Subgroups — C₂×C₇×#1837 #13197982

Order 1792.

870 conjugacy classes of subgroups.

The subgroup lattice has 212 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×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1837	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×C₄×C₇×C₈	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
13	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
15	C₇×#341	C₂²	1	16	yes	—
16	C₂×#1837	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	Q₈	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×M₁₆	—	2	4	no	—
20	C₂×C₇×M₁₆	—	2	16	no	—
21	C₂×C₇×M₁₆	C₂³	1	8	yes	—
22	C₂²×C₇×C₈	—	2	8	no	—
23	C₂²×C₇×C₈	—	2	4	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₇×C₈	C₂³	1	4	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₇×C₄⋊C₈	—	2	16	no	—
35	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
36	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
37	C₂×#423	C₂×C₇	1	2	yes	—
38	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
39	#1837	C₂×C₇	1	8	yes	direct product factor
40	C₂×#341	C₂×C₇	1	4	yes	—
41	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	#41	1	8	yes	—
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₇×M₁₆	—	2	16	no	—
55	C₇×M₁₆	—	4	8	no	—
56	C₂×C₇×C₈	—	2	8	no	—
57	C₂×C₇×C₈	—	2	8	no	—
58	C₂×C₇×C₈	—	2	8	no	—
59	C₂×C₇×C₈	—	2	8	no	—
60	C₂×C₇×C₈	—	4	8	no	—
61	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
62	C₄²×C₇	C₄⋊C₄	1	2	yes	—
63	C₄²×C₇	C₄⋊C₄	1	4	yes	—
64	C₄²×C₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	4	yes	—
68	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
69	C₂³×C₈	C₂²×C₇	1	2	yes	—
70	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
71	#423	—	2	4	no	—
72	#423	C₂²×C₇	1	8	yes	—
73	C₂×C₄⋊C₈	—	2	4	no	—
74	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
75	C₂×C₄×C₈	C₄×C₇	1	4	yes	—
76	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
77	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
78	#341	C₂²×C₇	1	16	yes	—
79	C₂³×C₇	—	2	4	no	—
80	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₇	#125	1	4	yes	—
83	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	2	8	no	—
86	C₂×C₄×C₇	—	2	8	no	—
87	C₂×C₄×C₇	#108	1	8	yes	—
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₇	#117	1	4	yes	—
93	C₂×C₄×C₇	#117	1	4	yes	—
94	C₂×C₄×C₇	#124	1	4	yes	—
95	C₂×C₄×C₇	#124	1	4	yes	—
96	C₂×C₄×C₇	#125	1	4	yes	—
97	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×#41	1	4	yes	—
100	C₇×C₈	—	2	8	no	—
101	C₇×C₈	—	4	8	no	—
102	C₂³×C₄	C₇×Q₈	1	2	yes	—
103	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	—	2	16	no	—
106	C₂×M₁₆	C₂³×C₇	1	8	yes	—
107	C₂²×C₈	—	2	8	no	—
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
110	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
111	C₂²×C₈	C₂³×C₇	1	4	yes	—
112	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
115	C₂×C₄²	C₇×D₄	1	2	yes	—
116	C₂×C₄²	C₇×D₄	1	2	yes	—
117	C₂×C₄²	C₇×Q₈	1	2	yes	—
118	C₂×C₄²	C₂³×C₇	1	2	yes	—
119	C₄⋊C₈	—	2	16	no	—
120	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
121	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
122	C₂²×C₇	—	2	8	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	#382	1	2	yes	—
126	C₂²×C₇	#382	1	2	yes	—
127	C₂²×C₇	C₂×#108	1	4	yes	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×#125	1	2	yes	—
130	C₄×C₇	—	2	4	no	—
131	C₄×C₇	#382	1	2	yes	—
132	C₄×C₇	#382	1	2	yes	—
133	C₄×C₇	C₂×#108	1	8	yes	—
134	C₄×C₇	C₂×#117	1	2	yes	—
135	C₄×C₇	C₂×#124	1	2	yes	—
136	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
137	C₂²×C₄	—	2	8	no	—
138	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
139	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
140	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
146	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₇×#41	1	8	yes	—
148	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
149	M₁₆	—	2	16	no	—
150	M₁₆	—	4	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	4	8	no	—
156	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
157	C₄²	C₇×C₄⋊C₄	1	2	yes	—
158	C₄²	C₇×C₄⋊C₄	1	4	yes	—
159	C₄²	C₇×C₄⋊C₄	1	2	yes	—
160	C₄²	C₂²×C₄×C₇	1	2	yes	—
161	C₄²	C₂²×C₄×C₇	1	2	yes	—
162	C₄²	C₂×C₇×D₄	1	4	yes	—
163	C₂×C₇	—	2	2	no	—
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	#1837	1	4	yes	direct product factor
166	C₂×C₇	C₂×#382	1	1	yes	characteristic
167	C₂×C₇	C₂²×#108	1	2	yes	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
170	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
171	C₂³	C₇×#125	1	4	yes	—
172	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	C₇×#108	1	8	yes	—
177	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
178	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
181	C₂×C₄	C₇×#117	1	4	yes	—
182	C₂×C₄	C₇×#117	1	4	yes	—
183	C₂×C₄	C₇×#124	1	4	yes	—
184	C₂×C₄	C₇×#124	1	4	yes	—
185	C₂×C₄	C₇×#125	1	4	yes	—
186	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
188	C₂×C₄	C₂×C₇×#41	1	4	yes	—
189	C₈	—	2	8	no	—
190	C₈	—	4	8	no	—
191	C₇	C₂×#1837	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₇×#382	1	2	yes	—
196	C₂²	C₇×#382	1	2	yes	—
197	C₂²	C₂×C₇×#108	1	4	yes	—
198	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
199	C₂²	C₂×C₇×#125	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	C₇×#382	1	2	yes	—
202	C₄	C₇×#382	1	2	yes	—
203	C₄	C₂×C₇×#108	1	8	yes	—
204	C₄	C₂×C₇×#117	1	2	yes	—
205	C₄	C₂×C₇×#124	1	2	yes	—
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₇×#1837	1	4	yes	direct product factor
209	C₂	C₂×C₇×#382	1	1	yes	characteristic
210	C₂	C₂²×C₇×#108	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×#1837	C₂	direct (N × H)
16. C₂×#1837	C₇	direct (N × H)
39. #1837	C₂×C₇	direct (N × H)
165. C₂×C₇	#1837	direct (N × H)
191. C₇	C₂×#1837	direct (N × H)
208. C₂	C₇×#1837	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×C₄×C₇×C₈	2
C₇×#1837	2
C₂×C₇×#341	2
C₂×#1837	7

Subgroups — C2×C7×#1837 #13197982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1837 #13197982