Subgroups — C₂×C₇×#1838 #13197983

Order 1792.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	C₂	1	2	yes	maximal
2	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
3	C₇×#1838	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#341	C₂	1	4	yes	maximal
5	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
6	C₂³×C₇×C₈	C₂²	1	2	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#423	—	2	4	no	—
9	C₇×#423	C₂²	1	8	yes	—
10	C₂×C₇×C₄⋊C₈	—	2	4	no	—
11	C₂×C₇×#97	C₄	1	4	yes	—
12	C₂×C₇×#97	C₄	1	4	yes	—
13	C₂×C₇×#97	C₂²	1	4	yes	—
14	C₇×#341	C₂²	1	16	yes	—
15	C₂×#1838	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	Q₈	1	2	yes	—
17	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₇×M₁₆	—	2	4	no	—
19	C₂×C₇×M₁₆	—	2	16	no	—
20	C₂×C₇×M₁₆	C₂³	1	8	yes	—
21	C₂²×C₇×C₈	—	2	8	no	—
22	C₂²×C₇×C₈	—	2	4	no	—
23	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₇×C₈	C₂³	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	2	yes	—
30	C₂×C₄²×C₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	2	yes	—
32	C₇×C₄⋊C₈	—	2	16	no	—
33	C₇×#97	C₂×C₄	1	16	yes	—
34	C₂×#423	C₂×C₇	1	2	yes	—
35	C₂²×#97	C₂×C₇	1	1	yes	characteristic
36	#1838	C₂×C₇	1	8	yes	direct product factor
37	C₂×#341	C₂×C₇	1	4	yes	—
38	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₇	—	4	4	no	—
40	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	#41	1	8	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
50	C₇×M₁₆	—	2	16	no	—
51	C₇×M₁₆	—	4	8	no	—
52	C₂×C₇×C₈	—	2	8	no	—
53	C₂×C₇×C₈	—	2	8	no	—
54	C₂×C₇×C₈	—	2	8	no	—
55	C₂×C₇×C₈	—	2	8	no	—
56	C₂×C₇×C₈	—	4	8	no	—
57	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
58	C₄²×C₇	C₄⋊C₄	1	4	yes	—
59	C₄²×C₇	C₄⋊C₄	1	4	yes	—
60	C₄²×C₇	C₂²×C₄	1	4	yes	—
61	C₄²×C₇	C₂×D₄	1	4	yes	—
62	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
63	C₂³×C₈	C₂²×C₇	1	2	yes	—
64	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
65	#423	—	2	4	no	—
66	#423	C₂²×C₇	1	8	yes	—
67	C₂×C₄⋊C₈	—	2	4	no	—
68	C₂×#97	C₄×C₇	1	4	yes	—
69	C₂×#97	C₄×C₇	1	4	yes	—
70	C₂×#97	C₂²×C₇	1	4	yes	—
71	#341	C₂²×C₇	1	16	yes	—
72	C₂³×C₇	—	2	4	no	—
73	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₇	#125	1	4	yes	—
76	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇	—	2	8	no	—
78	C₂×C₄×C₇	—	4	4	no	—
79	C₂×C₄×C₇	—	4	4	no	—
80	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	#117	1	4	yes	—
84	C₂×C₄×C₇	#117	1	4	yes	—
85	C₂×C₄×C₇	#124	1	4	yes	—
86	C₂×C₄×C₇	#124	1	4	yes	—
87	C₂×C₄×C₇	#125	1	4	yes	—
88	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×#41	1	4	yes	—
91	C₇×C₈	—	2	8	no	—
92	C₇×C₈	—	4	8	no	—
93	C₂³×C₄	C₇×Q₈	1	2	yes	—
94	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
95	C₂×M₁₆	—	2	4	no	—
96	C₂×M₁₆	—	2	16	no	—
97	C₂×M₁₆	C₂³×C₇	1	8	yes	—
98	C₂²×C₈	—	2	8	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
101	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
102	C₂²×C₈	C₂³×C₇	1	4	yes	—
103	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
104	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
105	C₂×C₄²	C₇×D₄	1	2	yes	—
106	C₂×C₄²	C₇×D₄	1	2	yes	—
107	C₂×C₄²	C₇×Q₈	1	2	yes	—
108	C₂×C₄²	C₂³×C₇	1	2	yes	—
109	C₄⋊C₈	—	2	16	no	—
110	#97	C₂×C₄×C₇	1	16	yes	—
111	C₂²×C₇	—	2	8	no	—
112	C₂²×C₇	—	2	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	#382	1	2	yes	—
115	C₂²×C₇	#382	1	2	yes	—
116	C₂²×C₇	#430	1	4	yes	—
117	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₇	C₂×#125	1	2	yes	—
119	C₄×C₇	—	2	8	no	—
120	C₄×C₇	#382	1	2	yes	—
121	C₄×C₇	#382	1	2	yes	—
122	C₄×C₇	C₂×#117	1	2	yes	—
123	C₄×C₇	C₂×#124	1	2	yes	—
124	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
127	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
128	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
133	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
134	C₂²×C₄	C₇×#41	1	8	yes	—
135	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
136	M₁₆	—	2	16	no	—
137	M₁₆	—	4	8	no	—
138	C₂×C₈	—	2	8	no	—
139	C₂×C₈	—	2	8	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	—	4	8	no	—
143	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
144	C₄²	C₇×C₄⋊C₄	1	4	yes	—
145	C₄²	C₇×C₄⋊C₄	1	4	yes	—
146	C₄²	C₂²×C₄×C₇	1	4	yes	—
147	C₄²	C₂×C₇×D₄	1	4	yes	—
148	C₂×C₇	—	2	2	no	—
149	C₂×C₇	—	2	2	no	—
150	C₂×C₇	#1838	1	4	yes	direct product factor
151	C₂×C₇	C₂×#382	1	1	yes	characteristic
152	C₂×C₇	C₂×#430	1	2	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
155	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
156	C₂³	C₇×#125	1	4	yes	—
157	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
162	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	C₇×#117	1	4	yes	—
165	C₂×C₄	C₇×#117	1	4	yes	—
166	C₂×C₄	C₇×#124	1	4	yes	—
167	C₂×C₄	C₇×#124	1	4	yes	—
168	C₂×C₄	C₇×#125	1	4	yes	—
169	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂×C₇×#41	1	4	yes	—
172	C₈	—	2	8	no	—
173	C₈	—	4	8	no	—
174	C₇	C₂×#1838	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	8	no	—
178	C₂²	C₇×#382	1	2	yes	—
179	C₂²	C₇×#382	1	2	yes	—
180	C₂²	C₇×#430	1	4	yes	—
181	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
182	C₂²	C₂×C₇×#125	1	2	yes	—
183	C₄	—	2	8	no	—
184	C₄	C₇×#382	1	2	yes	—
185	C₄	C₇×#382	1	2	yes	—
186	C₄	C₂×C₇×#117	1	2	yes	—
187	C₄	C₂×C₇×#124	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	C₇×#1838	1	4	yes	direct product factor
191	C₂	C₂×C₇×#382	1	1	yes	characteristic
192	C₂	C₂×C₇×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1838	C₂	direct (N × H)
15. C₂×#1838	C₇	direct (N × H)
36. #1838	C₂×C₇	direct (N × H)
150. C₂×C₇	#1838	direct (N × H)
174. C₇	C₂×#1838	direct (N × H)
190. C₂	C₇×#1838	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×#423	2
C₂²×C₇×#97	2
C₇×#1838	2
C₂×C₇×#341	2
C₂×#1838	7

Subgroups — C2×C7×#1838 #13197983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1838 #13197983