Subgroups — C₂×C₇×#909 #13061463

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#909	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×Dic₁₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#204	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#137	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#67	—	2	1	no	—
10	C₂²×C₇×Dic₃	—	2	1	no	—
11	C₇×#235	—	2	2	no	—
12	C₇×#235	C₂²	1	4	yes	—
13	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₇	—	2	1	no	—
15	C₂×C₇×Dic₆	—	2	2	no	—
16	C₂×C₇×Dic₆	—	2	1	no	—
17	C₂×C₇×Dic₆	C₂²	1	2	yes	—
18	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
19	C₂×C₇×Dic₆	C₂²	1	4	yes	complemented
20	C₃×C₇×M₁₆	C₂²	1	4	yes	complemented
21	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
22	C₇×Dic₁₂	C₂²	1	8	yes	complemented
23	C₇×#204	C₂²	1	8	yes	complemented
24	C₂×C₇×#41	—	3	1	no	—
25	C₂²×C₇×Q₈	—	3	1	no	—
26	C₇×#137	—	3	8	no	—
27	C₂×C₇×Q₁₆	—	3	2	no	—
28	C₂×C₇×SD₁₆	—	3	2	no	—
29	C₂×C₇×M₁₆	D₃	1	1	yes	characteristic, complemented
30	C₂×#909	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
32	C₂²×D₃×C₇	—	2	1	no	—
33	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
37	C₇×#67	—	2	4	no	—
38	C₂×C₇×Dic₃	—	2	2	no	—
39	C₂×C₇×Dic₃	—	2	4	no	—
40	C₂×C₇×Dic₃	—	2	1	no	—
41	C₇×D₁₂	—	2	1	no	—
42	C₇×D₁₂	C₂³	1	2	yes	—
43	C₄×D₃×C₇	—	2	4	no	—
44	C₇×Dic₆	—	2	2	no	—
45	C₇×Dic₆	—	2	4	no	—
46	C₇×Dic₆	—	2	1	no	—
47	C₇×Dic₆	C₂³	1	4	yes	—
48	C₇×Dic₆	C₂³	1	2	yes	—
49	C₃×C₇×C₈	C₂³	1	4	yes	—
50	C₇×#41	—	3	4	no	—
51	C₇×#41	—	6	2	no	—
52	C₂×C₇×Q₈	—	3	2	no	—
53	C₂×C₇×Q₈	—	3	1	no	—
54	C₂×C₇×Q₈	—	3	4	no	—
55	C₂×C₇×Q₈	—	6	2	no	—
56	C₂×C₇×Q₈	—	6	1	no	—
57	C₂×C₇×D₄	—	3	1	no	—
58	C₂×C₇×D₄	—	6	1	no	—
59	C₂²×C₄×C₇	—	6	1	no	—
60	C₂²×C₄×C₇	—	6	1	no	—
61	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
62	C₇×Q₁₆	—	3	8	no	—
63	C₇×SD₁₆	—	3	8	no	—
64	C₇×M₁₆	C₂×D₃	1	4	yes	complemented
65	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
66	C₂×#235	C₂×C₇	1	1	yes	characteristic
67	C₂²×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
69	#909	C₂×C₇	1	8	yes	direct product factor
70	C₂×Dic₁₂	C₂×C₇	1	2	yes	complemented
71	C₂×#204	C₂×C₇	1	2	yes	complemented
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂×D₃×C₇	—	2	2	no	—
76	C₂×D₃×C₇	—	4	2	no	—
77	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
79	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
80	C₇×Dic₃	—	2	4	no	—
81	C₇×Dic₃	—	2	2	no	—
82	C₂×#137	—	3	1	no	Sylow
83	C₂³×C₇	—	6	1	no	—
84	C₂³×C₇	D₁₂	1	1	yes	characteristic
85	C₇×Q₈	—	3	4	no	—
86	C₇×Q₈	—	3	2	no	—
87	C₇×Q₈	—	6	4	no	—
88	C₇×Q₈	—	6	2	no	—
89	C₇×Q₈	—	6	1	no	—
90	C₇×D₄	—	3	2	no	—
91	C₇×D₄	—	6	4	no	—
92	C₇×D₄	—	6	1	no	—
93	C₂×C₄×C₇	—	6	4	no	—
94	C₂×C₄×C₇	—	6	1	no	—
95	C₂×C₄×C₇	—	6	2	no	—
96	C₂×C₄×C₇	—	6	4	no	—
97	C₂×C₄×C₇	D₁₂	1	2	yes	—
98	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
100	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
101	C₇×C₈	C₂²×D₃	1	4	yes	—
102	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
103	C₂×#67	—	2	1	no	—
104	C₂²×Dic₃	—	2	1	no	—
105	#235	—	2	2	no	—
106	#235	C₂²×C₇	1	4	yes	—
107	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄×D₃	—	2	1	no	—
109	C₂×Dic₆	—	2	2	no	—
110	C₂×Dic₆	—	2	1	no	—
111	C₂×Dic₆	C₂²×C₇	1	2	yes	—
112	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic
113	C₂×Dic₆	C₂²×C₇	1	4	yes	complemented
114	C₃×M₁₆	C₂²×C₇	1	4	yes	complemented
115	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
116	Dic₁₂	C₂²×C₇	1	8	yes	complemented
117	#204	C₂²×C₇	1	8	yes	complemented
118	C₂×C₃×C₇	—	2	2	no	—
119	C₂×C₃×C₇	#137	1	2	yes	complemented
120	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
121	D₃×C₇	—	4	2	no	—
122	C₂×#41	—	3	1	no	—
123	C₂²×Q₈	—	3	1	no	—
124	#137	—	3	8	no	—
125	C₂×Q₁₆	—	3	2	no	—
126	C₂×SD₁₆	—	3	2	no	—
127	C₂×M₁₆	D₃×C₇	1	1	yes	characteristic, complemented
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	6	2	no	—
130	C₂²×C₇	—	12	2	no	—
131	C₂²×C₇	C₂×D₁₂	1	2	yes	—
132	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
133	C₄×C₇	—	6	2	no	—
134	C₄×C₇	—	6	4	no	—
135	C₄×C₇	C₂×D₁₂	1	2	yes	—
136	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
137	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
138	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
139	C₂²×D₃	—	2	1	no	—
140	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
144	#67	—	2	4	no	—
145	C₂×Dic₃	—	2	2	no	—
146	C₂×Dic₃	—	2	4	no	—
147	C₂×Dic₃	—	2	1	no	—
148	D₁₂	—	2	1	no	—
149	D₁₂	C₂³×C₇	1	2	yes	—
150	C₄×D₃	—	2	4	no	—
151	Dic₆	—	2	2	no	—
152	Dic₆	—	2	4	no	—
153	Dic₆	—	2	1	no	—
154	Dic₆	C₂³×C₇	1	4	yes	—
155	Dic₆	C₂³×C₇	1	2	yes	—
156	C₃×C₈	C₂³×C₇	1	4	yes	—
157	C₃×C₇	C₂×#137	1	1	yes	characteristic, complemented, Hall
158	#41	—	3	4	no	—
159	#41	—	6	2	no	—
160	C₂×Q₈	—	3	2	no	—
161	C₂×Q₈	—	3	1	no	—
162	C₂×Q₈	—	3	4	no	—
163	C₂×Q₈	—	6	1	no	—
164	C₂×Q₈	—	6	2	no	—
165	C₂×D₄	—	3	1	no	—
166	C₂×D₄	—	6	1	no	—
167	C₂²×C₄	—	6	1	no	—
168	C₂²×C₄	—	6	1	no	—
169	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
170	Q₁₆	—	3	8	no	—
171	SD₁₆	—	3	8	no	—
172	M₁₆	C₂×D₃×C₇	1	4	yes	complemented
173	C₂×C₈	C₂×D₃×C₇	1	2	yes	—
174	C₂×C₇	—	2	2	no	—
175	C₂×C₇	—	12	2	no	—
176	C₂×C₇	#909	1	2	yes	direct product factor
177	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
180	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂×D₃	—	2	2	no	—
182	C₂×D₃	—	4	2	no	—
183	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
184	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
185	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
186	Dic₃	—	2	4	no	—
187	Dic₃	—	2	2	no	—
188	C₂³	—	6	1	no	—
189	C₂³	C₇×D₁₂	1	1	yes	characteristic
190	Q₈	—	3	4	no	—
191	Q₈	—	3	2	no	—
192	Q₈	—	6	4	no	—
193	Q₈	—	6	2	no	—
194	Q₈	—	6	1	no	—
195	D₄	—	3	2	no	—
196	D₄	—	6	4	no	—
197	D₄	—	6	1	no	—
198	C₂×C₄	—	6	1	no	—
199	C₂×C₄	—	6	4	no	—
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	—	6	4	no	—
202	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₇×D₁₂	1	2	yes	—
204	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
206	C₈	C₂²×D₃×C₇	1	4	yes	—
207	C₇	C₂×#909	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	C₇×#137	1	2	yes	complemented
210	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
211	D₃	—	4	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	6	2	no	—
214	C₂²	—	12	2	no	—
215	C₂²	C₂×C₇×D₁₂	1	2	yes	—
216	C₂²	C₂×C₇×D₁₂	1	1	yes	characteristic
217	C₄	—	6	2	no	—
218	C₄	—	6	4	no	—
219	C₄	C₂×C₇×D₁₂	1	2	yes	—
220	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
221	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
222	C₃	C₂×C₇×#137	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	12	2	no	—
225	C₂	C₇×#909	1	2	yes	direct product factor
226	C₂	C₂²×C₇×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×#235	C₂
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₂×C₇×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×Dic₆	C₂	semidirect
3. C₂×C₃×C₇×M₁₆	C₂	semidirect
4. C₇×#909	C₂	direct (N × H)
5. C₂×C₇×Dic₁₂	C₂	semidirect
6. C₂×C₇×#204	C₂	semidirect
19. C₂×C₇×Dic₆	C₂²	semidirect
20. C₃×C₇×M₁₆	C₂²	semidirect
22. C₇×Dic₁₂	C₂²	semidirect
23. C₇×#204	C₂²	semidirect
29. C₂×C₇×M₁₆	D₃	semidirect
30. C₂×#909	C₇	direct (N × H)
64. C₇×M₁₆	C₂×D₃	semidirect
67. C₂²×Dic₆	C₂×C₇	semidirect
68. C₂×C₃×M₁₆	C₂×C₇	semidirect
69. #909	C₂×C₇	direct (N × H)
70. C₂×Dic₁₂	C₂×C₇	semidirect
71. C₂×#204	C₂×C₇	semidirect
113. C₂×Dic₆	C₂²×C₇	semidirect
114. C₃×M₁₆	C₂²×C₇	semidirect
116. Dic₁₂	C₂²×C₇	semidirect
117. #204	C₂²×C₇	semidirect
119. C₂×C₃×C₇	#137	semidirect
127. C₂×M₁₆	D₃×C₇	semidirect
157. C₃×C₇	C₂×#137	semidirect
172. M₁₆	C₂×D₃×C₇	semidirect
176. C₂×C₇	#909	direct (N × H)
207. C₇	C₂×#909	direct (N × H)
209. C₂×C₃	C₇×#137	semidirect
222. C₃	C₂×C₇×#137	semidirect
225. C₂	C₇×#909	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×C₇×#235	2
C₂²×C₇×Dic₆	2
C₂×C₃×C₇×M₁₆	2
C₇×#909	2
C₂×C₇×Dic₁₂	2
C₂×C₇×#204	2
C₂×C₇×#137	3
C₂×#909	7

Subgroups — C2×C7×#909 #13061463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#909 #13061463