Subgroups — C₂×#14530578 #17187379

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₂²×#10587490	C₂	1	1	yes	maximal, characteristic
3	#14530578	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762638	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63223	—	2	1	no	—
7	C₂×#10587483	C₄	1	2	yes	—
8	C₂×#10587483	C₄	1	2	yes	—
9	C₂×#10587483	C₄	1	4	yes	—
10	C₂×#10587483	C₂²	1	4	yes	—
11	C₂²×#63214	C₂²	1	2	yes	—
12	C₂×#10587490	—	2	2	no	—
13	C₂×#10587490	—	2	1	no	—
14	C₂×#10587490	C₂²	1	2	yes	—
15	C₂×#10587490	C₂²	1	4	yes	—
16	#11762638	C₂²	1	16	yes	complemented
17	C₂²×M₁₆×C₁₇	—	2	1	no	—
18	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×#108	—	2	1	no	—
20	C₂×C₁₇×#108	—	2	2	no	—
21	C₂×C₁₇×#108	C₂²	1	2	yes	—
22	C₂×C₁₇×#108	C₂²	1	4	yes	—
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₇	—	2	2	no	—
25	C₂×M₁₆×C₁₇	—	2	4	no	—
26	C₂×M₁₆×C₁₇	—	4	1	no	—
27	C₂×M₁₆×C₁₇	—	4	2	no	—
28	C₂²×C₈×C₁₇	—	4	1	no	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₇	D₄	1	2	yes	—
33	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
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×#108	—	2	2	no	—
45	C₁₇×#108	—	2	4	no	—
46	C₁₇×#108	C₂³	1	4	yes	—
47	#10587490	—	2	2	no	—
48	#10587490	—	2	4	no	—
49	#10587490	C₂³	1	4	yes	—
50	#10587483	C₂×C₄	1	8	yes	—
51	#10587483	C₂×C₄	1	8	yes	—
52	C₂×#63214	C₂×C₄	1	4	yes	—
53	C₂×#63214	C₂×C₄	1	4	yes	—
54	C₂×#63214	C₂³	1	4	yes	—
55	C₂⁴×C₁₇	Q₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	D₈	1	2	yes	—
62	C₂²×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	M₁₆×C₁₇	—	2	4	no	—
67	M₁₆×C₁₇	—	4	4	no	—
68	M₁₆×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	—	4	4	no	—
70	C₂×C₈×C₁₇	—	4	2	no	—
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	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
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	2	yes	—
82	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
89	#63223	—	2	4	no	—
90	#63223	—	4	4	no	—
91	#63223	—	4	2	no	—
92	C₂×#3622	—	4	4	no	—
93	C₂×#3622	—	4	2	no	—
94	#63214	C₂²×C₄	1	8	yes	—
95	C₂×#2161	—	17	1	no	maximal, Sylow
96	C₂³×C₁₇	—	4	2	no	—
97	C₂³×C₁₇	#107	1	1	yes	characteristic
98	C₂³×C₁₇	#107	1	1	yes	characteristic
99	C₂³×C₁₇	#107	1	2	yes	—
100	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂³×C₁₇	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₁₇	#107	1	2	yes	—
103	C₂×C₄×C₁₇	#107	1	2	yes	—
104	C₂×C₄×C₁₇	#107	1	2	yes	—
105	C₂×C₄×C₁₇	#107	1	2	yes	—
106	C₂×C₄×C₁₇	#107	1	2	yes	—
107	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
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×D₈	1	2	yes	—
114	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₁₇	—	4	4	no	—
121	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	2	yes	—
125	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
127	#3622	—	4	4	no	—
128	C₂²×M₃₂	—	17	1	no	—
129	C₂²×#108	—	17	1	no	—
130	C₂²×#108	D₁₇	1	1	yes	characteristic
131	#2161	—	17	8	no	—
132	C₂×#368	—	17	4	no	—
133	C₂²×C₁₇	—	2	2	no	—
134	C₂²×C₁₇	—	2	2	no	—
135	C₂²×C₁₇	—	2	2	no	—
136	C₂²×C₁₇	—	4	2	no	—
137	C₂²×C₁₇	—	4	2	no	—
138	C₂²×C₁₇	C₂×#107	1	2	yes	—
139	C₂²×C₁₇	C₂×#107	1	2	yes	—
140	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
141	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
142	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
143	C₄×C₁₇	C₂×#107	1	2	yes	—
144	C₄×C₁₇	C₂×#107	1	2	yes	—
145	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
146	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
147	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
149	C₂²×M₁₆	—	2	1	no	—
150	C₂²×M₁₆	—	34	1	no	—
151	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
152	C₂×M₃₂	—	17	2	no	—
153	C₂×M₃₂	—	17	2	no	—
154	C₂×M₃₂	—	17	4	no	—
155	C₂×M₃₂	—	17	4	no	—
156	C₂²×C₁₆	—	17	2	no	—
157	C₂×#108	—	2	2	no	—
158	C₂×#108	—	2	1	no	—
159	C₂×#108	—	17	2	no	—
160	C₂×#108	—	17	4	no	—
161	C₂×#108	—	34	2	no	—
162	C₂×#108	—	34	1	no	—
163	C₂×#108	C₂×D₁₇	1	2	yes	—
164	C₂×#108	C₂×D₁₇	1	4	yes	—
165	#368	—	17	16	no	—
166	C₂×C₁₇	—	2	1	no	—
167	C₂×C₁₇	—	2	2	no	—
168	C₂×C₁₇	—	2	2	no	—
169	C₂×C₁₇	—	2	1	no	—
170	C₂×C₁₇	#2161	1	2	yes	complemented
171	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	4	2	no	—
176	C₂×M₁₆	—	4	1	no	—
177	C₂×M₁₆	—	34	4	no	—
178	C₂×M₁₆	—	34	2	no	—
179	C₂×M₁₆	—	68	2	no	—
180	C₂×M₁₆	—	68	1	no	—
181	C₂²×C₈	—	4	1	no	—
182	C₂²×C₈	—	68	1	no	—
183	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
184	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
185	C₂²×C₈	Dic₃₄	1	2	yes	—
186	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
187	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₁₇	1	2	yes	—
189	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
193	M₃₂	—	17	8	no	—
194	M₃₂	—	17	8	no	—
195	C₂×C₁₆	—	17	4	no	—
196	C₂×C₁₆	—	17	4	no	—
197	C₂×C₁₆	—	17	4	no	—
198	#108	—	2	2	no	—
199	#108	—	2	4	no	—
200	#108	—	17	4	no	—
201	#108	—	34	2	no	—
202	#108	—	34	4	no	—
203	#108	C₂²×D₁₇	1	4	yes	—
204	C₁₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
205	C₂⁴	#63231	1	1	yes	characteristic
206	C₂²×C₄	#63225	1	1	yes	characteristic
207	C₂²×C₄	#63225	1	1	yes	characteristic
208	C₂²×C₄	#63225	1	2	yes	—
209	C₂²×C₄	#63228	1	2	yes	—
210	C₂²×C₄	#63228	1	1	yes	characteristic
211	C₂²×C₄	#63228	1	1	yes	characteristic
212	C₂²×C₄	#63231	1	1	yes	characteristic
213	C₂²×C₄	#63231	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	34	4	no	—
220	M₁₆	—	68	4	no	—
221	M₁₆	—	68	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	68	4	no	—
225	C₂×C₈	—	68	2	no	—
226	C₂×C₈	#63225	1	2	yes	—
227	C₂×C₈	#63225	1	2	yes	—
228	C₂×C₈	#63225	1	2	yes	—
229	C₂×C₈	#63225	1	2	yes	—
230	C₂×C₈	#63225	1	1	yes	characteristic
231	C₂×C₈	#63225	1	1	yes	characteristic
232	C₂×C₈	#63225	1	2	yes	—
233	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
234	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
235	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
236	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₁₇	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	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
243	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
244	C₁₆	—	17	8	no	—
245	C₂³	—	4	2	no	—
246	C₂³	#10587478	1	1	yes	characteristic
247	C₂³	#10587478	1	1	yes	characteristic
248	C₂³	#10587478	1	2	yes	—
249	C₂³	C₂×#63231	1	1	yes	characteristic
250	C₂³	C₂×#63231	1	2	yes	—
251	C₂×C₄	#10587478	1	2	yes	—
252	C₂×C₄	#10587478	1	2	yes	—
253	C₂×C₄	#10587478	1	2	yes	—
254	C₂×C₄	#10587478	1	1	yes	characteristic
255	C₂×C₄	#10587478	1	1	yes	characteristic
256	C₂×C₄	#10587478	1	2	yes	—
257	C₂×C₄	#10587478	1	2	yes	—
258	C₂×C₄	C₂×#63225	1	2	yes	—
259	C₂×C₄	C₂×#63225	1	2	yes	—
260	C₂×C₄	C₂×#63225	1	1	yes	characteristic
261	C₂×C₄	C₂×#63225	1	1	yes	characteristic
262	C₂×C₄	C₂×#63228	1	2	yes	—
263	C₂×C₄	C₂×#63228	1	2	yes	—
264	C₂×C₄	C₂×#63228	1	1	yes	characteristic
265	C₂×C₄	C₂×#63228	1	1	yes	characteristic
266	C₂×C₄	C₂×#63231	1	2	yes	—
267	C₂×C₄	C₂×#63231	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	—	68	4	no	—
271	C₈	C₂×#63225	1	1	yes	characteristic
272	C₈	C₂×#63225	1	1	yes	characteristic
273	C₈	C₂×#63225	1	2	yes	—
274	C₈	C₂×#63225	1	2	yes	—
275	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
276	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂×#10587478	1	2	yes	—
283	C₂²	C₂×#10587478	1	2	yes	—
284	C₂²	C₂×#10587478	1	1	yes	characteristic
285	C₂²	C₂×#10587478	1	1	yes	characteristic
286	C₂²	C₂²×#63231	1	1	yes	characteristic
287	C₄	C₂×#10587478	1	2	yes	—
288	C₄	C₂×#10587478	1	2	yes	—
289	C₄	C₂×#10587478	1	1	yes	characteristic
290	C₄	C₂×#10587478	1	1	yes	characteristic
291	C₄	C₂²×#63225	1	1	yes	characteristic
292	C₄	C₂²×#63228	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	#14530578	1	2	yes	direct product factor
298	C₂	C₂²×#10587478	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂⁴×C₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×#108	D₁₇
hypocenter	C₁₇	C₂×#2161
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #14530578	C₂	direct (N × H)
4. C₂×#11762638	C₂	semidirect
16. #11762638	C₂²	semidirect
170. C₂×C₁₇	#2161	semidirect
204. C₁₇	C₂×#2161	semidirect
297. C₂	#14530578	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
C₂²×#10587490	2
#14530578	2
C₂×#11762638	2
C₂²×C₁₇×#108	2
C₂×#2161	17

Subgroups — C2×#14530578 #17187379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530578 #17187379