Subgroups — C₂×#14544996 #17978132

Order 4352.

854 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587529	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763772	C₂	1	1	yes	maximal, characteristic
3	#14544996	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	C₂×M₃₂×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×#10587529	C₄	1	4	yes	complemented
10	C₂×#10587529	C₄	1	2	yes	complemented
11	C₂×#10587529	C₂²	1	1	yes	characteristic
12	C₂×#10587529	C₂²	1	1	yes	characteristic
13	C₂×#10587529	C₂²	1	4	yes	—
14	C₂²×#63216	C₄	1	2	yes	complemented
15	#11763772	C₄	1	4	yes	—
16	#11763772	C₂²	1	4	yes	—
17	C₂×#10587532	C₄	1	2	yes	—
18	C₂×C₈×#595	C₂²	1	2	yes	—
19	#11763696	C₂²	1	8	yes	complemented
20	C₁₆×#595	C₂²	1	8	yes	complemented
21	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₂×#10587483	C₄	1	1	yes	characteristic, complemented
23	C₂×#10587469	C₄	1	2	yes	complemented
24	M₃₂×D₁₇	C₄	1	4	yes	complemented
25	M₃₂×D₁₇	C₂²	1	4	yes	—
26	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
27	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊#595	C₈	1	2	yes	—
30	C₂×#63247	C₈	1	2	yes	—
31	#10587677	C₈	1	4	yes	—
32	#10587677	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊#595	C₈	1	2	yes	—
34	C₂×C₄×#595	C₂×C₄	1	2	yes	—
35	D₁₇⋊M₁₆	C₈	1	4	yes	—
36	D₁₇⋊M₁₆	C₂×C₄	1	4	yes	—
37	C₂×#63243	C₈	1	2	yes	—
38	C₂×#63242	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₇	C₂³	1	2	yes	—
49	C₂×#63216	C₂×C₄	1	2	yes	—
50	C₂×#63216	C₂×C₄	1	2	yes	—
51	C₂×#63216	C₂×C₄	1	8	yes	complemented
52	#10587532	C₂×C₄	1	8	yes	—
53	C₈×#595	C₂×C₄	1	4	yes	—
54	C₈×#595	C₂³	1	4	yes	—
55	#10587529	C₂×C₄	1	2	yes	—
56	#10587529	C₂×C₄	1	2	yes	—
57	#10587529	C₂×C₄	1	8	yes	complemented
58	#10587529	C₂³	1	2	yes	—
59	#10587529	C₂³	1	2	yes	—
60	C₁₇×M₃₂	C₂×C₄	1	4	yes	complemented
61	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
62	#10587483	C₂×C₄	1	4	yes	complemented
63	C₂×#63214	C₂×C₄	1	2	yes	—
64	#10587469	C₂×C₄	1	8	yes	complemented
65	C₁₆×D₁₇	C₂×C₄	1	4	yes	—
66	C₁₆×D₁₇	C₂³	1	4	yes	—
67	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×#595	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
70	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₁₇	C₄²	1	2	yes	—
72	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
79	C₂²⋊#595	C₂×C₈	1	8	yes	—
80	#63247	C₂×C₈	1	8	yes	—
81	C₂×#3624	C₂×C₈	1	4	yes	—
82	C₄⋊#595	C₂×C₈	1	8	yes	—
83	C₄×#595	C₂×C₈	1	4	yes	—
84	C₄×#595	C₂²×C₄	1	4	yes	—
85	#63243	C₂×C₈	1	8	yes	—
86	#63242	C₂×C₈	1	4	yes	—
87	#63242	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₄²	1	2	yes	—
90	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×#3622	C₄²	1	1	yes	characteristic
93	C₂×#3622	C₄²	1	2	yes	—
94	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
95	C₂×#3622	C₂²×C₄	1	2	yes	—
96	C₈×D₁₇	C₄²	1	2	yes	—
97	C₈×D₁₇	C₄²	1	2	yes	—
98	C₈×D₁₇	C₄²	1	2	yes	—
99	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₁₇	C₂²×C₄	1	2	yes	—
101	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₁₇	C₂²×C₄	1	2	yes	—
103	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₁₇	C₂²×C₄	1	2	yes	—
105	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
106	#63216	C₂²×C₄	1	4	yes	—
107	#63216	C₂²×C₄	1	4	yes	—
108	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
109	#63214	C₂²×C₄	1	4	yes	—
110	C₂×#2108	—	17	1	no	maximal, Sylow
111	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
112	C₂²×D₁₇	—	2	4	no	—
113	C₂²×D₁₇	C₄×C₈	1	1	yes	characteristic
114	C₂²×D₁₇	C₄×C₈	1	2	yes	—
115	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
116	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×#595	—	2	8	no	—
118	C₂×#595	C₂²×C₈	1	8	yes	—
119	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
120	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
124	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
127	C₄×D₁₇	C₄×C₈	1	2	yes	—
128	C₄×D₁₇	C₄×C₈	1	2	yes	—
129	C₄×D₁₇	C₄×C₈	1	2	yes	—
130	C₄×D₁₇	C₂×C₄²	1	2	yes	—
131	C₄×D₁₇	C₂×C₄²	1	1	yes	characteristic
132	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
133	C₄×D₁₇	C₂²×C₈	1	2	yes	—
134	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
135	C₄×D₁₇	C₂²×C₈	1	2	yes	—
136	C₄×D₁₇	C₂³×C₄	1	1	yes	characteristic
137	#3624	C₂²×C₈	1	8	yes	—
138	C₈×C₁₇	C₂×C₄²	1	2	yes	—
139	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
140	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
141	#3622	C₂×C₄²	1	1	yes	characteristic
142	#3622	C₂×C₄²	1	2	yes	—
143	#3622	C₂³×C₄	1	1	yes	characteristic
144	C₂²×M₃₂	—	17	1	no	—
145	C₂³×C₁₆	—	17	1	no	—
146	C₂×#405	—	17	1	no	—
147	#2108	—	17	8	no	—
148	C₂×#346	—	17	2	no	—
149	C₂×C₄×C₁₆	—	17	2	no	—
150	C₂²×C₁₇	—	2	2	no	—
151	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
152	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂×D₁₇	—	2	2	no	—
154	C₂×D₁₇	—	2	4	no	—
155	C₂×D₁₇	—	2	4	no	—
156	C₂×D₁₇	C₂×C₄×C₈	1	2	yes	—
157	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
158	C₂×D₁₇	#504	1	4	yes	—
159	C₂×D₁₇	C₂³×C₈	1	1	yes	characteristic
160	#595	—	2	8	no	—
161	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
162	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
163	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
164	Dic₁₇	C₂×C₄×C₈	1	1	yes	characteristic
165	Dic₁₇	C₂×C₄×C₈	1	2	yes	—
166	Dic₁₇	C₂³×C₈	1	1	yes	characteristic
167	C₂²×M₁₆	—	17	1	no	—
168	C₂³×C₈	—	17	1	no	—
169	C₂×#117	—	17	1	no	—
170	C₂×M₃₂	—	17	1	no	—
171	C₂×M₃₂	—	17	4	no	—
172	C₂×M₃₂	—	17	4	no	—
173	C₂×M₃₂	—	17	2	no	—
174	C₂×M₃₂	#595	1	1	yes	characteristic, complemented
175	C₂²×C₁₆	—	17	1	no	—
176	C₂²×C₁₆	—	17	1	no	—
177	C₂²×C₁₆	—	17	4	no	—
178	C₂²×C₁₆	—	17	2	no	—
179	C₂²×C₁₆	—	17	4	no	—
180	C₂²×C₁₆	—	17	2	no	—
181	C₂²×C₁₆	—	17	2	no	—
182	#405	—	17	4	no	—
183	#405	—	17	4	no	—
184	C₂×#97	—	17	2	no	—
185	C₂×C₄×C₈	—	17	2	no	—
186	#346	—	17	8	no	—
187	C₄×C₁₆	—	17	8	no	—
188	C₂×C₁₇	—	2	2	no	—
189	C₂×C₁₇	#2108	1	2	yes	complemented
190	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
191	D₁₇	—	2	2	no	—
192	D₁₇	#2108	1	2	yes	complemented
193	D₁₇	C₂×#504	1	2	yes	—
194	C₂³×C₄	—	17	1	no	—
195	C₂×M₁₆	—	17	2	no	—
196	C₂×M₁₆	—	17	4	no	—
197	C₂×M₁₆	—	17	4	no	—
198	C₂×M₁₆	—	17	2	no	—
199	C₂²×C₈	—	17	2	no	—
200	C₂²×C₈	—	17	1	no	—
201	C₂²×C₈	—	17	1	no	—
202	C₂²×C₈	—	17	2	no	—
203	C₂²×C₈	—	17	2	no	—
204	C₂²×C₈	—	17	1	no	—
205	C₂²×C₈	—	17	2	no	—
206	C₂²×C₈	—	17	2	no	—
207	C₂²×C₈	—	17	1	no	—
208	C₂²×C₈	—	17	1	no	—
209	C₂²×C₈	C₂×#595	1	1	yes	characteristic
210	#117	—	17	4	no	—
211	#117	—	17	4	no	—
212	C₂×C₄⋊C₄	—	17	2	no	—
213	C₂×C₂²⋊C₄	—	17	2	no	—
214	C₂×C₄²	—	17	2	no	—
215	M₃₂	—	17	4	no	—
216	M₃₂	—	17	8	no	—
217	M₃₂	C₂×#595	1	4	yes	complemented
218	C₂×C₁₆	—	17	2	no	—
219	C₂×C₁₆	—	17	2	no	—
220	C₂×C₁₆	—	17	2	no	—
221	C₂×C₁₆	—	17	2	no	—
222	C₂×C₁₆	—	17	8	no	—
223	C₂×C₁₆	—	17	2	no	—
224	C₂×C₁₆	—	17	2	no	—
225	C₂×C₁₆	—	17	8	no	—
226	C₂×C₁₆	—	17	2	no	—
227	C₂×C₁₆	—	17	4	no	—
228	C₂×C₁₆	—	17	4	no	—
229	C₂×C₁₆	C₂×#595	1	2	yes	—
230	#97	—	17	8	no	—
231	C₄×C₈	—	17	4	no	—
232	C₄×C₈	—	17	4	no	—
233	C₁₇	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
234	C₂⁴	—	17	1	no	—
235	C₂²×C₄	—	17	1	no	—
236	C₂²×C₄	—	17	2	no	—
237	C₂²×C₄	—	17	1	no	—
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	—	17	2	no	—
240	C₂²×C₄	—	17	1	no	—
241	C₂²×C₄	—	17	2	no	—
242	C₂²×C₄	—	17	1	no	—
243	C₂²×C₄	—	17	2	no	—
244	C₂²×C₄	—	17	4	no	—
245	C₂²×C₄	C₄×#595	1	1	yes	characteristic
246	M₁₆	—	17	8	no	—
247	M₁₆	—	17	8	no	—
248	C₂×C₈	—	17	4	no	—
249	C₂×C₈	—	17	4	no	—
250	C₂×C₈	—	17	4	no	—
251	C₂×C₈	—	17	2	no	—
252	C₂×C₈	—	17	1	no	—
253	C₂×C₈	—	17	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	2	no	—
257	C₂×C₈	—	17	2	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	2	no	—
260	C₂×C₈	—	17	2	no	—
261	C₂×C₈	—	17	2	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	17	2	no	—
265	C₂×C₈	C₄×#595	1	1	yes	characteristic
266	C₂×C₈	C₄×#595	1	2	yes	—
267	C₂×C₈	C₂²×#595	1	2	yes	—
268	C₂×C₈	C₂²×#595	1	1	yes	characteristic
269	C₄⋊C₄	—	17	8	no	—
270	C₂²⋊C₄	—	17	8	no	—
271	C₄²	—	17	4	no	—
272	C₄²	—	17	4	no	—
273	C₁₆	—	17	4	no	—
274	C₁₆	—	17	4	no	—
275	C₁₆	—	17	4	no	—
276	C₁₆	C₂²×#595	1	4	yes	—
277	C₂³	—	17	1	no	—
278	C₂³	—	17	2	no	—
279	C₂³	—	17	2	no	—
280	C₂³	—	17	1	no	—
281	C₂³	—	34	4	no	—
282	C₂³	C₈×#595	1	1	yes	characteristic
283	C₂×C₄	—	17	1	no	—
284	C₂×C₄	—	17	2	no	—
285	C₂×C₄	—	17	2	no	—
286	C₂×C₄	—	17	1	no	—
287	C₂×C₄	—	17	1	no	—
288	C₂×C₄	—	17	2	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	17	2	no	—
291	C₂×C₄	—	17	2	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	8	no	—
298	C₂×C₄	—	34	8	no	—
299	C₂×C₄	C₈×#595	1	2	yes	—
300	C₂×C₄	C₈×#595	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄×#595	1	2	yes	—
302	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
303	C₈	—	17	8	no	—
304	C₈	—	17	1	no	—
305	C₈	—	17	1	no	—
306	C₈	—	17	2	no	—
307	C₈	C₂×C₄×#595	1	2	yes	—
308	C₈	C₂×C₄×#595	1	1	yes	characteristic
309	C₈	C₂³×#595	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	17	4	no	—
312	C₂²	—	17	1	no	—
313	C₂²	—	17	1	no	—
314	C₂²	—	17	2	no	—
315	C₂²	—	34	4	no	—
316	C₂²	—	34	4	no	—
317	C₂²	—	34	2	no	—
318	C₂²	C₂×C₈×#595	1	2	yes	—
319	C₂²	C₂×C₈×#595	1	1	yes	characteristic
320	C₄	—	17	1	no	—
321	C₄	—	17	1	no	—
322	C₄	—	17	2	no	—
323	C₄	—	34	8	no	—
324	C₄	C₂×C₈×#595	1	1	yes	characteristic
325	C₄	C₂×C₈×#595	1	2	yes	—
326	C₄	C₂²×C₄×#595	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	17	2	no	—
329	C₂	—	17	2	no	—
330	C₂	—	34	2	no	—
331	C₂	#14544996	1	2	yes	direct product factor
332	C₂	C₂²×C₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₈	C₂³×#595
Fitting	C₂×C₁₇×M₃₂	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂×M₃₂	#595
hypocenter	C₁₇	C₂×#2108
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14544996	C₂	direct (N × H)
4. C₂×#11763696	C₂	semidirect
5. C₂×C₁₆×#595	C₂	semidirect
9. C₂×#10587529	C₄	semidirect
10. C₂×#10587529	C₄	semidirect
14. C₂²×#63216	C₄	semidirect
19. #11763696	C₂²	semidirect
20. C₁₆×#595	C₂²	semidirect
22. C₂×#10587483	C₄	semidirect
23. C₂×#10587469	C₄	semidirect
24. M₃₂×D₁₇	C₄	semidirect
27. C₂×C₁₇×M₃₂	C₄	semidirect
51. C₂×#63216	C₂×C₄	semidirect
57. #10587529	C₂×C₄	semidirect
60. C₁₇×M₃₂	C₂×C₄	semidirect
62. #10587483	C₂×C₄	semidirect
64. #10587469	C₂×C₄	semidirect
174. C₂×M₃₂	#595	semidirect
189. C₂×C₁₇	#2108	semidirect
192. D₁₇	#2108	semidirect
217. M₃₂	C₂×#595	semidirect
233. C₁₇	C₂×#2108	semidirect
331. C₂	#14544996	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×#10587529	2
C₂×#11763772	2
#14544996	2
C₂×#11763696	2
C₂×C₁₆×#595	2
C₂×M₃₂×D₁₇	2
C₂×#2108	17

Subgroups — C2×#14544996 #17978132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14544996 #17978132