Subgroups — C₂×C₇×#805 #13055972

Order 1344.

340 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#805	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×#104	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₄×C₇	—	2	1	no	—
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×#67	—	2	1	no	—
9	C₇×#235	—	2	1	no	—
10	C₇×#235	—	2	1	no	—
11	C₇×#235	C₄	1	2	yes	complemented
12	C₇×#235	C₂²	1	2	yes	—
13	C₂×C₇×D₁₂	C₄	1	1	yes	characteristic, complemented
14	C₂×C₄×D₃×C₇	—	2	1	no	—
15	C₂×C₇×Dic₆	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄²×C₇	—	2	1	no	—
17	C₃×C₄²×C₇	C₂²	1	2	yes	complemented
18	C₇×#208	—	2	1	no	—
19	C₇×#208	C₂²	1	2	yes	complemented
20	C₂×C₇×C₃⋊C₈	—	2	1	no	—
21	C₂×C₇×#41	—	3	1	no	—
22	C₂×C₇×M₁₆	—	3	1	no	—
23	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
24	C₇×#104	—	3	4	no	—
25	C₂×#805	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
27	C₂²×D₃×C₇	—	2	1	no	—
28	C₂×C₃×C₄×C₇	—	2	4	no	—
29	C₂×C₃×C₄×C₇	—	2	2	no	—
30	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₇×#67	—	2	2	no	—
37	C₇×#67	—	2	2	no	—
38	C₂×C₇×Dic₃	—	2	1	no	—
39	C₇×D₁₂	—	2	1	no	—
40	C₇×D₁₂	C₂×C₄	1	2	yes	complemented
41	C₄×D₃×C₇	—	2	2	no	—
42	C₄×D₃×C₇	—	2	2	no	—
43	C₇×Dic₆	—	2	1	no	—
44	C₇×Dic₆	C₂×C₄	1	2	yes	complemented
45	C₇×C₃⋊C₈	—	2	2	no	—
46	C₇×#41	—	3	2	no	—
47	C₇×#41	—	3	2	no	—
48	C₇×#41	—	6	1	no	—
49	C₇×#41	—	6	1	no	—
50	C₂×C₇×Q₈	—	3	1	no	—
51	C₂×C₇×D₄	—	3	1	no	—
52	C₂×C₇×D₄	—	6	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	6	1	no	—
55	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
56	C₇×M₁₆	—	3	2	no	—
57	C₇×M₁₆	—	6	1	no	—
58	C₂×C₇×C₈	—	6	1	no	—
59	C₄²×C₇	—	2	1	no	—
60	C₄²×C₇	C₂×D₃	1	2	yes	complemented
61	C₂×#235	C₂×C₇	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂×#208	C₂×C₇	1	1	yes	characteristic, complemented
64	#805	C₂×C₇	1	4	yes	direct product factor
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
69	C₂×D₃×C₇	—	2	2	no	—
70	C₂×D₃×C₇	—	4	2	no	—
71	C₃×C₄×C₇	—	2	4	no	—
72	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₃×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₇×Dic₃	—	2	2	no	—
77	C₂×#104	—	3	1	no	Sylow
78	C₂³×C₇	—	6	1	no	—
79	C₂³×C₇	#67	1	1	yes	characteristic
80	C₇×Q₈	—	3	2	no	—
81	C₇×Q₈	—	6	1	no	—
82	C₇×D₄	—	3	2	no	—
83	C₇×D₄	—	6	2	no	—
84	C₇×D₄	—	6	2	no	—
85	C₇×D₄	—	6	1	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	6	1	no	—
89	C₂×C₄×C₇	—	6	2	no	—
90	C₂×C₄×C₇	—	6	2	no	—
91	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
93	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
94	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
95	C₂×C₄×C₇	#67	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
97	C₇×C₈	—	6	2	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
100	C₂×#67	—	2	1	no	—
101	#235	—	2	1	no	—
102	#235	—	2	1	no	—
103	#235	C₄×C₇	1	2	yes	complemented
104	#235	C₂²×C₇	1	2	yes	—
105	C₂×D₁₂	C₄×C₇	1	1	yes	characteristic, complemented
106	C₂×C₄×D₃	—	2	1	no	—
107	C₂×Dic₆	C₄×C₇	1	1	yes	characteristic, complemented
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	C₂²×C₇	1	2	yes	complemented
110	#208	—	2	1	no	—
111	#208	C₂²×C₇	1	2	yes	complemented
112	C₂×C₃⋊C₈	—	2	1	no	—
113	C₂×C₃×C₇	—	2	1	no	—
114	C₂×C₃×C₇	—	2	1	no	—
115	C₂×C₃×C₇	#104	1	2	yes	complemented
116	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	D₃×C₇	—	4	2	no	—
118	C₂×#41	—	3	1	no	—
119	C₂×M₁₆	—	3	1	no	—
120	C₂×C₄²	D₃×C₇	1	1	yes	characteristic, complemented
121	#104	—	3	4	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	6	2	no	—
124	C₂²×C₇	—	12	2	no	—
125	C₂²×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
126	C₂²×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
127	C₂²×C₇	C₂×#67	1	1	yes	characteristic
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	—	6	2	no	—
130	C₄×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
131	C₄×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
133	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
134	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
135	C₂²×D₃	—	2	1	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
140	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
144	#67	—	2	2	no	—
145	#67	—	2	2	no	—
146	C₂×Dic₃	—	2	1	no	—
147	D₁₂	—	2	1	no	—
148	D₁₂	C₂×C₄×C₇	1	2	yes	complemented
149	C₄×D₃	—	2	2	no	—
150	C₄×D₃	—	2	2	no	—
151	Dic₆	—	2	1	no	—
152	Dic₆	C₂×C₄×C₇	1	2	yes	complemented
153	C₃⋊C₈	—	2	2	no	—
154	C₃×C₇	C₂×#104	1	1	yes	characteristic, complemented, Hall
155	#41	—	3	2	no	—
156	#41	—	3	2	no	—
157	#41	—	6	1	no	—
158	#41	—	6	1	no	—
159	C₂×Q₈	—	3	1	no	—
160	C₂×D₄	—	3	1	no	—
161	C₂×D₄	—	6	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	6	1	no	—
164	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
165	M₁₆	—	3	2	no	—
166	M₁₆	—	6	1	no	—
167	C₂×C₈	—	6	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	C₂×D₃×C₇	1	2	yes	complemented
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	—	2	1	no	—
172	C₂×C₇	—	12	2	no	—
173	C₂×C₇	#805	1	2	yes	direct product factor
174	C₂×C₇	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
179	C₂×D₃	—	2	2	no	—
180	C₂×D₃	—	4	2	no	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
183	C₃×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
184	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
185	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
186	Dic₃	—	2	2	no	—
187	C₂³	—	6	1	no	—
188	C₂³	C₇×#67	1	1	yes	characteristic
189	Q₈	—	3	2	no	—
190	Q₈	—	6	1	no	—
191	D₄	—	3	2	no	—
192	D₄	—	6	2	no	—
193	D₄	—	6	2	no	—
194	D₄	—	6	1	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	6	1	no	—
198	C₂×C₄	—	6	2	no	—
199	C₂×C₄	—	6	2	no	—
200	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
201	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
202	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
204	C₂×C₄	C₇×#67	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
206	C₈	—	6	2	no	—
207	C₇	C₂×#805	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	C₇×#104	1	2	yes	complemented
211	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
212	D₃	—	4	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	6	2	no	—
215	C₂²	—	12	2	no	—
216	C₂²	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₂²	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
218	C₂²	C₂×C₇×#67	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	6	2	no	—
221	C₄	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
222	C₄	C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
224	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
225	C₃	C₂×C₇×#104	1	1	yes	characteristic, complemented, Sylow
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	12	2	no	—
229	C₂	C₇×#805	1	2	yes	direct product factor
230	C₂	C₂×C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂²×C₃×C₇	C₂²⋊C₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₄²×C₇	D₃
hypocenter	C₃	C₂×C₇×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄²×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₇	C₂	semidirect
3. C₂×C₇×#208	C₂	semidirect
4. C₇×#805	C₂	direct (N × H)
11. C₇×#235	C₄	semidirect
13. C₂×C₇×D₁₂	C₄	semidirect
15. C₂×C₇×Dic₆	C₄	semidirect
17. C₃×C₄²×C₇	C₂²	semidirect
19. C₇×#208	C₂²	semidirect
23. C₂×C₄²×C₇	D₃	semidirect
25. C₂×#805	C₇	direct (N × H)
40. C₇×D₁₂	C₂×C₄	semidirect
44. C₇×Dic₆	C₂×C₄	semidirect
60. C₄²×C₇	C₂×D₃	semidirect
62. C₂×C₃×C₄²	C₂×C₇	semidirect
63. C₂×#208	C₂×C₇	semidirect
64. #805	C₂×C₇	direct (N × H)
103. #235	C₄×C₇	semidirect
105. C₂×D₁₂	C₄×C₇	semidirect
107. C₂×Dic₆	C₄×C₇	semidirect
109. C₃×C₄²	C₂²×C₇	semidirect
111. #208	C₂²×C₇	semidirect
115. C₂×C₃×C₇	#104	semidirect
120. C₂×C₄²	D₃×C₇	semidirect
148. D₁₂	C₂×C₄×C₇	semidirect
152. Dic₆	C₂×C₄×C₇	semidirect
154. C₃×C₇	C₂×#104	semidirect
169. C₄²	C₂×D₃×C₇	semidirect
173. C₂×C₇	#805	direct (N × H)
207. C₇	C₂×#805	direct (N × H)
210. C₂×C₃	C₇×#104	semidirect
225. C₃	C₂×C₇×#104	semidirect
229. C₂	C₇×#805	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
C₂×C₃×C₄²×C₇	2
C₂×C₇×#208	2
C₇×#805	2
C₂×C₇×#104	3
C₂×#805	7

Subgroups — C2×C7×#805 #13055972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#805 #13055972