Subgroups — C₂×#10621582 #13062258

Order 1344.

1796 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#239	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1161	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621582	C₂	1	16	yes	maximal, direct product factor
5	C₂×#66681	C₂	1	3	yes	maximal, complemented
6	C₂×D₃×D₂₈	C₂	1	3	yes	maximal, complemented
7	C₂×D₇×D₁₂	C₂	1	3	yes	maximal, complemented
8	C₂×#66675	C₂	1	3	yes	maximal, complemented
9	C₂×#6553	—	3	1	no	maximal, Hall
10	C₂²×D₃×D₇	C₂²	1	6	yes	complemented
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	#66732	C₂²	1	8	yes	complemented
13	C₂×D₈₄	C₂²	1	3	yes	complemented
14	C₂×C₄×D₂₁	C₂²	1	3	yes	complemented
15	C₇×#239	C₂²	1	8	yes	complemented
16	C₂×C₇×D₁₂	C₂²	1	3	yes	complemented
17	C₂×C₄×D₃×C₇	C₂²	1	3	yes	complemented
18	C₃×#1161	C₂²	1	8	yes	complemented
19	C₂×C₃×D₂₈	C₂²	1	3	yes	complemented
20	C₂×C₃×C₄×D₇	C₂²	1	3	yes	complemented
21	C₂×#4248	C₂²	1	1	yes	characteristic, complemented
22	C₂×#4247	C₂²	1	3	yes	complemented
23	C₂×#4246	C₂²	1	3	yes	complemented
24	C₂×#4245	C₂²	1	3	yes	complemented
25	#66681	C₂²	1	24	yes	complemented
26	D₃×D₂₈	C₂²	1	24	yes	complemented
27	D₇×D₁₂	C₂²	1	24	yes	complemented
28	#66675	C₂²	1	24	yes	complemented
29	C₂×C₇×#41	—	3	1	no	—
30	#6553	—	3	16	no	—
31	C₂×#1161	—	3	1	no	—
32	C₂×#1161	D₃	1	1	yes	characteristic, complemented
33	C₂×D₄×D₇	—	3	3	no	—
34	C₂×D₄×D₇	—	3	3	no	—
35	C₂×#1157	—	3	3	no	—
36	C₂²×D₂₈	—	3	3	no	—
37	C₂×#1009	—	7	1	no	maximal, Hall
38	C₂²×D₂₁	C₂³	1	3	yes	—
39	C₂²×D₃×C₇	C₂³	1	3	yes	—
40	C₂²×C₃×D₇	C₂³	1	3	yes	—
41	C₂×D₃×D₇	—	4	12	no	—
42	C₂×D₃×D₇	C₂³	1	24	yes	complemented
43	C₃×C₇×Q₈	C₂³	1	4	yes	complemented
44	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
45	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
46	D₈₄	C₂³	1	12	yes	complemented
47	C₄×D₂₁	C₂³	1	12	yes	complemented
48	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
49	C₇×D₁₂	C₂³	1	12	yes	complemented
50	C₄×D₃×C₇	C₂³	1	12	yes	complemented
51	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
52	C₃×D₂₈	C₂³	1	12	yes	complemented
53	C₃×C₄×D₇	C₂³	1	12	yes	complemented
54	#4248	C₂³	1	4	yes	complemented
55	#4247	C₂³	1	12	yes	complemented
56	#4246	C₂³	1	12	yes	complemented
57	#4245	C₂³	1	12	yes	complemented
58	C₂³×D₇	—	3	6	no	—
59	C₇×#41	—	3	8	no	—
60	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
61	C₂×C₇×D₄	—	3	3	no	—
62	C₂²×C₄×C₇	—	3	3	no	—
63	C₂×C₂²⋊D₇	—	3	3	no	—
64	C₂×C₂²⋊D₇	—	3	3	no	—
65	#1161	—	3	8	no	—
66	#1161	C₂×D₃	1	8	yes	complemented
67	D₄×D₇	—	3	24	no	—
68	D₄×D₇	—	3	24	no	—
69	#1157	—	3	24	no	—
70	C₂×D₂₈	—	3	3	no	—
71	C₂×D₂₈	—	3	3	no	—
72	C₂×D₂₈	—	3	24	no	—
73	C₂×D₂₈	C₂×D₃	1	3	yes	complemented
74	C₂×C₄×D₇	—	3	3	no	—
75	C₂×C₄×D₇	C₂×D₃	1	3	yes	complemented
76	C₂×Dic₁₄	—	3	1	no	—
77	C₂×C₃×#41	—	7	1	no	—
78	#1009	—	7	16	no	—
79	C₂×#239	—	7	1	no	—
80	C₂×#239	D₇	1	1	yes	characteristic, complemented
81	C₂×D₃×D₄	—	7	3	no	—
82	C₂×D₃×D₄	—	7	3	no	—
83	C₂×#235	—	7	3	no	—
84	C₂²×D₁₂	—	7	3	no	—
85	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
86	C₂×D₂₁	—	2	6	no	—
87	C₂×D₂₁	C₂⁴	1	6	yes	—
88	C₂×D₃×C₇	—	2	6	no	—
89	C₂×D₃×C₇	C₂⁴	1	6	yes	—
90	C₂×C₃×D₇	—	2	6	no	—
91	C₂×C₃×D₇	C₂⁴	1	6	yes	—
92	D₃×D₇	—	4	24	no	—
93	C₃×C₄×C₇	C₂⁴	1	6	yes	—
94	Dic₂₁	C₂⁴	1	2	yes	—
95	C₃×Dic₇	C₂⁴	1	2	yes	—
96	C₇×Dic₃	C₂⁴	1	2	yes	—
97	C₂×H(5,𝔽₂)	—	21	1	no	Sylow
98	C₂³×C₇	—	3	3	no	—
99	C₂²×D₇	—	3	3	no	—
100	C₂²×D₇	—	3	24	no	—
101	C₂²×D₇	—	12	12	no	—
102	C₂²×D₇	C₂²×D₃	1	3	yes	—
103	C₇×Q₈	C₂²×D₃	1	4	yes	complemented
104	C₇×D₄	—	3	12	no	—
105	C₂×C₄×C₇	—	3	1	no	—
106	C₂×C₄×C₇	—	3	12	no	—
107	C₂×C₄×C₇	C₂²×D₃	1	3	yes	—
108	C₂²⋊D₇	—	3	12	no	—
109	C₂²⋊D₇	—	3	12	no	—
110	C₂×Dic₇	—	3	1	no	—
111	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
112	D₂₈	—	3	12	no	—
113	D₂₈	—	3	12	no	—
114	D₂₈	C₂²×D₃	1	12	yes	complemented
115	C₄×D₇	—	3	12	no	—
116	C₄×D₇	C₂²×D₃	1	12	yes	complemented
117	Dic₁₄	—	3	4	no	—
118	C₂³×D₃	—	7	6	no	—
119	C₃×#41	—	7	8	no	—
120	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
121	C₂×C₃×D₄	—	7	3	no	—
122	C₂²×C₃×C₄	—	7	3	no	—
123	C₂×#67	—	7	3	no	—
124	C₂×#67	—	7	3	no	—
125	#239	—	7	8	no	—
126	#239	C₂×D₇	1	8	yes	complemented
127	D₃×D₄	—	7	24	no	—
128	D₃×D₄	—	7	24	no	—
129	#235	—	7	24	no	—
130	C₂×D₁₂	—	7	3	no	—
131	C₂×D₁₂	—	7	3	no	—
132	C₂×D₁₂	—	7	24	no	—
133	C₂×D₁₂	C₂×D₇	1	3	yes	complemented
134	C₂×C₄×D₃	—	7	3	no	—
135	C₂×C₄×D₃	C₂×D₇	1	3	yes	complemented
136	C₂×Dic₆	—	7	1	no	—
137	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	complemented
138	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
139	D₂₁	—	2	6	no	—
140	C₃×D₇	—	2	6	no	—
141	D₃×C₇	—	2	6	no	—
142	H(5,𝔽₂)	—	21	16	no	—
143	C₂×#41	—	3	1	no	—
144	C₂×#41	—	7	1	no	—
145	C₂×#41	—	21	3	no	—
146	C₂×#41	—	21	1	no	—
147	C₂²×D₄	—	21	3	no	—
148	C₂²×D₄	—	21	3	no	—
149	C₂²×D₄	—	21	3	no	—
150	C₂²×C₇	—	3	6	no	—
151	C₂²×C₇	—	6	6	no	—
152	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
153	C₂×D₇	—	2	6	no	—
154	C₂×D₇	—	3	6	no	—
155	C₂×D₇	—	6	6	no	—
156	C₂×D₇	—	12	24	no	—
157	C₂×D₇	C₂³×D₃	1	6	yes	—
158	C₄×C₇	—	3	2	no	—
159	C₄×C₇	C₂³×D₃	1	6	yes	—
160	Dic₇	—	3	2	no	—
161	Dic₇	C₂³×D₃	1	2	yes	—
162	C₂³×C₃	—	7	3	no	—
163	C₂²×D₃	—	7	3	no	—
164	C₂²×D₃	—	7	24	no	—
165	C₂²×D₃	—	28	12	no	—
166	C₂²×D₃	C₂²×D₇	1	3	yes	—
167	C₃×Q₈	C₂²×D₇	1	4	yes	complemented
168	C₃×D₄	—	7	12	no	—
169	C₂×C₃×C₄	—	7	1	no	—
170	C₂×C₃×C₄	—	7	12	no	—
171	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
172	#67	—	7	12	no	—
173	#67	—	7	12	no	—
174	C₂×Dic₃	—	7	1	no	—
175	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
176	D₁₂	—	7	12	no	—
177	D₁₂	—	7	12	no	—
178	D₁₂	C₂²×D₇	1	12	yes	complemented
179	C₄×D₃	—	7	12	no	—
180	C₄×D₃	C₂²×D₇	1	12	yes	complemented
181	Dic₆	—	7	4	no	—
182	C₃×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
183	C₂⁴	—	21	6	no	—
184	#41	—	3	8	no	—
185	#41	—	7	8	no	—
186	#41	—	21	24	no	—
187	#41	—	21	8	no	—
188	C₂×Q₈	—	21	1	no	—
189	C₂×Q₈	D₃×D₇	1	1	yes	characteristic, complemented
190	C₂×D₄	—	3	3	no	—
191	C₂×D₄	—	7	3	no	—
192	C₂×D₄	—	21	24	no	—
193	C₂×D₄	—	21	3	no	—
194	C₂×D₄	—	21	24	no	—
195	C₂×D₄	—	21	3	no	—
196	C₂×D₄	—	21	24	no	—
197	C₂×D₄	—	21	3	no	—
198	C₂×D₄	—	21	3	no	—
199	C₂²×C₄	—	3	3	no	—
200	C₂²×C₄	—	7	3	no	—
201	C₂²×C₄	—	21	3	no	—
202	C₂×C₇	—	6	6	no	—
203	C₂×C₇	#1009	1	2	yes	complemented
204	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
205	D₇	—	2	6	no	—
206	D₇	—	6	6	no	—
207	C₂²×C₃	—	7	6	no	—
208	C₂²×C₃	—	14	6	no	—
209	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
210	C₂×D₃	—	2	6	no	—
211	C₂×D₃	—	7	6	no	—
212	C₂×D₃	—	14	6	no	—
213	C₂×D₃	—	28	24	no	—
214	C₂×D₃	C₂³×D₇	1	6	yes	—
215	C₃×C₄	—	7	2	no	—
216	C₃×C₄	C₂³×D₇	1	6	yes	—
217	Dic₃	—	7	2	no	—
218	Dic₃	C₂³×D₇	1	2	yes	—
219	C₂³	—	3	3	no	—
220	C₂³	—	7	3	no	—
221	C₂³	—	21	24	no	—
222	C₂³	—	21	3	no	—
223	C₂³	—	84	12	no	—
224	Q₈	—	21	4	no	—
225	Q₈	C₂×D₃×D₇	1	4	yes	complemented
226	D₄	—	3	12	no	—
227	D₄	—	7	12	no	—
228	D₄	—	21	12	no	—
229	D₄	—	21	12	no	—
230	D₄	—	21	12	no	—
231	D₄	—	21	12	no	—
232	C₂×C₄	—	3	1	no	—
233	C₂×C₄	—	3	12	no	—
234	C₂×C₄	—	7	1	no	—
235	C₂×C₄	—	7	12	no	—
236	C₂×C₄	—	21	1	no	—
237	C₂×C₄	—	21	12	no	—
238	C₂×C₄	C₂×D₃×D₇	1	3	yes	—
239	C₇	C₂×#1009	1	1	yes	characteristic, complemented, Sylow
240	C₂×C₃	—	14	6	no	—
241	C₂×C₃	#6553	1	2	yes	complemented
242	C₂×C₃	C₂⁴×D₇	1	1	yes	characteristic
243	D₃	—	2	6	no	—
244	D₃	—	14	6	no	—
245	C₂²	—	3	6	no	—
246	C₂²	—	6	6	no	—
247	C₂²	—	7	6	no	—
248	C₂²	—	14	6	no	—
249	C₂²	—	21	6	no	—
250	C₂²	—	42	6	no	—
251	C₂²	—	84	24	no	—
252	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
253	C₄	—	3	2	no	—
254	C₄	—	7	2	no	—
255	C₄	—	21	2	no	—
256	C₄	C₂²×D₃×D₇	1	6	yes	—
257	C₃	C₂×#6553	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	6	6	no	—
259	C₂	—	14	6	no	—
260	C₂	—	42	6	no	—
261	C₂	#10621582	1	2	yes	direct product factor
262	C₂	C₂³×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂⁵
Frattini	C₂	C₂³×D₃×D₇
Fitting	C₂×C₃×C₇×Q₈	C₂²
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×Q₈	D₃×D₇
hypocenter	C₃×C₇	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#66732	C₂	semidirect
2. C₂×C₇×#239	C₂	semidirect
3. C₂×C₃×#1161	C₂	semidirect
4. #10621582	C₂	direct (N × H)
5. C₂×#66681	C₂	semidirect
6. C₂×D₃×D₂₈	C₂	semidirect
7. C₂×D₇×D₁₂	C₂	semidirect
8. C₂×#66675	C₂	semidirect
10. C₂²×D₃×D₇	C₂²	semidirect
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
12. #66732	C₂²	semidirect
13. C₂×D₈₄	C₂²	semidirect
14. C₂×C₄×D₂₁	C₂²	semidirect
15. C₇×#239	C₂²	semidirect
16. C₂×C₇×D₁₂	C₂²	semidirect
17. C₂×C₄×D₃×C₇	C₂²	semidirect
18. C₃×#1161	C₂²	semidirect
19. C₂×C₃×D₂₈	C₂²	semidirect
20. C₂×C₃×C₄×D₇	C₂²	semidirect
21. C₂×#4248	C₂²	semidirect
22. C₂×#4247	C₂²	semidirect
23. C₂×#4246	C₂²	semidirect
24. C₂×#4245	C₂²	semidirect
25. #66681	C₂²	semidirect
26. D₃×D₂₈	C₂²	semidirect
27. D₇×D₁₂	C₂²	semidirect
28. #66675	C₂²	semidirect
32. C₂×#1161	D₃	semidirect
42. C₂×D₃×D₇	C₂³	semidirect
43. C₃×C₇×Q₈	C₂³	semidirect
46. D₈₄	C₂³	semidirect
47. C₄×D₂₁	C₂³	semidirect
49. C₇×D₁₂	C₂³	semidirect
50. C₄×D₃×C₇	C₂³	semidirect
52. C₃×D₂₈	C₂³	semidirect
53. C₃×C₄×D₇	C₂³	semidirect
54. #4248	C₂³	semidirect
55. #4247	C₂³	semidirect
56. #4246	C₂³	semidirect
57. #4245	C₂³	semidirect
60. C₂×C₇×Q₈	C₂×D₃	semidirect
66. #1161	C₂×D₃	semidirect
73. C₂×D₂₈	C₂×D₃	semidirect
75. C₂×C₄×D₇	C₂×D₃	semidirect
80. C₂×#239	D₇	semidirect
103. C₇×Q₈	C₂²×D₃	semidirect
114. D₂₈	C₂²×D₃	semidirect
116. C₄×D₇	C₂²×D₃	semidirect
120. C₂×C₃×Q₈	C₂×D₇	semidirect
126. #239	C₂×D₇	semidirect
133. C₂×D₁₂	C₂×D₇	semidirect
135. C₂×C₄×D₃	C₂×D₇	semidirect
137. C₂×C₃×C₇	H(5,𝔽₂)	semidirect
167. C₃×Q₈	C₂²×D₇	semidirect
178. D₁₂	C₂²×D₇	semidirect
180. C₄×D₃	C₂²×D₇	semidirect
182. C₃×C₇	C₂×H(5,𝔽₂)	semidirect
189. C₂×Q₈	D₃×D₇	semidirect
203. C₂×C₇	#1009	semidirect
225. Q₈	C₂×D₃×D₇	semidirect
239. C₇	C₂×#1009	semidirect
241. C₂×C₃	#6553	semidirect
257. C₃	C₂×#6553	semidirect
261. C₂	#10621582	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×#66732	2
C₂×C₇×#239	2
C₂×C₃×#1161	2
#10621582	2
C₂×#66681	2
C₂×D₃×D₂₈	2
C₂×D₇×D₁₂	2
C₂×#66675	2
C₂×#6553	3
C₂×#1009	7

Subgroups — C2×#10621582 #13062258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10621582 #13062258