Subgroups — C₂×C₇×#938 #13061518

Order 1344.

852 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#67	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#217	C₂	1	2	yes	maximal, complemented
5	C₇×#938	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#121	—	3	1	no	maximal, Hall
8	C₂⁴×C₃×C₇	C₂²	1	2	yes	—
9	C₂³×D₃×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	—	2	4	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×#67	—	2	8	no	—
14	C₂×C₇×#67	C₂²	1	8	yes	complemented
15	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃×C₇	—	2	4	no	—
18	C₂×C₄×D₃×C₇	C₂²	1	4	yes	complemented
19	C₇×#217	C₂²	1	8	yes	complemented
20	C₇×C₄⋊Dic₃	C₂²	1	4	yes	complemented
21	C₂²×C₇×D₄	—	3	2	no	—
22	C₂²×C₇×D₄	D₃	1	1	yes	characteristic, complemented
23	C₂³×C₄×C₇	—	3	1	no	—
24	C₇×#121	—	3	8	no	—
25	C₂×C₇×C₄⋊C₄	—	3	1	no	—
26	C₂×C₇×C₂²⋊C₄	—	3	2	no	—
27	C₂×#938	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂³×C₃×C₇	—	2	4	no	—
29	C₂³×C₃×C₇	—	2	4	no	—
30	C₂³×C₃×C₇	—	2	4	no	—
31	C₂³×C₃×C₇	C₂³	1	4	yes	—
32	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
33	C₂²×D₃×C₇	—	2	2	no	—
34	C₂²×D₃×C₇	—	2	2	no	—
35	C₂²×D₃×C₇	D₄	1	4	yes	complemented
36	C₂²×D₃×C₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	—	2	8	no	—
38	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₇×#67	—	2	16	no	—
41	C₂×C₇×Dic₃	—	2	2	no	—
42	C₂×C₇×Dic₃	—	2	4	no	—
43	C₂×C₇×Dic₃	C₂³	1	4	yes	—
44	C₂×C₇×Dic₃	C₂³	1	2	yes	—
45	C₄×D₃×C₇	—	2	8	no	—
46	C₂⁴×C₇	—	3	1	no	—
47	C₂⁴×C₇	C₂×D₃	1	2	yes	—
48	C₂×C₇×D₄	—	2	4	no	—
49	C₂×C₇×D₄	—	3	8	no	—
50	C₂×C₇×D₄	—	6	8	no	—
51	C₂×C₇×D₄	C₂×D₃	1	4	yes	complemented
52	C₂²×C₄×C₇	—	3	1	no	—
53	C₂²×C₄×C₇	—	3	2	no	—
54	C₂²×C₄×C₇	—	3	4	no	—
55	C₂²×C₄×C₇	—	6	4	no	—
56	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
57	C₇×C₄⋊C₄	—	3	4	no	—
58	C₇×C₂²⋊C₄	—	3	8	no	—
59	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂²×#67	C₂×C₇	1	2	yes	complemented
61	C₂²×C₄×D₃	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂×#217	C₂×C₇	1	2	yes	complemented
63	#938	C₂×C₇	1	8	yes	direct product factor
64	C₂×C₄⋊Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂²×C₃×C₇	—	2	4	no	—
66	C₂²×C₃×C₇	—	2	4	no	—
67	C₂²×C₃×C₇	—	2	4	no	—
68	C₂²×C₃×C₇	—	4	8	no	—
69	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
70	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₇	#41	1	2	yes	—
72	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
73	C₂×D₃×C₇	—	2	8	no	—
74	C₂×D₃×C₇	—	2	2	no	—
75	C₂×D₃×C₇	—	2	2	no	—
76	C₂×D₃×C₇	C₂×D₄	1	4	yes	complemented
77	C₃×C₄×C₇	C₂×D₄	1	4	yes	complemented
78	C₇×Dic₃	—	2	2	no	—
79	C₇×Dic₃	—	2	4	no	—
80	C₂×#121	—	3	1	no	Sylow
81	C₂³×C₇	—	2	4	no	—
82	C₂³×C₇	—	2	4	no	—
83	C₂³×C₇	—	2	4	no	—
84	C₂³×C₇	—	3	4	no	—
85	C₂³×C₇	—	3	2	no	—
86	C₂³×C₇	—	6	2	no	—
87	C₂³×C₇	—	6	2	no	—
88	C₂³×C₇	C₂²×D₃	1	4	yes	—
89	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
90	C₇×D₄	—	2	8	no	—
91	C₇×D₄	—	6	16	no	—
92	C₂×C₄×C₇	—	3	2	no	—
93	C₂×C₄×C₇	—	3	4	no	—
94	C₂×C₄×C₇	—	6	4	no	—
95	C₂×C₄×C₇	—	6	2	no	—
96	C₂×C₄×C₇	—	6	8	no	—
97	C₂×C₄×C₇	#67	1	4	yes	complemented
98	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
99	C₂⁴×C₃	C₂²×C₇	1	2	yes	—
100	C₂³×D₃	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₃×D₄	—	2	4	no	—
102	C₂×C₃×D₄	C₂²×C₇	1	4	yes	complemented
103	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
104	C₂×#67	—	2	8	no	—
105	C₂×#67	C₂²×C₇	1	8	yes	complemented
106	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
107	C₂²×Dic₃	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄×D₃	—	2	4	no	—
109	C₂×C₄×D₃	C₂²×C₇	1	4	yes	complemented
110	#217	C₂²×C₇	1	8	yes	complemented
111	C₄⋊Dic₃	C₂²×C₇	1	4	yes	complemented
112	C₂×C₃×C₇	—	4	4	no	—
113	C₂×C₃×C₇	#121	1	4	yes	complemented
114	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
117	D₃×C₇	—	2	4	no	—
118	C₂²×D₄	—	3	2	no	—
119	C₂²×D₄	D₃×C₇	1	1	yes	characteristic, complemented
120	C₂³×C₄	—	3	1	no	—
121	#121	—	3	8	no	—
122	C₂×C₄⋊C₄	—	3	1	no	—
123	C₂×C₂²⋊C₄	—	3	2	no	—
124	C₂²×C₇	—	2	4	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	2	4	no	—
127	C₂²×C₇	—	3	4	no	—
128	C₂²×C₇	—	4	8	no	—
129	C₂²×C₇	—	6	8	no	—
130	C₂²×C₇	—	6	2	no	—
131	C₂²×C₇	—	6	2	no	—
132	C₂²×C₇	D₃×D₄	1	2	yes	—
133	C₂²×C₇	#237	1	2	yes	—
134	C₂²×C₇	C₂×#67	1	2	yes	—
135	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
136	C₄×C₇	—	6	4	no	—
137	C₄×C₇	—	6	2	no	—
138	C₄×C₇	C₂×#67	1	4	yes	complemented
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	C₂³×C₇	1	4	yes	—
143	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
144	C₂²×D₃	—	2	2	no	—
145	C₂²×D₃	—	2	2	no	—
146	C₂²×D₃	C₇×D₄	1	4	yes	complemented
147	C₂²×D₃	C₂³×C₇	1	2	yes	—
148	C₃×D₄	—	2	8	no	—
149	C₂×C₃×C₄	C₇×D₄	1	4	yes	complemented
150	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
151	#67	—	2	16	no	—
152	C₂×Dic₃	—	2	2	no	—
153	C₂×Dic₃	—	2	4	no	—
154	C₂×Dic₃	C₂³×C₇	1	4	yes	—
155	C₂×Dic₃	C₂³×C₇	1	2	yes	—
156	C₄×D₃	—	2	8	no	—
157	C₃×C₇	C₂×#121	1	1	yes	characteristic, complemented, Hall
158	C₂⁴	—	3	1	no	—
159	C₂⁴	C₂×D₃×C₇	1	2	yes	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	3	8	no	—
162	C₂×D₄	—	6	8	no	—
163	C₂×D₄	C₂×D₃×C₇	1	4	yes	complemented
164	C₂²×C₄	—	3	1	no	—
165	C₂²×C₄	—	3	2	no	—
166	C₂²×C₄	—	3	4	no	—
167	C₂²×C₄	—	6	4	no	—
168	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
169	C₄⋊C₄	—	3	4	no	—
170	C₂²⋊C₄	—	3	8	no	—
171	C₂×C₇	—	4	4	no	—
172	C₂×C₇	—	6	4	no	—
173	C₂×C₇	#938	1	4	yes	direct product factor
174	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
175	C₂×C₇	C₂×#237	1	1	yes	characteristic
176	C₂×C₇	C₂²×#67	1	1	yes	characteristic
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	4	8	no	—
181	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
182	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
183	C₂²×C₃	C₇×#41	1	2	yes	—
184	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
185	C₂×D₃	—	2	8	no	—
186	C₂×D₃	—	2	2	no	—
187	C₂×D₃	—	2	2	no	—
188	C₂×D₃	C₂×C₇×D₄	1	4	yes	complemented
189	C₃×C₄	C₂×C₇×D₄	1	4	yes	complemented
190	Dic₃	—	2	2	no	—
191	Dic₃	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	3	4	no	—
196	C₂³	—	3	2	no	—
197	C₂³	—	6	2	no	—
198	C₂³	—	6	2	no	—
199	C₂³	C₂²×D₃×C₇	1	4	yes	—
200	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	D₄	—	6	16	no	—
203	C₂×C₄	—	3	2	no	—
204	C₂×C₄	—	3	4	no	—
205	C₂×C₄	—	6	4	no	—
206	C₂×C₄	—	6	2	no	—
207	C₂×C₄	—	6	8	no	—
208	C₂×C₄	C₇×#67	1	4	yes	complemented
209	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
210	C₇	C₂×#938	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×C₃	—	4	4	no	—
212	C₂×C₃	C₇×#121	1	4	yes	complemented
213	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
214	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
215	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
216	D₃	—	2	4	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	3	4	no	—
221	C₂²	—	4	8	no	—
222	C₂²	—	6	8	no	—
223	C₂²	—	6	2	no	—
224	C₂²	—	6	2	no	—
225	C₂²	D₃×C₇×D₄	1	2	yes	—
226	C₂²	C₇×#237	1	2	yes	—
227	C₂²	C₂×C₇×#67	1	2	yes	—
228	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
229	C₄	—	6	4	no	—
230	C₄	—	6	2	no	—
231	C₄	C₂×C₇×#67	1	4	yes	complemented
232	C₃	C₂×C₇×#121	1	1	yes	characteristic, complemented, Sylow
233	C₂	—	4	4	no	—
234	C₂	—	6	4	no	—
235	C₂	C₇×#938	1	4	yes	direct product factor
236	C₂	C₂×D₃×C₇×D₄	1	1	yes	characteristic
237	C₂	C₂×C₇×#237	1	1	yes	characteristic
238	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂³×C₃×C₇	C₂³
hypercenter	C₂²×C₇×D₄	D₃
hypocenter	C₃	C₂×C₇×#121
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂²×C₇×#67	C₂	semidirect
3. C₂²×C₄×D₃×C₇	C₂	semidirect
4. C₂×C₇×#217	C₂	semidirect
5. C₇×#938	C₂	direct (N × H)
6. C₂×C₇×C₄⋊Dic₃	C₂	semidirect
11. C₂×C₃×C₇×D₄	C₂²	semidirect
14. C₂×C₇×#67	C₂²	semidirect
18. C₂×C₄×D₃×C₇	C₂²	semidirect
19. C₇×#217	C₂²	semidirect
20. C₇×C₄⋊Dic₃	C₂²	semidirect
22. C₂²×C₇×D₄	D₃	semidirect
27. C₂×#938	C₇	direct (N × H)
35. C₂²×D₃×C₇	D₄	semidirect
38. C₂×C₃×C₄×C₇	D₄	semidirect
51. C₂×C₇×D₄	C₂×D₃	semidirect
59. C₂²×C₃×D₄	C₂×C₇	semidirect
60. C₂²×#67	C₂×C₇	semidirect
61. C₂²×C₄×D₃	C₂×C₇	semidirect
62. C₂×#217	C₂×C₇	semidirect
63. #938	C₂×C₇	direct (N × H)
64. C₂×C₄⋊Dic₃	C₂×C₇	semidirect
76. C₂×D₃×C₇	C₂×D₄	semidirect
77. C₃×C₄×C₇	C₂×D₄	semidirect
97. C₂×C₄×C₇	#67	semidirect
102. C₂×C₃×D₄	C₂²×C₇	semidirect
105. C₂×#67	C₂²×C₇	semidirect
109. C₂×C₄×D₃	C₂²×C₇	semidirect
110. #217	C₂²×C₇	semidirect
111. C₄⋊Dic₃	C₂²×C₇	semidirect
113. C₂×C₃×C₇	#121	semidirect
119. C₂²×D₄	D₃×C₇	semidirect
138. C₄×C₇	C₂×#67	semidirect
146. C₂²×D₃	C₇×D₄	semidirect
149. C₂×C₃×C₄	C₇×D₄	semidirect
157. C₃×C₇	C₂×#121	semidirect
163. C₂×D₄	C₂×D₃×C₇	semidirect
173. C₂×C₇	#938	direct (N × H)
188. C₂×D₃	C₂×C₇×D₄	semidirect
189. C₃×C₄	C₂×C₇×D₄	semidirect
208. C₂×C₄	C₇×#67	semidirect
210. C₇	C₂×#938	direct (N × H)
212. C₂×C₃	C₇×#121	semidirect
231. C₄	C₂×C₇×#67	semidirect
232. C₃	C₂×C₇×#121	semidirect
235. C₂	C₇×#938	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×C₃×C₇×D₄	2
C₂²×C₇×#67	2
C₂²×C₄×D₃×C₇	2
C₂×C₇×#217	2
C₇×#938	2
C₂×C₇×C₄⋊Dic₃	2
C₂×C₇×#121	3
C₂×#938	7

Subgroups — C2×C7×#938 #13061518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#938 #13061518