	—
326	C₂²	C₂×C₈×#595	1	2	yes	—
327	C₂²	C₂×C₈×#595	1	1	yes	characteristic
328	C₄	—	17	1	no	—
329	C₄	—	17	1	no	—
330	C₄	—	17	2	no	—
331	C₄	—	34	8	no	—
332	C₄	C₂×C₈×#595	1	1	yes	characteristic
333	C₄	C₂×C₈×#595	1	2	yes	—
334	C₄	C₂²×C₄×#595	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	17	2	no	—
337	C₂	—	17	2	no	—
338	C₂	—	34	2	no	—
339	C₂	#14544992	1	2	yes	direct product factor
340	C₂	C₂²×C₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₈	C₂³×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	C₂×#2108
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14544992	C₂	direct (N × H)
4. C₂×#11763696	C₂	semidirect
5. C₂×C₁₆×#595	C₂	semidirect
9. C₂×#10587530	C₄	semidirect
10. #11763865	C₄	semidirect
12. C₂×#10587538	C₄	semidirect
18. #11763696	C₂²	semidirect
19. C₁₆×#595	C₂²	semidirect
21. C₂²×#63214	C₄	semidirect
22. C₂×C₁₆×D₁₇	C₄	semidirect
23. C₂×C₁₆×D₁₇	C₄	semidirect
24. C₂×C₁₆×D₁₇	C₄	semidirect
28. C₂²×C₁₆×C₁₇	C₄	semidirect
50. #10587538	C₂×C₄	semidirect
55. #10587530	C₂×C₄	semidirect
60. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
63. C₂×#63214	C₂×C₄	semidirect
66. C₁₆×D₁₇	C₂×C₄	semidirect
67. C₁₆×D₁₇	C₂×C₄	semidirect
186. C₂²×C₁₆	#595	semidirect
194. C₂×C₁₇	#2108	semidirect
197. D₁₇	#2108	semidirect
236. C₂×C₁₆	C₂×#595	semidirect
240. C₁₇	C₂×#2108	semidirect
339. C₂	#14544992	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763865	2
C₂×#11763772	2
#14544992	2
C₂×#11763696	2
C₂×C₁₆×#595	2
C₂²×C₁₆×D₁₇	2
C₂×#2108	17

Subgroups — C2×#14544992 #17978127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14544992 #17978127