Subgroups — C₂×#11762565 #14528484

Order 2176.

404 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63223	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587612	C₂	1	1	yes	maximal, characteristic
3	#11762565	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₇×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₁₇×#117	—	2	1	no	—
7	C₁₇×#117	—	2	1	no	—
8	C₁₇×#117	C₄	1	2	yes	complemented
9	C₁₇×#117	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
11	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
12	C₂×C₄²×C₁₇	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Dic₁₇	—	2	1	no	—
14	#10587612	—	2	1	no	—
15	#10587612	—	2	1	no	—
16	#10587612	C₄	1	2	yes	complemented
17	#10587612	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₇	—	2	1	no	—
19	C₂×C₄×Dic₁₇	C₄	1	2	yes	complemented
20	C₂×#63223	—	2	1	no	—
21	C₂×#63223	—	2	1	no	—
22	C₂×#63223	—	2	1	no	—
23	C₂×#63223	—	2	1	no	—
24	C₂×#63223	C₄	1	2	yes	complemented
25	C₂×#63223	C₂²	1	2	yes	—
26	C₂²×#3622	C₄	1	2	yes	complemented
27	C₂⁴×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	—	2	2	no	—
29	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	2	2	no	—
44	C₄⋊C₄×C₁₇	—	2	2	no	—
45	C₄⋊C₄×C₁₇	—	2	2	no	—
46	C₂²⋊C₄×C₁₇	—	2	2	no	—
47	C₂²⋊C₄×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	C₂×C₄	1	4	yes	complemented
50	C₂²⋊Dic₁₇	—	2	2	no	—
51	C₂²⋊Dic₁₇	—	2	2	no	—
52	C₄⋊Dic₁₇	—	2	2	no	—
53	C₄⋊Dic₁₇	—	2	2	no	—
54	C₄×Dic₁₇	—	2	2	no	—
55	C₄×Dic₁₇	C₂×C₄	1	4	yes	complemented
56	#63223	—	2	2	no	—
57	#63223	—	2	2	no	—
58	#63223	—	2	2	no	—
59	#63223	—	2	2	no	—
60	C₂×#3622	—	2	2	no	—
61	C₂×#3622	—	2	2	no	—
62	C₂×#3622	C₂×C₄	1	4	yes	complemented
63	C₂³×C₁₇	—	4	2	no	—
64	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
101	C₂×Dic₁₇	—	2	4	no	—
102	C₂×Dic₁₇	—	4	4	no	—
103	#3622	—	2	4	no	—
104	C₂×H(3,ℤ/4)	—	17	1	no	maximal, Sylow
105	C₂²×C₁₇	—	2	2	no	—
106	C₂²×C₁₇	—	2	2	no	—
107	C₂²×C₁₇	—	2	2	no	—
108	C₂²×C₁₇	—	4	1	no	—
109	C₂²×C₁₇	—	4	1	no	—
110	C₂²×C₁₇	—	4	1	no	—
111	C₂²×C₁₇	—	4	1	no	—
112	C₂²×C₁₇	#95	1	1	yes	characteristic
113	C₂²×C₁₇	#95	1	1	yes	characteristic
114	C₂²×C₁₇	#95	1	1	yes	characteristic
115	C₂²×C₁₇	#95	1	1	yes	characteristic
116	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₁₇	—	4	4	no	—
120	C₄×C₁₇	#95	1	1	yes	characteristic
121	C₄×C₁₇	#95	1	1	yes	characteristic
122	C₄×C₁₇	#95	1	1	yes	characteristic
123	C₄×C₁₇	#95	1	1	yes	characteristic
124	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
125	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	Dic₁₇	—	4	4	no	—
129	C₂²×M₁₆	—	17	1	no	—
130	C₂×#117	—	17	1	no	—
131	C₂×#117	D₁₇	1	1	yes	characteristic
132	H(3,ℤ/4)	—	17	4	no	—
133	C₂×C₁₇	—	2	1	no	—
134	C₂×C₁₇	—	2	1	no	—
135	C₂×C₁₇	—	2	1	no	—
136	C₂×C₁₇	—	2	1	no	—
137	C₂×C₁₇	—	2	1	no	—
138	C₂×C₁₇	—	2	1	no	—
139	C₂×C₁₇	H(3,ℤ/4)	1	2	yes	complemented
140	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
141	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
142	C₂×M₁₆	—	17	2	no	—
143	C₂×M₁₆	—	17	2	no	—
144	C₂×M₁₆	—	34	1	no	—
145	C₂×M₁₆	—	34	1	no	—
146	C₂×M₁₆	—	34	1	no	—
147	C₂×M₁₆	—	34	1	no	—
148	C₂²×C₈	—	17	2	no	—
149	#117	—	2	1	no	—
150	#117	—	2	1	no	—
151	#117	—	17	2	no	—
152	#117	—	17	2	no	—
153	#117	—	34	1	no	—
154	#117	—	34	1	no	—
155	#117	Dic₁₇	1	2	yes	complemented
156	#117	C₂×D₁₇	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	34	1	no	—
159	C₂×C₂²⋊C₄	—	2	1	no	—
160	C₂×C₂²⋊C₄	—	34	1	no	—
161	C₂×C₄²	—	17	2	no	—
162	C₂×C₄²	Dic₁₇	1	2	yes	complemented
163	C₁₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
164	C₂⁴	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	34	2	no	—
167	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
168	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
170	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
171	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
172	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
173	C₂²×C₄	D₆₈	1	1	yes	characteristic
174	C₂²×C₄	D₆₈	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
181	M₁₆	—	34	2	no	—
182	M₁₆	—	34	2	no	—
183	M₁₆	—	34	2	no	—
184	M₁₆	—	34	2	no	—
185	C₂×C₈	—	17	4	no	—
186	C₂×C₈	—	34	2	no	—
187	C₂×C₈	—	34	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	34	2	no	—
191	C₄⋊C₄	—	34	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	34	2	no	—
195	C₂²⋊C₄	—	34	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	17	4	no	—
198	C₄²	—	34	2	no	—
199	C₄²	C₂×Dic₁₇	1	4	yes	complemented
200	C₂³	—	4	2	no	—
201	C₂³	#63225	1	1	yes	characteristic
202	C₂³	#63225	1	1	yes	characteristic
203	C₂³	#63227	1	1	yes	characteristic
204	C₂³	#63227	1	1	yes	characteristic
205	C₂³	C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₂³	C₂²⋊Dic₁₇	1	1	yes	characteristic
207	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	34	4	no	—
211	C₂×C₄	—	68	4	no	—
212	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
213	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
215	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
216	C₂×C₄	#63225	1	1	yes	characteristic
217	C₂×C₄	#63225	1	1	yes	characteristic
218	C₂×C₄	#63225	1	1	yes	characteristic
219	C₂×C₄	#63225	1	1	yes	characteristic
220	C₂×C₄	#63225	1	1	yes	characteristic
221	C₂×C₄	#63225	1	1	yes	characteristic
222	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
223	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
224	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
225	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
226	C₂×C₄	#63227	1	1	yes	characteristic
227	C₂×C₄	#63227	1	1	yes	characteristic
228	C₂×C₄	#63227	1	1	yes	characteristic
229	C₂×C₄	#63227	1	1	yes	characteristic
230	C₂×C₄	#63227	1	1	yes	characteristic
231	C₂×C₄	#63227	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
234	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
237	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
240	C₈	—	34	4	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	#10587502	1	1	yes	characteristic
249	C₂²	#10587502	1	1	yes	characteristic
250	C₂²	#10587502	1	1	yes	characteristic
251	C₂²	#10587502	1	1	yes	characteristic
252	C₂²	C₂×#63225	1	1	yes	characteristic
253	C₂²	C₂×#63227	1	1	yes	characteristic
254	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
255	C₄	—	4	4	no	—
256	C₄	—	68	4	no	—
257	C₄	#10587502	1	1	yes	characteristic
258	C₄	#10587502	1	1	yes	characteristic
259	C₄	#10587502	1	1	yes	characteristic
260	C₄	#10587502	1	1	yes	characteristic
261	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
262	C₄	C₂×#63225	1	1	yes	characteristic
263	C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
264	C₄	C₂×#63227	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#11762565	1	2	yes	direct product factor
272	C₂	C₂×#10587502	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#117	C₂
socle	C₂²×C₁₇	#95
prime radical	C₂⁴×C₁₇	D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#117	D₁₇
hypocenter	C₁₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #11762565	C₂	direct (N × H)
8. C₁₇×#117	C₄	semidirect
12. C₂×C₄²×C₁₇	C₄	semidirect
16. #10587612	C₄	semidirect
19. C₂×C₄×Dic₁₇	C₄	semidirect
24. C₂×#63223	C₄	semidirect
26. C₂²×#3622	C₄	semidirect
49. C₄²×C₁₇	C₂×C₄	semidirect
55. C₄×Dic₁₇	C₂×C₄	semidirect
62. C₂×#3622	C₂×C₄	semidirect
139. C₂×C₁₇	H(3,ℤ/4)	semidirect
155. #117	Dic₁₇	semidirect
162. C₂×C₄²	Dic₁₇	semidirect
163. C₁₇	C₂×H(3,ℤ/4)	semidirect
199. C₄²	C₂×Dic₁₇	semidirect
271. C₂	#11762565	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×#63223	2
C₂×#10587612	2
#11762565	2
C₂×C₁₇×#117	2
C₂×H(3,ℤ/4)	17

Subgroups — C2×#11762565 #14528484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762565 #14528484