Subgroups — C₂×C₇×#2241 #13210911

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
3	C₇×#2241	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#358	C₂	1	4	yes	maximal
5	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	—	2	1	no	—
7	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#500	—	2	2	no	—
12	C₇×#500	C₂²	1	4	yes	—
13	C₇×#494	—	2	2	no	—
14	C₇×#494	C₂²	1	4	yes	—
15	C₂×C₇×#107	—	2	1	no	—
16	C₂×C₇×#107	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#358	C₂²	1	16	yes	—
19	C₄×C₇×C₁₆	C₂²	1	4	yes	—
20	C₂×#2241	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂²×C₇×Q₈	—	4	1	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	2	4	no	—
25	C₂×C₇×Q₁₆	—	2	8	no	—
26	C₂×C₇×Q₁₆	—	4	4	no	—
27	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×C₄⋊Q₈	—	2	4	no	—
31	C₇×C₄⋊Q₈	—	2	4	no	—
32	C₇×C₄⋊Q₈	—	4	2	no	—
33	C₇×C₄⋊Q₈	—	4	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	4	1	no	—
36	C₂×C₇×C₄⋊C₄	—	4	1	no	—
37	C₂×C₇×C₄⋊C₄	—	4	1	no	—
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₇×C₁₆	—	2	4	no	—
40	C₂×C₇×C₁₆	C₂³	1	4	yes	—
41	C₇×#107	—	2	2	no	—
42	C₇×#107	—	2	4	no	—
43	C₇×#107	C₂³	1	4	yes	—
44	C₄×C₇×C₈	D₄	1	2	yes	—
45	C₄×C₇×C₈	C₂³	1	2	yes	—
46	C₂×#500	C₂×C₇	1	1	yes	characteristic
47	C₂×#494	C₂×C₇	1	1	yes	characteristic
48	#2241	C₂×C₇	1	8	yes	direct product factor
49	C₂×#358	C₂×C₇	1	4	yes	—
50	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic
51	C₂×C₇×Q₈	—	2	4	no	—
52	C₂×C₇×Q₈	—	4	4	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	4	no	—
55	C₂×C₇×Q₈	—	4	4	no	—
56	C₂×C₇×Q₈	—	8	4	no	—
57	C₂²×C₄×C₇	—	4	2	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₇×Q₁₆	—	2	8	no	—
63	C₇×Q₁₆	—	4	4	no	—
64	C₇×Q₁₆	—	4	8	no	—
65	C₂×C₇×C₈	SD₁₆	1	4	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₇×C₈	#41	1	4	yes	—
69	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
70	C₇×C₄⋊C₄	—	2	4	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₇×C₄⋊C₄	—	4	4	no	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₇×C₄⋊C₄	—	4	4	no	—
75	C₄²×C₇	D₈	1	2	yes	—
76	C₄²×C₇	C₂×D₄	1	2	yes	—
77	C₇×C₁₆	—	2	4	no	—
78	C₂²×Q₁₆	—	2	1	no	—
79	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
80	C₂×C₄⋊Q₈	—	2	1	no	—
81	C₂×C₄⋊Q₈	—	2	1	no	—
82	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
83	#500	—	2	2	no	—
84	#500	C₂²×C₇	1	4	yes	—
85	#494	—	2	2	no	—
86	#494	C₂²×C₇	1	4	yes	—
87	C₂×#107	—	2	1	no	—
88	C₂×#107	C₂²×C₇	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
90	#358	C₂²×C₇	1	16	yes	—
91	C₄×C₁₆	C₂²×C₇	1	4	yes	—
92	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
93	C₇×Q₈	—	4	4	no	—
94	C₇×Q₈	—	4	4	no	—
95	C₇×Q₈	—	8	4	no	—
96	C₇×Q₈	—	8	8	no	—
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	—	8	4	no	—
100	C₂×C₄×C₇	—	8	4	no	—
101	C₂×C₄×C₇	SD₃₂	1	4	yes	—
102	C₂×C₄×C₇	Q₃₂	1	4	yes	—
103	C₂×C₄×C₇	#124	1	2	yes	—
104	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
109	C₇×C₈	#124	1	2	yes	—
110	C₇×C₈	C₂×SD₁₆	1	4	yes	—
111	C₇×C₈	C₂×#41	1	2	yes	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×Q₈	—	4	1	no	—
114	C₂×Q₁₆	—	2	4	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	8	no	—
117	C₂×Q₁₆	—	4	4	no	—
118	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
119	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	2	no	—
124	C₄⋊Q₈	—	4	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	C₂³×C₇	1	4	yes	—
132	#107	—	2	4	no	—
133	#107	—	2	2	no	—
134	#107	C₂³×C₇	1	4	yes	—
135	C₄×C₈	C₇×D₄	1	2	yes	—
136	C₄×C₈	C₂³×C₇	1	2	yes	—
137	C₂²×C₇	#486	1	2	yes	—
138	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
139	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
140	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₇	—	8	4	no	—
142	C₄×C₇	—	8	4	no	—
143	C₄×C₇	#486	1	2	yes	—
144	C₄×C₇	C₂×SD₃₂	1	4	yes	—
145	C₄×C₇	C₂×Q₃₂	1	4	yes	—
146	C₄×C₇	C₂×#124	1	1	yes	characteristic
147	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	4	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	8	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
157	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	4	4	no	—
161	Q₁₆	—	4	8	no	—
162	C₂×C₈	C₇×SD₁₆	1	4	yes	—
163	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
164	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
165	C₂×C₈	C₇×#41	1	4	yes	—
166	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄²	C₇×D₈	1	2	yes	—
173	C₄²	C₂×C₇×D₄	1	2	yes	—
174	C₁₆	—	2	4	no	—
175	C₂×C₇	#2241	1	4	yes	direct product factor
176	C₂×C₇	C₂×#486	1	1	yes	characteristic
177	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
178	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
179	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
180	Q₈	—	4	4	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	8	4	no	—
183	Q₈	—	8	8	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	8	4	no	—
188	C₂×C₄	C₇×SD₃₂	1	4	yes	—
189	C₂×C₄	C₇×Q₃₂	1	4	yes	—
190	C₂×C₄	C₇×#124	1	2	yes	—
191	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
193	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
194	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
195	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
196	C₈	C₇×#124	1	2	yes	—
197	C₈	C₂×C₇×SD₁₆	1	4	yes	—
198	C₈	C₂×C₇×#41	1	2	yes	—
199	C₇	C₂×#2241	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	C₇×#486	1	2	yes	—
201	C₂²	C₂×C₇×SD₃₂	1	2	yes	—
202	C₂²	C₂×C₇×Q₃₂	1	2	yes	—
203	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	8	4	no	—
206	C₄	C₇×#486	1	2	yes	—
207	C₄	C₂×C₇×SD₃₂	1	4	yes	—
208	C₄	C₂×C₇×Q₃₂	1	4	yes	—
209	C₄	C₂×C₇×#124	1	1	yes	characteristic
210	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
211	C₂	C₇×#2241	1	4	yes	direct product factor
212	C₂	C₂×C₇×#486	1	1	yes	characteristic
213	C₂	C₂²×C₇×SD₃₂	1	1	yes	characteristic
214	C₂	C₂²×C₇×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2241	C₂	direct (N × H)
20. C₂×#2241	C₇	direct (N × H)
48. #2241	C₂×C₇	direct (N × H)
175. C₂×C₇	#2241	direct (N × H)
199. C₇	C₂×#2241	direct (N × H)
211. C₂	C₇×#2241	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×#500	2
C₂×C₇×#494	2
C₇×#2241	2
C₂×C₇×#358	2
C₂×C₄×C₇×C₁₆	2
C₂×#2241	7

Subgroups — C2×C7×#2241 #13210911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2241 #13210911