Subgroups — C₂×#14531009 #17189609

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763186	C₂	1	1	yes	maximal, characteristic
2	#14531009	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762609	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₃₆	—	2	1	no	—
7	C₂²×D₁₃₆	C₂²	1	2	yes	—
8	C₂×#10587560	—	2	1	no	—
9	C₂²×#63214	C₂²	1	2	yes	—
10	#11763186	—	2	2	no	—
11	#11763186	C₂²	1	4	yes	—
12	#11762609	C₂²	1	16	yes	complemented
13	C₄×#63214	C₂²	1	4	yes	complemented
14	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
15	C₁₇×#500	—	2	2	no	—
16	C₁₇×#500	C₂²	1	4	yes	complemented
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₁₇×#107	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	4	1	no	—
21	C₂²×D₆₈	—	2	2	no	—
22	C₂²×D₆₈	—	4	1	no	—
23	C₂²×D₆₈	—	4	1	no	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₁₇×C₄⋊Q₈	—	2	4	no	—
27	C₁₇×C₄⋊Q₈	—	4	2	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
30	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
31	C₂×D₁₃₆	—	2	4	no	—
32	C₂×D₁₃₆	—	2	2	no	—
33	C₂×D₁₃₆	—	2	8	no	—
34	C₂×D₁₃₆	—	4	4	no	—
35	C₂×D₁₃₆	C₂³	1	4	yes	—
36	#10587560	—	2	4	no	—
37	#10587560	—	4	2	no	—
38	C₁₇×#107	—	2	2	no	—
39	C₁₇×#107	—	2	4	no	—
40	C₁₇×#107	C₂³	1	4	yes	—
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	C₄×C₈×C₁₇	C₂³	1	2	yes	—
43	C₂×#63214	—	2	4	no	—
44	C₂×#63214	C₂³	1	4	yes	—
45	C₂³×D₁₇	—	4	2	no	—
46	C₂×Q₈×C₁₇	—	4	4	no	—
47	C₂×Q₈×C₁₇	—	8	4	no	—
48	C₂²×C₄×C₁₇	—	4	2	no	—
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×D₆₈	—	2	4	no	—
53	C₂×D₆₈	—	4	4	no	—
54	C₂×D₆₈	—	4	4	no	—
55	C₂×D₆₈	—	4	2	no	—
56	C₂×D₆₈	—	4	4	no	—
57	C₂×D₆₈	—	4	4	no	—
58	C₂×D₆₈	—	8	4	no	—
59	C₂×D₆₈	—	8	4	no	—
60	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
61	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₇	#41	1	4	yes	—
64	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇	—	2	4	no	—
66	C₄⋊C₄×C₁₇	—	4	4	no	—
67	C₄⋊C₄×C₁₇	—	4	2	no	—
68	C₄²×C₁₇	D₈	1	2	yes	—
69	C₄²×C₁₇	C₂×D₄	1	2	yes	—
70	D₁₃₆	—	2	8	no	—
71	D₁₃₆	—	4	4	no	—
72	D₁₃₆	—	4	8	no	—
73	#63214	—	2	4	no	—
74	C₂×#2240	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₁₇	—	4	4	no	—
77	C₂²×D₁₇	—	8	4	no	—
78	C₂²×D₁₇	—	8	4	no	—
79	C₂²×D₁₇	—	8	4	no	—
80	Q₈×C₁₇	—	8	8	no	—
81	C₂×C₄×C₁₇	—	4	4	no	—
82	C₂×C₄×C₁₇	—	8	4	no	—
83	C₂×C₄×C₁₇	D₁₆	1	4	yes	—
84	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
85	C₂×C₄×C₁₇	#124	1	2	yes	—
86	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
91	D₆₈	—	4	4	no	—
92	D₆₈	—	4	4	no	—
93	D₆₈	—	8	4	no	—
94	D₆₈	—	8	8	no	—
95	D₆₈	—	8	8	no	—
96	C₈×C₁₇	#124	1	2	yes	—
97	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
98	C₈×C₁₇	C₂×#41	1	2	yes	—
99	C₂×#500	D₁₇	1	1	yes	characteristic, complemented
100	C₂×#493	—	17	1	no	—
101	#2240	—	17	8	no	—
102	C₂×#357	—	17	4	no	—
103	C₂×C₄×C₁₆	—	17	1	no	—
104	C₂²×C₁₇	#486	1	2	yes	—
105	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
106	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
107	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
108	C₂×D₁₇	—	8	4	no	—
109	C₂×D₁₇	—	8	4	no	—
110	C₂×D₁₇	—	8	4	no	—
111	C₂×D₁₇	—	16	8	no	—
112	C₄×C₁₇	—	8	4	no	—
113	C₄×C₁₇	#486	1	2	yes	—
114	C₄×C₁₇	C₂×D₁₆	1	4	yes	—
115	C₄×C₁₇	C₂×SD₃₂	1	4	yes	—
116	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
117	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
118	C₂²×D₈	—	17	2	no	—
119	C₂²×D₈	—	34	1	no	—
120	C₂×C₄⋊Q₈	—	2	1	no	—
121	C₂×#127	—	34	1	no	—
122	C₂²×C₁₆	—	17	2	no	—
123	#500	—	2	2	no	—
124	#500	C₂×D₁₇	1	4	yes	complemented
125	#493	—	17	4	no	—
126	#493	—	34	2	no	—
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂×D₁₇	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
130	#357	—	17	16	no	—
131	C₄×C₁₆	—	17	4	no	—
132	C₂×C₁₇	#2240	1	4	yes	complemented
133	C₂×C₁₇	C₂×#486	1	1	yes	characteristic
134	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
135	C₂×C₁₇	C₂²×SD₃₂	1	1	yes	characteristic
136	D₁₇	—	16	4	no	—
137	C₂²×Q₈	—	4	1	no	—
138	C₂²×D₄	—	34	2	no	—
139	C₂²×D₄	—	68	1	no	—
140	C₂²×D₄	—	68	1	no	—
141	C₂×D₈	—	17	4	no	—
142	C₂×D₈	—	34	8	no	—
143	C₂×D₈	—	34	2	no	—
144	C₂×D₈	—	34	4	no	—
145	C₂×D₈	—	68	4	no	—
146	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	4	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	#127	—	34	4	no	—
151	#127	—	68	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
155	C₂×C₁₆	—	17	4	no	—
156	C₂×C₁₆	—	34	4	no	—
157	#107	—	2	4	no	—
158	#107	—	2	2	no	—
159	#107	C₂²×D₁₇	1	4	yes	—
160	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
161	C₄×C₈	C₂²×D₁₇	1	2	yes	—
162	C₁₇	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
163	C₂⁴	—	68	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	34	4	no	—
167	C₂×D₄	—	68	4	no	—
168	C₂×D₄	—	68	4	no	—
169	C₂×D₄	—	68	2	no	—
170	C₂×D₄	—	68	4	no	—
171	C₂×D₄	—	68	4	no	—
172	C₂×D₄	—	136	4	no	—
173	C₂×D₄	—	136	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	#63228	1	1	yes	characteristic
176	C₂²×C₄	#63228	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
178	D₈	—	34	8	no	—
179	D₈	—	68	8	no	—
180	D₈	—	68	4	no	—
181	C₂×C₈	#63230	1	4	yes	—
182	C₂×C₈	#63256	1	4	yes	—
183	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
184	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
185	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	#63228	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
191	C₁₆	—	34	4	no	—
192	C₂³	—	68	4	no	—
193	C₂³	—	136	4	no	—
194	C₂³	—	136	4	no	—
195	C₂³	—	136	4	no	—
196	C₂³	C₂×#63228	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	68	4	no	—
199	D₄	—	68	4	no	—
200	D₄	—	136	8	no	—
201	D₄	—	136	8	no	—
202	D₄	—	136	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#10587497	1	4	yes	—
206	C₂×C₄	#10587499	1	4	yes	—
207	C₂×C₄	C₂×#63228	1	2	yes	—
208	C₂×C₄	C₂×#63228	1	2	yes	—
209	C₂×C₄	C₂×#63228	1	1	yes	characteristic
210	C₂×C₄	C₂×#63230	1	2	yes	—
211	C₂×C₄	#10587633	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
213	C₈	C₂×#63230	1	4	yes	—
214	C₈	#10587633	1	2	yes	—
215	C₈	C₂×#63256	1	2	yes	—
216	C₂²	—	272	8	no	—
217	C₂²	—	136	4	no	—
218	C₂²	—	136	4	no	—
219	C₂²	—	136	4	no	—
220	C₂²	#11763157	1	2	yes	—
221	C₂²	C₂×#10587497	1	2	yes	—
222	C₂²	C₂×#10587499	1	2	yes	—
223	C₂²	C₂²×#63228	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#11763157	1	2	yes	—
226	C₄	C₂×#10587497	1	4	yes	—
227	C₄	C₂×#10587499	1	4	yes	—
228	C₄	C₂×#10587633	1	1	yes	characteristic
229	C₄	C₂²×#63230	1	1	yes	characteristic
230	C₂	—	272	4	no	—
231	C₂	#14531009	1	4	yes	direct product factor
232	C₂	C₂×#11763157	1	1	yes	characteristic
233	C₂	C₂²×#10587497	1	1	yes	characteristic
234	C₂	C₂²×#10587499	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂×#11763186	C₂
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	C₂×#2240
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #14531009	C₂	direct (N × H)
3. C₂×#11762609	C₂	semidirect
4. C₂×C₄×#63214	C₂	semidirect
5. C₂×C₁₇×#500	C₂	semidirect
12. #11762609	C₂²	semidirect
13. C₄×#63214	C₂²	semidirect
16. C₁₇×#500	C₂²	semidirect
99. C₂×#500	D₁₇	semidirect
124. #500	C₂×D₁₇	semidirect
132. C₂×C₁₇	#2240	semidirect
162. C₁₇	C₂×#2240	semidirect
231. C₂	#14531009	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×#11763186	2
#14531009	2
C₂×#11762609	2
C₂×C₄×#63214	2
C₂×C₁₇×#500	2
C₂×#2240	17

Subgroups — C2×#14531009 #17189609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531009 #17189609