Subgroups — C₂×#14530675 #17187757

Order 4352.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763249	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762917	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530675	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762633	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762624	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762619	C₂	1	1	yes	maximal, characteristic
7	C₂×#11762587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#444	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₃₄	—	2	1	no	—
11	#11763249	C₄	1	4	yes	complemented
12	#11763249	C₂²	1	4	yes	—
13	C₂×#10587500	C₄	1	2	yes	complemented
14	C₂×#10587498	C₄	1	2	yes	complemented
15	C₂×#10587483	C₂²	1	1	yes	characteristic
16	C₂×#10587483	C₂²	1	1	yes	characteristic
17	#11762917	—	2	2	no	—
18	#11762917	C₂²	1	4	yes	complemented
19	C₂×#10587488	C₂²	1	1	yes	characteristic
20	C₂×#10587487	—	2	1	no	—
21	#11762633	C₂²	1	4	yes	complemented
22	#11762624	C₂²	1	4	yes	complemented
23	#11762619	C₂²	1	4	yes	—
24	#11762587	C₂²	1	4	yes	complemented
25	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×#131	—	2	1	no	—
27	C₁₇×#444	—	2	2	no	—
28	C₁₇×#444	C₂²	1	4	yes	—
29	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
30	C₂×C₁₇×#104	—	2	1	no	—
31	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
32	C₂×#41×C₁₇	—	2	1	no	—
33	C₂²×Dic₃₄	—	2	1	no	—
34	C₁₇×#135	—	2	1	no	—
35	C₁₇×#135	—	2	1	no	—
36	C₁₇×#135	C₂×C₄	1	2	yes	—
37	C₁₇×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₁₆×C₁₇	—	2	1	no	—
40	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₁₇×#131	—	2	4	no	—
42	C₁₇×#131	—	4	2	no	—
43	C₂×M₁₆×C₁₇	—	2	1	no	—
44	C₂×M₁₆×C₁₇	—	4	1	no	—
45	C₂²×C₈×C₁₇	—	4	1	no	—
46	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₁₇	—	2	1	no	—
50	C₂×#63225	—	4	1	no	—
51	C₂×C₄×Dic₁₇	—	4	1	no	—
52	C₂×Dic₆₈	—	2	1	no	—
53	C₂×Dic₆₈	—	2	1	no	—
54	C₂×Dic₆₈	—	2	1	no	—
55	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
56	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
57	C₂×Dic₆₈	C₂³	1	2	yes	—
58	C₄×Dic₃₄	—	2	4	no	—
59	C₄×Dic₃₄	—	4	2	no	—
60	C₁₇×#108	—	2	1	no	—
61	C₁₇×#108	C₂³	1	2	yes	—
62	C₁₇×#104	—	2	4	no	—
63	C₁₇×#97	D₄	1	2	yes	—
64	C₁₇×#97	C₂³	1	2	yes	—
65	#10587500	C₂×C₄	1	8	yes	complemented
66	#10587498	C₂×C₄	1	8	yes	complemented
67	#10587488	—	2	1	no	—
68	#10587488	C₂³	1	2	yes	—
69	#10587487	—	2	4	no	—
70	#10587483	—	2	1	no	—
71	#10587483	C₂×C₄	1	2	yes	—
72	#10587483	C₂³	1	2	yes	—
73	#10587483	C₂³	1	2	yes	—
74	C₂×#63214	—	2	1	no	—
75	C₂×#63214	C₂×C₄	1	2	yes	—
76	#41×C₁₇	—	2	2	no	—
77	#41×C₁₇	—	2	2	no	—
78	#41×C₁₇	—	4	1	no	—
79	#41×C₁₇	—	4	1	no	—
80	C₂×Q₈×C₁₇	—	2	1	no	—
81	C₂×D₄×C₁₇	—	2	1	no	—
82	C₂×D₄×C₁₇	—	4	1	no	—
83	C₂²×C₄×C₁₇	—	2	1	no	—
84	C₂²×C₄×C₁₇	—	4	1	no	—
85	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₁₇	—	4	1	no	—
87	C₂²×Dic₁₇	—	4	1	no	—
88	C₂×Dic₃₄	—	2	2	no	—
89	C₂×Dic₃₄	—	2	2	no	—
90	C₂×Dic₃₄	—	2	2	no	—
91	C₂×Dic₃₄	—	4	1	no	—
92	C₂×Dic₃₄	—	4	1	no	—
93	C₂×Dic₃₄	—	4	1	no	—
94	Q₁₆×C₁₇	—	2	1	no	—
95	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
96	SD₁₆×C₁₇	—	2	2	no	—
97	SD₁₆×C₁₇	—	2	2	no	—
98	D₈×C₁₇	—	2	1	no	—
99	D₈×C₁₇	C₂²×C₄	1	2	yes	—
100	M₁₆×C₁₇	—	2	2	no	—
101	M₁₆×C₁₇	—	4	4	no	—
102	M₁₆×C₁₇	—	4	1	no	—
103	C₂×C₈×C₁₇	—	2	1	no	—
104	C₂×C₈×C₁₇	—	2	1	no	—
105	C₂×C₈×C₁₇	—	4	4	no	—
106	C₂×C₈×C₁₇	—	4	1	no	—
107	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
114	C₄²×C₁₇	—	2	1	no	—
115	C₄²×C₁₇	C₂×D₄	1	2	yes	—
116	C₄⋊Dic₁₇	—	2	2	no	—
117	C₄⋊Dic₁₇	—	4	1	no	—
118	#63225	—	4	4	no	—
119	C₄×Dic₁₇	—	4	4	no	—
120	Dic₆₈	—	2	2	no	—
121	Dic₆₈	—	2	2	no	—
122	Dic₆₈	—	2	2	no	—
123	Dic₆₈	C₂²×C₄	1	4	yes	—
124	#63214	—	2	2	no	—
125	#63214	C₂²×C₄	1	4	yes	—
126	C₂×#2180	—	17	1	no	maximal, Sylow
127	C₂³×C₁₇	—	4	1	no	—
128	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
129	Q₈×C₁₇	—	2	2	no	—
130	Q₈×C₁₇	—	4	1	no	—
131	D₄×C₁₇	—	2	2	no	—
132	D₄×C₁₇	—	4	2	no	—
133	D₄×C₁₇	—	4	2	no	—
134	D₄×C₁₇	—	4	1	no	—
135	C₂×C₄×C₁₇	—	2	2	no	—
136	C₂×C₄×C₁₇	—	4	2	no	—
137	C₂×C₄×C₁₇	—	4	1	no	—
138	C₂×C₄×C₁₇	—	4	2	no	—
139	C₂×C₄×C₁₇	—	4	2	no	—
140	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₁₇	—	4	2	no	—
147	C₂×Dic₁₇	—	4	2	no	—
148	C₂×Dic₁₇	—	4	2	no	—
149	C₂×Dic₁₇	—	4	2	no	—
150	C₂×Dic₁₇	—	8	2	no	—
151	Dic₃₄	—	2	4	no	—
152	Dic₃₄	—	4	2	no	—
153	Dic₃₄	—	4	2	no	—
154	Dic₃₄	—	4	2	no	—
155	C₈×C₁₇	—	2	2	no	—
156	C₈×C₁₇	—	4	2	no	—
157	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
158	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
159	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
160	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₂×#510	—	17	1	no	—
162	C₂×#444	D₁₇	1	1	yes	characteristic
163	C₂×#441	—	17	1	no	—
164	#2180	—	17	8	no	—
165	C₂×#365	—	17	1	no	—
166	C₂×#362	—	17	1	no	—
167	C₂×#360	—	17	1	no	—
168	C₂×#347	—	17	1	no	—
169	C₂²×C₁₇	—	2	2	no	—
170	C₂²×C₁₇	—	4	2	no	—
171	C₂²×C₁₇	—	8	2	no	—
172	C₂²×C₁₇	#442	1	1	yes	characteristic
173	C₂²×C₁₇	#442	1	1	yes	characteristic
174	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄×C₁₇	—	4	2	no	—
176	C₄×C₁₇	—	4	2	no	—
177	C₄×C₁₇	#442	1	1	yes	characteristic
178	C₄×C₁₇	#442	1	1	yes	characteristic
179	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	Dic₁₇	—	4	4	no	—
182	Dic₁₇	—	8	2	no	—
183	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
184	C₂²×Q₁₆	—	17	1	no	—
185	C₂×#131	—	2	1	no	—
186	C₂×C₄×Q₈	—	34	1	no	—
187	#510	—	17	4	no	—
188	#510	—	17	4	no	—
189	C₂×Q₃₂	—	17	2	no	—
190	C₂×SD₃₂	—	17	2	no	—
191	C₂×M₃₂	—	17	1	no	—
192	C₂×M₃₂	—	17	1	no	—
193	#444	—	2	2	no	—
194	#444	C₂×D₁₇	1	4	yes	—
195	#441	—	17	4	no	—
196	#441	—	34	2	no	—
197	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
198	C₂×#106	—	17	1	no	—
199	C₂×#104	—	2	1	no	—
200	C₂×#103	—	34	1	no	—
201	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
202	#365	—	17	4	no	—
203	#362	—	17	4	no	—
204	#360	—	17	4	no	—
205	#347	—	17	4	no	—
206	C₂×C₁₇	—	2	1	no	—
207	C₂×C₁₇	—	2	1	no	—
208	C₂×C₁₇	—	8	2	no	—
209	C₂×C₁₇	#2180	1	2	yes	complemented
210	C₂×C₁₇	C₂×#442	1	1	yes	characteristic
211	C₂×#41	—	2	1	no	—
212	C₂²×Q₈	—	34	1	no	—
213	#135	—	2	1	no	—
214	#135	—	2	1	no	—
215	#135	C₄×D₁₇	1	2	yes	—
216	#135	C₂²×D₁₇	1	2	yes	—
217	C₂×Q₁₆	—	17	2	no	—
218	C₂×Q₁₆	—	17	2	no	—
219	C₂×Q₁₆	—	17	2	no	—
220	C₂×Q₁₆	—	34	1	no	—
221	C₂×Q₁₆	—	34	1	no	—
222	C₂×Q₁₆	—	34	1	no	—
223	C₂×Q₁₆	C₄×D₁₇	1	1	yes	characteristic
224	C₂×SD₁₆	—	2	1	no	—
225	C₂×D₈	C₄×D₁₇	1	1	yes	characteristic
226	#131	—	2	4	no	—
227	#131	—	4	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	4	1	no	—
230	C₂²×C₈	—	4	1	no	—
231	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₄×Q₈	—	34	4	no	—
234	C₄×Q₈	—	68	2	no	—
235	C₂×C₄⋊C₄	—	34	1	no	—
236	C₂×C₄⋊C₄	—	68	1	no	—
237	C₂×C₄²	—	68	1	no	—
238	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
239	Q₃₂	—	17	8	no	—
240	SD₃₂	—	17	8	no	—
241	M₃₂	—	17	2	no	—
242	M₃₂	—	17	2	no	—
243	M₃₂	—	17	2	no	—
244	M₃₂	—	34	1	no	—
245	C₂×C₁₆	—	17	2	no	—
246	C₂×C₁₆	—	34	1	no	—
247	#108	—	2	1	no	—
248	#108	C₂²×D₁₇	1	2	yes	—
249	#106	—	17	2	no	—
250	#106	—	34	1	no	—
251	#104	—	2	4	no	—
252	#103	—	34	4	no	—
253	#97	C₂²⋊D₁₇	1	2	yes	—
254	#97	C₂²×D₁₇	1	2	yes	—
255	C₁₇	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	34	2	no	—
262	C₂×Q₈	—	34	2	no	—
263	C₂×Q₈	—	34	2	no	—
264	C₂×Q₈	—	68	1	no	—
265	C₂×Q₈	—	68	1	no	—
266	C₂×Q₈	—	68	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	68	1	no	—
272	C₂²×C₄	—	68	1	no	—
273	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	17	4	no	—
276	Q₁₆	—	34	2	no	—
277	Q₁₆	—	34	2	no	—
278	Q₁₆	—	34	2	no	—
279	Q₁₆	C₂×C₄×D₁₇	1	2	yes	—
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	C₂×C₄×D₁₇	1	2	yes	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	4	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
293	C₂×C₈	#63252	1	1	yes	characteristic
294	C₂×C₈	#63252	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
298	C₄⋊C₄	—	34	2	no	—
299	C₄⋊C₄	—	68	4	no	—
300	C₄⋊C₄	—	68	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	68	4	no	—
303	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
304	C₁₆	—	17	4	no	—
305	C₁₆	—	34	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	#10587635	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	34	4	no	—
311	Q₈	—	68	2	no	—
312	Q₈	—	68	2	no	—
313	Q₈	—	68	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	68	2	no	—
324	C₂×C₄	—	68	2	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	136	2	no	—
328	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
330	C₂×C₄	#10587618	1	1	yes	characteristic
331	C₂×C₄	#10587618	1	1	yes	characteristic
332	C₂×C₄	#10587635	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
339	C₈	C₂×#63252	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	#11762919	1	1	yes	characteristic
344	C₂²	#11762919	1	1	yes	characteristic
345	C₂²	C₂×#10587635	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	4	2	no	—
348	C₄	—	68	4	no	—
349	C₄	—	136	2	no	—
350	C₄	#11762919	1	1	yes	characteristic
351	C₄	#11762919	1	1	yes	characteristic
352	C₄	C₂×C₄×C₂²⋊D₁₇	1	1	yes	characteristic
353	C₄	C₂×#10587618	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	8	2	no	—
357	C₂	#14530675	1	2	yes	direct product factor
358	C₂	C₂×#11762919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762919
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#444	C₂
socle	C₂²×C₁₇	#442
prime radical	C₂×C₁₇×#135	C₂²
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#444	D₁₇
hypocenter	C₁₇	C₂×#2180
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
2. C₂×#11762917	C₂	semidirect
3. #14530675	C₂	direct (N × H)
4. C₂×#11762633	C₂	semidirect
5. C₂×#11762624	C₂	semidirect
7. C₂×#11762587	C₂	semidirect
11. #11763249	C₄	semidirect
13. C₂×#10587500	C₄	semidirect
14. C₂×#10587498	C₄	semidirect
18. #11762917	C₂²	semidirect
21. #11762633	C₂²	semidirect
22. #11762624	C₂²	semidirect
24. #11762587	C₂²	semidirect
65. #10587500	C₂×C₄	semidirect
66. #10587498	C₂×C₄	semidirect
209. C₂×C₁₇	#2180	semidirect
255. C₁₇	C₂×#2180	semidirect
357. C₂	#14530675	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×#11763249	2
C₂×#11762917	2
#14530675	2
C₂×#11762633	2
C₂×#11762624	2
C₂×#11762619	2
C₂×#11762587	2
C₂×C₁₇×#444	2
C₂×#2180	17

Subgroups — C2×#14530675 #17187757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530675 #17187757