Subgroups — C₂×#10620854 #13059426

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66559	C₂	1	2	yes	maximal, complemented
5	#10620854	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₂₁⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×D₇×M₁₆	—	3	1	no	maximal, Hall
8	C₂³×D₂₁	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₁	C₄	1	1	yes	characteristic
11	C₂×C₄×D₂₁	C₄	1	1	yes	characteristic
12	C₂×C₄×D₂₁	C₄	1	4	yes	—
13	C₂×C₄×D₂₁	C₄	1	1	yes	characteristic
14	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁	C₂²	1	4	yes	—
16	C₂×C₄×D₂₁	C₂²	1	1	yes	characteristic
17	C₇×#208	C₂²	1	4	yes	complemented
18	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
19	C₃×#1136	C₂²	1	4	yes	complemented
20	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
21	#66559	C₂²	1	8	yes	complemented
22	D₂₁⋊C₈	C₂²	1	8	yes	complemented
23	C₂²×C₄×D₇	—	3	1	no	—
24	C₂×C₇×M₁₆	—	3	1	no	—
25	C₂×#1136	D₃	1	1	yes	characteristic, complemented
26	D₇×M₁₆	—	3	8	no	—
27	C₂×#1131	—	3	2	no	—
28	C₂×C₈×D₇	—	3	2	no	—
29	C₂×D₃×M₁₆	—	7	1	no	maximal, Hall
30	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₂₁	—	2	4	no	—
32	C₂²×D₂₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₂₁	C₂×C₄	1	4	yes	—
34	C₂²×D₂₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
39	C₂×Dic₂₁	C₂×C₄	1	4	yes	—
40	C₂×Dic₂₁	C₂×C₄	1	1	yes	characteristic
41	C₂×Dic₂₁	C₂×C₄	1	1	yes	characteristic
42	C₄×D₂₁	C₂×C₄	1	4	yes	—
43	C₄×D₂₁	C₂×C₄	1	2	yes	—
44	C₄×D₂₁	C₂×C₄	1	2	yes	—
45	C₄×D₂₁	C₂×C₄	1	4	yes	—
46	C₄×D₂₁	C₂³	1	2	yes	—
47	C₄×D₂₁	C₂³	1	2	yes	—
48	C₃×C₇⋊C₈	C₂³	1	4	yes	complemented
49	C₇×C₃⋊C₈	C₂³	1	4	yes	complemented
50	C₂³×D₇	—	3	1	no	—
51	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
52	C₂²×Dic₇	—	3	1	no	—
53	C₂×C₄×D₇	—	3	1	no	—
54	C₂×C₄×D₇	—	3	4	no	—
55	C₂×C₄×D₇	—	3	1	no	—
56	C₂×C₄×D₇	—	3	1	no	—
57	C₂×C₄×D₇	—	3	1	no	—
58	C₂×C₄×D₇	—	3	4	no	—
59	C₇×M₁₆	—	3	4	no	—
60	C₂×C₇×C₈	—	3	2	no	—
61	#1136	C₂×D₃	1	4	yes	complemented
62	C₂×C₇⋊C₈	C₂×D₃	1	2	yes	complemented
63	#1131	—	3	8	no	—
64	C₈×D₇	—	3	8	no	—
65	C₂²×C₄×D₃	—	7	1	no	—
66	C₂×C₃×M₁₆	—	7	1	no	—
67	C₂×#208	D₇	1	1	yes	characteristic, complemented
68	D₃×M₁₆	—	7	8	no	—
69	C₂×ΓL(1,𝔽₂₅)	—	7	2	no	—
70	C₂×D₃×C₈	—	7	2	no	—
71	C₂²×C₃×C₇	—	2	2	no	—
72	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×D₂₁	—	2	2	no	—
75	C₂×D₂₁	—	2	8	no	—
76	C₂×D₂₁	M₁₆	1	4	yes	complemented
77	C₂×D₂₁	C₂²×C₄	1	2	yes	—
78	C₂×D₂₁	C₂²×C₄	1	2	yes	—
79	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
81	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
82	Dic₂₁	C₂²×C₄	1	2	yes	—
83	Dic₂₁	C₂²×C₄	1	2	yes	—
84	C₂²×M₁₆	—	21	1	no	Sylow
85	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
86	C₂²×D₇	—	3	1	no	—
87	C₂²×D₇	—	3	4	no	—
88	C₂²×D₇	—	3	1	no	—
89	C₂²×D₇	—	6	4	no	—
90	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
91	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
93	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₂×Dic₇	—	3	4	no	—
95	C₂×Dic₇	—	3	1	no	—
96	C₂×Dic₇	—	3	1	no	—
97	C₄×D₇	—	3	2	no	—
98	C₄×D₇	—	3	4	no	—
99	C₄×D₇	—	3	2	no	—
100	C₄×D₇	—	3	4	no	—
101	C₄×D₇	—	3	2	no	—
102	C₄×D₇	—	3	2	no	—
103	C₇×C₈	—	3	4	no	—
104	C₇⋊C₈	C₂²×D₃	1	4	yes	complemented
105	C₂³×D₃	—	7	1	no	—
106	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
107	C₂²×Dic₃	—	7	1	no	—
108	C₂×C₄×D₃	—	7	1	no	—
109	C₂×C₄×D₃	—	7	4	no	—
110	C₂×C₄×D₃	—	7	1	no	—
111	C₂×C₄×D₃	—	7	1	no	—
112	C₂×C₄×D₃	—	7	1	no	—
113	C₂×C₄×D₃	—	7	4	no	—
114	C₃×M₁₆	—	7	4	no	—
115	C₂×C₃×C₈	—	7	2	no	—
116	#208	C₂×D₇	1	4	yes	complemented
117	C₂×C₃⋊C₈	C₂×D₇	1	2	yes	complemented
118	ΓL(1,𝔽₂₅)	—	7	8	no	—
119	D₃×C₈	—	7	8	no	—
120	C₂×C₃×C₇	—	2	2	no	—
121	C₂×C₃×C₇	C₂×M₁₆	1	2	yes	complemented
122	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
123	D₂₁	—	2	2	no	—
124	D₂₁	C₂×M₁₆	1	4	yes	complemented
125	C₂³×C₄	—	21	1	no	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂×M₁₆	—	7	1	no	—
128	C₂×M₁₆	—	21	2	no	—
129	C₂×M₁₆	—	21	8	no	—
130	C₂²×C₈	—	21	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
133	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×D₇	—	3	4	no	—
135	C₂×D₇	—	3	2	no	—
136	C₂×D₇	—	3	2	no	—
137	C₂×D₇	—	6	8	no	—
138	C₂×D₇	—	6	2	no	—
139	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
140	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
141	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
142	Dic₇	—	3	2	no	—
143	Dic₇	—	3	2	no	—
144	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
145	C₂²×D₃	—	7	1	no	—
146	C₂²×D₃	—	7	4	no	—
147	C₂²×D₃	—	7	1	no	—
148	C₂²×D₃	—	14	4	no	—
149	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
150	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
152	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
153	C₂×Dic₃	—	7	4	no	—
154	C₂×Dic₃	—	7	1	no	—
155	C₂×Dic₃	—	7	1	no	—
156	C₄×D₃	—	7	2	no	—
157	C₄×D₃	—	7	4	no	—
158	C₄×D₃	—	7	2	no	—
159	C₄×D₃	—	7	4	no	—
160	C₄×D₃	—	7	2	no	—
161	C₄×D₃	—	7	2	no	—
162	C₃×C₈	—	7	4	no	—
163	C₃⋊C₈	C₂²×D₇	1	4	yes	complemented
164	C₃×C₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Hall
165	C₂⁴	—	21	1	no	—
166	C₂²×C₄	—	21	1	no	—
167	C₂²×C₄	—	21	1	no	—
168	C₂²×C₄	—	21	4	no	—
169	C₂²×C₄	—	21	1	no	—
170	C₂²×C₄	—	21	4	no	—
171	C₂²×C₄	—	21	1	no	—
172	C₂²×C₄	—	21	1	no	—
173	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
174	M₁₆	—	3	4	no	—
175	M₁₆	—	7	4	no	—
176	M₁₆	—	21	8	no	—
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	7	2	no	—
179	C₂×C₈	—	21	8	no	—
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	D₃×M₁₆	1	2	yes	complemented
182	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
183	D₇	—	3	4	no	—
184	D₇	—	6	2	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
187	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
188	C₂×D₃	—	7	4	no	—
189	C₂×D₃	—	7	2	no	—
190	C₂×D₃	—	7	2	no	—
191	C₂×D₃	—	14	8	no	—
192	C₂×D₃	—	14	2	no	—
193	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
194	C₃×C₄	C₂×C₄×D₇	1	1	yes	characteristic
195	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
196	Dic₃	—	7	2	no	—
197	Dic₃	—	7	2	no	—
198	C₂³	—	21	1	no	—
199	C₂³	—	21	4	no	—
200	C₂³	—	21	1	no	—
201	C₂³	—	42	4	no	—
202	C₂³	D₂₁⋊C₄	1	1	yes	characteristic
203	C₂×C₄	—	21	4	no	—
204	C₂×C₄	—	21	1	no	—
205	C₂×C₄	—	21	1	no	—
206	C₂×C₄	—	21	4	no	—
207	C₂×C₄	—	21	4	no	—
208	C₂×C₄	—	21	2	no	—
209	C₂×C₄	—	21	2	no	—
210	C₂×C₄	—	21	2	no	—
211	C₂×C₄	—	21	2	no	—
212	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
213	C₂×C₄	D₂₁⋊C₄	1	2	yes	—
214	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
215	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
216	C₈	—	3	4	no	—
217	C₈	—	7	4	no	—
218	C₇	C₂×D₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	D₇×M₁₆	1	2	yes	complemented
221	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
222	D₃	—	7	4	no	—
223	D₃	—	14	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	21	2	no	—
226	C₂²	—	21	4	no	—
227	C₂²	—	21	2	no	—
228	C₂²	—	42	8	no	—
229	C₂²	—	42	2	no	—
230	C₂²	C₂×D₂₁⋊C₄	1	2	yes	—
231	C₂²	C₂×D₂₁⋊C₄	1	1	yes	characteristic
232	C₄	—	21	2	no	—
233	C₄	—	21	2	no	—
234	C₄	C₂×D₂₁⋊C₄	1	2	yes	—
235	C₄	C₂×D₂₁⋊C₄	1	1	yes	characteristic
236	C₄	C₂²×D₃×D₇	1	1	yes	characteristic
237	C₃	C₂×D₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	21	4	no	—
240	C₂	—	42	2	no	—
241	C₂	#10620854	1	2	yes	direct product factor
242	C₂	C₂²×D₂₁⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₄	C₂²×D₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂²
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×D₂₁	C₄
cosocle	C₃×C₄×C₇	C₂⁴
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂²×M₁₆
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#208	C₂	semidirect
3. C₂×C₃×#1136	C₂	semidirect
4. C₂×#66559	C₂	semidirect
5. #10620854	C₂	direct (N × H)
6. C₂×D₂₁⋊C₈	C₂	semidirect
17. C₇×#208	C₂²	semidirect
18. C₂×C₇×C₃⋊C₈	C₂²	semidirect
19. C₃×#1136	C₂²	semidirect
20. C₂×C₃×C₇⋊C₈	C₂²	semidirect
21. #66559	C₂²	semidirect
22. D₂₁⋊C₈	C₂²	semidirect
25. C₂×#1136	D₃	semidirect
48. C₃×C₇⋊C₈	C₂³	semidirect
49. C₇×C₃⋊C₈	C₂³	semidirect
61. #1136	C₂×D₃	semidirect
62. C₂×C₇⋊C₈	C₂×D₃	semidirect
67. C₂×#208	D₇	semidirect
76. C₂×D₂₁	M₁₆	semidirect
104. C₇⋊C₈	C₂²×D₃	semidirect
116. #208	C₂×D₇	semidirect
117. C₂×C₃⋊C₈	C₂×D₇	semidirect
121. C₂×C₃×C₇	C₂×M₁₆	semidirect
124. D₂₁	C₂×M₁₆	semidirect
163. C₃⋊C₈	C₂²×D₇	semidirect
164. C₃×C₇	C₂²×M₁₆	semidirect
181. C₂×C₇	D₃×M₁₆	semidirect
218. C₇	C₂×D₃×M₁₆	semidirect
220. C₂×C₃	D₇×M₁₆	semidirect
237. C₃	C₂×D₇×M₁₆	semidirect
241. C₂	#10620854	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×C₄×D₂₁	2
C₂×C₇×#208	2
C₂×C₃×#1136	2
C₂×#66559	2
#10620854	2
C₂×D₂₁⋊C₈	2
C₂×D₇×M₁₆	3
C₂×D₃×M₁₆	7

Subgroups — C2×#10620854 #13059426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10620854 #13059426