Subgroups — C₂×#14527572 #17003408

Order 4352.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762862	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762833	C₂	1	1	yes	maximal, characteristic
3	#14527572	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762561	C₂	1	2	yes	maximal, complemented
5	C₂×#11762559	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63223	—	2	1	no	—
8	#11762862	C₄	1	4	yes	—
9	#11762862	C₂²	1	4	yes	—
10	#11762833	—	2	2	no	—
11	#11762833	C₂²	1	4	yes	—
12	C₂×C₄⋊#3622	—	2	1	no	—
13	C₂×C₄⋊#3622	C₄	1	2	yes	—
14	C₂×C₄⋊#3622	C₂²	1	2	yes	—
15	C₂×C₂²⋊#3622	C₄	1	2	yes	—
16	C₂×C₄×#3622	C₂²	1	2	yes	—
17	#11762561	C₂²	1	8	yes	complemented
18	#11762559	C₂²	1	8	yes	complemented
19	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#431	—	2	2	no	—
21	C₁₇×#431	C₂²	1	4	yes	—
22	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
25	C₂×C₄×C₈×C₁₇	—	2	1	no	—
26	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	—	2	1	no	—
30	C₂²×C₈×C₁₇	—	2	1	no	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	D₄	1	2	yes	—
35	C₂×C₄²×C₁₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
39	C₂×#63223	—	2	4	no	—
40	C₂×#63223	—	4	4	no	—
41	C₂²×#3622	—	2	2	no	—
42	C₂²×#3622	C₂×C₄	1	4	yes	—
43	C₁₇×C₄⋊C₈	—	2	1	no	—
44	C₁₇×C₄⋊C₈	—	2	1	no	—
45	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
46	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
47	C₁₇×C₂²⋊C₈	—	2	4	no	—
48	C₄×C₈×C₁₇	—	2	4	no	—
49	C₂²⋊#3622	C₂×C₄	1	8	yes	—
50	C₄⋊#3622	—	2	4	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₄⋊#3622	C₂×C₄	1	8	yes	—
53	C₄⋊#3622	C₂³	1	4	yes	—
54	C₄×#3622	C₂×C₄	1	4	yes	—
55	C₄×#3622	C₂³	1	4	yes	—
56	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	—	2	4	no	—
58	C₂²×C₄×C₁₇	—	2	4	no	—
59	C₂²×C₄×C₁₇	—	2	2	no	—
60	C₂²×C₄×C₁₇	—	2	2	no	—
61	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	2	2	no	—
75	C₂×C₈×C₁₇	—	4	2	no	—
76	C₂×C₈×C₁₇	—	4	2	no	—
77	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₇	C₂×D₄	1	2	yes	—
83	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
85	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
87	#63223	—	4	8	no	—
88	C₂×#3622	—	2	8	no	—
89	C₂×#3622	—	2	4	no	—
90	C₂×#3622	—	4	4	no	—
91	C₂×#3622	C₂²×C₄	1	8	yes	—
92	C₂×#1574	—	17	1	no	maximal, Sylow
93	C₂³×C₁₇	—	2	2	no	—
94	C₂³×C₁₇	—	2	2	no	—
95	C₂³×C₁₇	—	2	2	no	—
96	C₂³×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	4	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	4	no	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	—	2	4	no	—
107	C₂×C₄×C₁₇	—	2	4	no	—
108	C₂×C₄×C₁₇	#108	1	4	yes	—
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×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₇	#131	1	4	yes	—
116	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₁₇	#137	1	2	yes	—
118	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
121	C₈×C₁₇	—	4	2	no	—
122	C₈×C₁₇	—	4	2	no	—
123	#3622	—	2	8	no	—
124	#3622	—	4	4	no	—
125	C₂×#431	—	17	1	no	—
126	C₂×#431	D₁₇	1	1	yes	characteristic
127	C₂×#423	—	17	1	no	—
128	#1574	—	17	8	no	—
129	C₂×#335	—	17	2	no	—
130	C₂×#334	—	17	2	no	—
131	C₂²×C₁₇	—	2	2	no	—
132	C₂²×C₁₇	—	2	2	no	—
133	C₂²×C₁₇	—	2	2	no	—
134	C₂²×C₁₇	—	4	4	no	—
135	C₂²×C₁₇	#424	1	2	yes	—
136	C₂²×C₁₇	#428	1	2	yes	—
137	C₂²×C₁₇	C₂×#108	1	2	yes	—
138	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	#424	1	2	yes	—
144	C₄×C₁₇	#428	1	2	yes	—
145	C₄×C₁₇	C₂×#108	1	4	yes	—
146	C₄×C₁₇	C₂×#131	1	2	yes	—
147	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
149	C₂²×M₁₆	—	34	1	no	—
150	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
151	#431	—	2	2	no	—
152	#431	—	17	4	no	—
153	#431	—	17	4	no	—
154	#431	C₂×D₁₇	1	4	yes	—
155	#423	—	17	4	no	—
156	#423	—	34	2	no	—
157	C₂×C₄⋊C₈	—	17	2	no	—
158	C₂×C₄⋊C₈	—	17	2	no	—
159	C₂×C₄⋊C₈	—	34	1	no	—
160	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×C₂²⋊C₈	—	17	2	no	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	—	17	2	no	—
166	#335	—	17	8	no	—
167	#334	—	17	8	no	—
168	C₂×C₁₇	—	4	2	no	—
169	C₂×C₁₇	#1574	1	4	yes	complemented
170	C₂×C₁₇	C₂×#424	1	1	yes	characteristic
171	C₂×C₁₇	C₂×#428	1	1	yes	characteristic
172	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
173	C₂³×C₄	Dic₃₄	1	1	yes	characteristic
174	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
176	C₂×M₁₆	—	34	4	no	—
177	C₂×M₁₆	—	68	4	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	17	4	no	—
181	C₂²×C₈	—	34	2	no	—
182	C₂×C₄²	Dic₃₄	1	1	yes	characteristic
183	C₂×C₄²	Dic₃₄	1	2	yes	—
184	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₁₇	1	2	yes	—
186	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
189	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	17	8	no	—
193	C₄⋊C₈	—	17	4	no	—
194	C₄⋊C₈	—	34	2	no	—
195	C₄⋊C₈	—	34	4	no	—
196	C₄⋊C₈	C₂²×D₁₇	1	2	yes	—
197	C₄⋊C₈	C₂²×D₁₇	1	2	yes	—
198	C₂²⋊C₈	—	2	4	no	—
199	C₂²⋊C₈	—	17	8	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	17	4	no	—
202	C₄×C₈	—	17	4	no	—
203	C₁₇	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
204	C₂⁴	#63225	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	#63225	1	1	yes	characteristic
210	C₂²×C₄	#63225	1	2	yes	—
211	C₂²×C₄	#63225	1	1	yes	characteristic
212	C₂²×C₄	#63225	1	2	yes	—
213	C₂²×C₄	#63225	1	2	yes	—
214	C₂²×C₄	#63225	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₃₄	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₇	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
221	M₁₆	—	68	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	17	8	no	—
227	C₂×C₈	—	34	8	no	—
228	C₂×C₈	—	34	4	no	—
229	C₂×C₈	—	68	4	no	—
230	C₄²	#63225	1	2	yes	—
231	C₄²	#63225	1	2	yes	—
232	C₄²	#63225	1	2	yes	—
233	C₄²	C₂×Dic₃₄	1	2	yes	—
234	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
235	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₁₇	1	2	yes	—
237	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
238	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
239	C₄²	C₂³×D₁₇	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×#63225	1	2	yes	—
244	C₂³	C₂×#63225	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	#10587493	1	4	yes	—
256	C₂×C₄	#10587587	1	4	yes	—
257	C₂×C₄	C₂×#63225	1	2	yes	—
258	C₂×C₄	C₂×#63225	1	2	yes	—
259	C₂×C₄	C₂×#63225	1	1	yes	characteristic
260	C₂×C₄	C₂×#63225	1	1	yes	characteristic
261	C₂×C₄	C₂×#63225	1	1	yes	characteristic
262	C₂×C₄	C₂×#63225	1	2	yes	—
263	C₂×C₄	#10587635	1	2	yes	—
264	C₂×C₄	#10587637	1	2	yes	—
265	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	4	2	no	—
270	C₈	—	34	8	no	—
271	C₈	—	68	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#11762835	1	2	yes	—
277	C₂²	#11762852	1	2	yes	—
278	C₂²	C₂×#10587493	1	2	yes	—
279	C₂²	C₂²×#63225	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#11762835	1	2	yes	—
285	C₄	#11762852	1	2	yes	—
286	C₄	C₂×#10587493	1	4	yes	—
287	C₄	C₂×#10587587	1	2	yes	—
288	C₄	C₂×#10587635	1	1	yes	characteristic
289	C₄	C₂×#10587637	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#14527572	1	4	yes	direct product factor
292	C₂	C₂×#11762835	1	1	yes	characteristic
293	C₂	C₂×#11762852	1	1	yes	characteristic
294	C₂	C₂²×#10587493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#431	C₂
socle	C₂³×C₁₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	C₂×#431	D₁₇
hypocenter	C₁₇	C₂×#1574
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #14527572	C₂	direct (N × H)
4. C₂×#11762561	C₂	semidirect
5. C₂×#11762559	C₂	semidirect
17. #11762561	C₂²	semidirect
18. #11762559	C₂²	semidirect
169. C₂×C₁₇	#1574	semidirect
203. C₁₇	C₂×#1574	semidirect
291. C₂	#14527572	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×#11762862	2
C₂×#11762833	2
#14527572	2
C₂×#11762561	2
C₂×#11762559	2
C₂×C₁₇×#431	2
C₂×#1574	17

Subgroups — C2×#14527572 #17003408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14527572 #17003408