Subgroups — C₂×#10621364 #13061576

Order 1344.

852 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4268	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#66631	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#66629	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#10621364	C₂	1	8	yes	maximal, direct product factor
9	C₂×(C₂²⋊D₇)⋊C₄	—	3	1	no	maximal, Hall
10	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
11	C₂³×D₂₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×#4268	C₄	1	8	yes	complemented
15	C₂×#4268	C₂²	1	4	yes	—
16	C₂²×Dic₂₁	—	2	4	no	—
17	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
20	C₂²×Dic₂₁	C₂²	1	4	yes	complemented
21	C₂×C₄×D₂₁	—	2	4	no	—
22	C₂×C₄×D₂₁	C₂²	1	4	yes	complemented
23	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	complemented
24	#66631	C₂²	1	4	yes	complemented
25	#66629	C₂²	1	4	yes	complemented
26	C₄×Dic₂₁	C₂²	1	4	yes	complemented
27	C₂²×C₂²⋊D₇	—	3	1	no	—
28	C₂³×Dic₇	—	3	1	no	—
29	C₂²×C₄×D₇	—	3	1	no	—
30	C₂×C₇×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
31	C₂×(C₂×D₇)⋊C₄	—	3	1	no	—
32	C₂×Dic₇⋊C₄	—	3	1	no	—
33	C₂×C₄×Dic₇	—	3	1	no	—
34	(C₂²⋊D₇)⋊C₄	—	3	8	no	—
35	C₂×#67⋊C₄	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	—	2	2	no	—
37	C₂³×C₃×C₇	—	2	2	no	—
38	C₂³×C₃×C₇	C₂×C₄	1	4	yes	complemented
39	C₂³×C₃×C₇	C₂³	1	2	yes	—
40	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
41	C₂²×D₂₁	—	2	2	no	—
42	C₂²×D₂₁	—	2	2	no	—
43	C₂²×D₂₁	C₂×C₄	1	4	yes	complemented
44	C₂²×D₂₁	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₇	—	2	2	no	—
46	C₂×C₃×C₄×C₇	—	2	2	no	—
47	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	#4268	C₂×C₄	1	16	yes	complemented
50	C₂×Dic₂₁	—	2	2	no	—
51	C₂×Dic₂₁	—	2	8	no	—
52	C₂×Dic₂₁	C₂×C₄	1	4	yes	complemented
53	C₂×Dic₂₁	D₄	1	4	yes	complemented
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	C₄×D₂₁	—	2	8	no	—
58	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
59	C₂³×D₇	—	3	1	no	—
60	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
62	C₂×C₂²⋊D₇	—	3	8	no	—
63	C₂×C₂²⋊D₇	—	3	4	no	—
64	C₂²×Dic₇	—	3	1	no	—
65	C₂²×Dic₇	—	3	1	no	—
66	C₂²×Dic₇	—	3	4	no	—
67	C₂²×Dic₇	—	3	1	no	—
68	C₂²×Dic₇	—	6	4	no	—
69	C₂×C₄×D₇	—	3	4	no	—
70	C₂×C₄×D₇	—	6	4	no	—
71	C₇×C₂²⋊C₄	C₂×D₃	1	4	yes	complemented
72	(C₂×D₇)⋊C₄	—	3	4	no	—
73	Dic₇⋊C₄	—	3	4	no	—
74	C₄×Dic₇	—	3	4	no	—
75	C₂²×#67	—	7	1	no	—
76	C₂³×Dic₃	—	7	1	no	—
77	C₂²×C₄×D₃	—	7	1	no	—
78	C₂×C₃×C₂²⋊C₄	D₇	1	1	yes	characteristic, complemented
79	C₂×(C₂×D₃)⋊C₄	—	7	1	no	—
80	C₂×Dic₃⋊C₄	—	7	1	no	—
81	C₂×C₄×Dic₃	—	7	1	no	—
82	#67⋊C₄	—	7	8	no	—
83	C₂²×C₃×C₇	—	2	8	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	—	2	2	no	—
86	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₃×C₇	C₂²×C₄	1	4	yes	complemented
88	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
89	C₂²×C₃×C₇	#41	1	2	yes	—
90	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
91	C₂×D₂₁	—	2	8	no	—
92	C₂×D₂₁	—	2	2	no	—
93	C₂×D₂₁	—	2	2	no	—
94	C₂×D₂₁	C₂²×C₄	1	4	yes	complemented
95	C₃×C₄×C₇	—	2	2	no	—
96	C₃×C₄×C₇	—	2	2	no	—
97	Dic₂₁	—	2	2	no	—
98	Dic₂₁	C₂²×C₄	1	4	yes	complemented
99	Dic₂₁	C₂×D₄	1	4	yes	complemented
100	C₂×C₄×D₄	—	21	1	no	Sylow
101	C₂³×C₇	—	2	2	no	—
102	C₂³×C₇	—	2	2	no	—
103	C₂³×C₇	C₄×D₃	1	4	yes	complemented
104	C₂³×C₇	C₂²×D₃	1	2	yes	—
105	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
106	C₂²×D₇	—	3	4	no	—
107	C₂²×D₇	—	3	2	no	—
108	C₂²×D₇	—	6	2	no	—
109	C₂²×D₇	—	6	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
113	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
114	C₂²⋊D₇	—	3	16	no	—
115	C₂×Dic₇	—	3	4	no	—
116	C₂×Dic₇	—	3	4	no	—
117	C₂×Dic₇	—	3	2	no	—
118	C₂×Dic₇	—	3	2	no	—
119	C₂×Dic₇	—	3	2	no	—
120	C₂×Dic₇	—	6	8	no	—
121	C₂×Dic₇	—	6	2	no	—
122	C₄×D₇	—	6	8	no	—
123	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
124	C₂³×D₃	—	7	1	no	—
125	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
127	C₂×#67	—	7	8	no	—
128	C₂×#67	—	7	4	no	—
129	C₂²×Dic₃	—	7	1	no	—
130	C₂²×Dic₃	—	7	1	no	—
131	C₂²×Dic₃	—	7	4	no	—
132	C₂²×Dic₃	—	7	1	no	—
133	C₂²×Dic₃	—	14	4	no	—
134	C₂×C₄×D₃	—	7	4	no	—
135	C₂×C₄×D₃	—	14	4	no	—
136	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	complemented
137	(C₂×D₃)⋊C₄	—	7	4	no	—
138	Dic₃⋊C₄	—	7	4	no	—
139	C₄×Dic₃	—	7	4	no	—
140	C₂×C₃×C₇	—	2	4	no	—
141	C₂×C₃×C₇	C₄×D₄	1	4	yes	complemented
142	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
145	D₂₁	—	2	4	no	—
146	C₂²×D₄	—	21	1	no	—
147	C₂³×C₄	—	21	1	no	—
148	C₂³×C₄	—	21	1	no	—
149	C₄×D₄	—	21	8	no	—
150	C₂×C₄⋊C₄	—	21	1	no	—
151	C₂×C₂²⋊C₄	—	21	1	no	—
152	C₂×C₂²⋊C₄	D₂₁	1	1	yes	characteristic, complemented
153	C₂×C₄²	—	21	1	no	—
154	C₂²×C₇	—	2	8	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
158	C₂²×C₇	C₂×C₄×D₃	1	4	yes	complemented
159	C₂²×C₇	D₃×D₄	1	2	yes	—
160	C₂²×C₇	#237	1	2	yes	—
161	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
162	C₂×D₇	—	3	4	no	—
163	C₂×D₇	—	6	8	no	—
164	C₂×D₇	—	6	2	no	—
165	C₂×D₇	—	6	2	no	—
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	2	no	—
168	Dic₇	—	3	4	no	—
169	Dic₇	—	3	4	no	—
170	Dic₇	—	6	2	no	—
171	C₂³×C₃	—	2	2	no	—
172	C₂³×C₃	—	2	2	no	—
173	C₂³×C₃	C₄×D₇	1	4	yes	complemented
174	C₂³×C₃	C₂²×D₇	1	2	yes	—
175	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
176	C₂²×D₃	—	7	4	no	—
177	C₂²×D₃	—	7	2	no	—
178	C₂²×D₃	—	14	2	no	—
179	C₂²×D₃	—	14	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
184	#67	—	7	16	no	—
185	C₂×Dic₃	—	7	4	no	—
186	C₂×Dic₃	—	7	4	no	—
187	C₂×Dic₃	—	7	2	no	—
188	C₂×Dic₃	—	7	2	no	—
189	C₂×Dic₃	—	7	2	no	—
190	C₂×Dic₃	—	14	8	no	—
191	C₂×Dic₃	—	14	2	no	—
192	C₄×D₃	—	14	8	no	—
193	C₃×C₇	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Hall
194	C₂⁴	—	21	1	no	—
195	C₂⁴	C₂×D₂₁	1	1	yes	characteristic
196	C₂×D₄	—	21	8	no	—
197	C₂×D₄	—	21	4	no	—
198	C₂²×C₄	—	21	1	no	—
199	C₂²×C₄	—	21	1	no	—
200	C₂²×C₄	—	21	4	no	—
201	C₂²×C₄	—	21	1	no	—
202	C₂²×C₄	—	21	4	no	—
203	C₂²×C₄	—	42	4	no	—
204	C₂²×C₄	—	42	4	no	—
205	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
207	C₄⋊C₄	—	21	4	no	—
208	C₂²⋊C₄	—	21	4	no	—
209	C₂²⋊C₄	C₂×D₂₁	1	4	yes	complemented
210	C₄²	—	21	4	no	—
211	C₂×C₇	—	2	4	no	—
212	C₂×C₇	#67⋊C₄	1	4	yes	complemented
213	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
214	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
215	C₂×C₇	C₂×#237	1	1	yes	characteristic
216	D₇	—	6	4	no	—
217	C₂²×C₃	—	2	8	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
221	C₂²×C₃	C₂×C₄×D₇	1	4	yes	complemented
222	C₂²×C₃	D₄×D₇	1	2	yes	—
223	C₂²×C₃	#1159	1	2	yes	—
224	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
225	C₂×D₃	—	7	4	no	—
226	C₂×D₃	—	14	8	no	—
227	C₂×D₃	—	14	2	no	—
228	C₂×D₃	—	14	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	Dic₃	—	7	4	no	—
232	Dic₃	—	7	4	no	—
233	Dic₃	—	14	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	21	4	no	—
237	C₂³	—	21	2	no	—
238	C₂³	—	42	2	no	—
239	C₂³	—	42	2	no	—
240	C₂³	C₄×D₂₁	1	4	yes	complemented
241	C₂³	C₂²×D₂₁	1	2	yes	—
242	C₂³	C₂²×D₂₁	1	1	yes	characteristic
243	D₄	—	21	16	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	21	4	no	—
247	C₂×C₄	—	21	4	no	—
248	C₂×C₄	—	21	2	no	—
249	C₂×C₄	—	21	2	no	—
250	C₂×C₄	—	21	2	no	—
251	C₂×C₄	—	42	8	no	—
252	C₂×C₄	—	42	8	no	—
253	C₂×C₄	—	42	2	no	—
254	C₂×C₄	C₂²×D₂₁	1	2	yes	—
255	C₂×C₄	C₂²×D₂₁	1	2	yes	—
256	C₇	C₂×#67⋊C₄	1	1	yes	characteristic, complemented, Sylow
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	(C₂²⋊D₇)⋊C₄	1	4	yes	complemented
259	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
260	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
261	C₂×C₃	C₂×#1159	1	1	yes	characteristic
262	D₃	—	14	4	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	21	4	no	—
267	C₂²	—	42	8	no	—
268	C₂²	—	42	2	no	—
269	C₂²	—	42	2	no	—
270	C₂²	C₂×C₄×D₂₁	1	2	yes	—
271	C₂²	C₂×C₄×D₂₁	1	4	yes	complemented
272	C₂²	D₄×D₂₁	1	2	yes	—
273	C₂²	#66730	1	2	yes	—
274	C₂²	C₂³×D₂₁	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	21	4	no	—
278	C₄	—	21	4	no	—
279	C₄	—	42	2	no	—
280	C₃	C₂×(C₂²⋊D₇)⋊C₄	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	2	4	no	—
282	C₂	—	42	4	no	—
283	C₂	#10621364	1	4	yes	direct product factor
284	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
285	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
286	C₂	C₂×#66730	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂³×D₂₁
Fitting	C₂×C₃×C₇×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂²×#4268	C₂
cosocle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₂²⋊C₄	D₂₁
hypocenter	C₃×C₇	C₂×C₄×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₂₁	C₂	semidirect
3. C₂²×C₄×D₂₁	C₂	semidirect
4. C₂×C₃×C₇×C₂²⋊C₄	C₂	semidirect
5. C₂×#66631	C₂	semidirect
6. C₂×#66629	C₂	semidirect
7. C₂×C₄×Dic₂₁	C₂	semidirect
8. #10621364	C₂	direct (N × H)
14. C₂×#4268	C₄	semidirect
20. C₂²×Dic₂₁	C₂²	semidirect
22. C₂×C₄×D₂₁	C₂²	semidirect
23. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
24. #66631	C₂²	semidirect
25. #66629	C₂²	semidirect
26. C₄×Dic₂₁	C₂²	semidirect
30. C₂×C₇×C₂²⋊C₄	D₃	semidirect
38. C₂³×C₃×C₇	C₂×C₄	semidirect
43. C₂²×D₂₁	C₂×C₄	semidirect
49. #4268	C₂×C₄	semidirect
52. C₂×Dic₂₁	C₂×C₄	semidirect
53. C₂×Dic₂₁	D₄	semidirect
71. C₇×C₂²⋊C₄	C₂×D₃	semidirect
78. C₂×C₃×C₂²⋊C₄	D₇	semidirect
87. C₂²×C₃×C₇	C₂²×C₄	semidirect
94. C₂×D₂₁	C₂²×C₄	semidirect
98. Dic₂₁	C₂²×C₄	semidirect
99. Dic₂₁	C₂×D₄	semidirect
103. C₂³×C₇	C₄×D₃	semidirect
136. C₃×C₂²⋊C₄	C₂×D₇	semidirect
141. C₂×C₃×C₇	C₄×D₄	semidirect
152. C₂×C₂²⋊C₄	D₂₁	semidirect
158. C₂²×C₇	C₂×C₄×D₃	semidirect
173. C₂³×C₃	C₄×D₇	semidirect
193. C₃×C₇	C₂×C₄×D₄	semidirect
209. C₂²⋊C₄	C₂×D₂₁	semidirect
212. C₂×C₇	#67⋊C₄	semidirect
221. C₂²×C₃	C₂×C₄×D₇	semidirect
240. C₂³	C₄×D₂₁	semidirect
256. C₇	C₂×#67⋊C₄	semidirect
258. C₂×C₃	(C₂²⋊D₇)⋊C₄	semidirect
271. C₂²	C₂×C₄×D₂₁	semidirect
280. C₃	C₂×(C₂²⋊D₇)⋊C₄	semidirect
283. C₂	#10621364	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×#4268	2
C₂³×Dic₂₁	2
C₂²×C₄×D₂₁	2
C₂×C₃×C₇×C₂²⋊C₄	2
C₂×#66631	2
C₂×#66629	2
C₂×C₄×Dic₂₁	2
#10621364	2
C₂×(C₂²⋊D₇)⋊C₄	3
C₂×#67⋊C₄	7

Subgroups — C2×#10621364 #13061576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10621364 #13061576