Subgroups — C₂×C₇×#1579 #13195063

Order 1792.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1579	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
9	C₇×C₈×Q₈	—	2	4	no	—
10	C₇×C₈×Q₈	C₂²	1	8	yes	—
11	C₄×C₇×Q₁₆	C₄	1	4	yes	—
12	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₇×#107	C₄	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	—	2	2	no	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×#103	C₄	1	2	yes	—
17	C₂×C₄×C₇×C₈	—	2	2	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#334	C₂²	1	4	yes	—
22	C₇×Q₈⋊C₈	C₂²	1	8	yes	—
23	C₇×#322	C₂²	1	4	yes	—
24	C₂×#1579	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
26	C₂×C₇×Q₁₆	C₈	1	8	yes	complemented
27	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	D₄	1	2	yes	—
33	C₄×C₇×Q₈	—	2	2	no	—
34	C₄×C₇×Q₈	—	2	2	no	—
35	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
36	C₄×C₇×Q₈	C₂³	1	4	yes	—
37	C₂×C₇×C₄⋊C₄	—	2	2	no	—
38	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇	—	2	2	no	—
40	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
41	C₇×#107	C₂×C₄	1	4	yes	—
42	C₇×C₄⋊C₈	—	2	8	no	—
43	C₇×C₄⋊C₈	—	2	2	no	—
44	C₇×C₄⋊C₈	C₂³	1	4	yes	—
45	C₇×#103	C₂×C₄	1	8	yes	—
46	C₄×C₇×C₈	—	2	8	no	—
47	C₄×C₇×C₈	—	2	1	no	—
48	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₇×C₈	D₄	1	2	yes	—
50	C₄×C₇×C₈	C₂³	1	2	yes	—
51	C₄×C₇×C₈	C₂³	1	2	yes	—
52	C₄×C₇×C₈	C₂³	1	2	yes	—
53	C₂×C₈×Q₈	C₂×C₇	1	2	yes	—
54	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
55	#1579	C₂×C₇	1	8	yes	direct product factor
56	C₂×#334	C₂×C₇	1	1	yes	characteristic
57	C₂×Q₈⋊C₈	C₂×C₇	1	2	yes	—
58	C₂×#322	C₂×C₇	1	1	yes	characteristic
59	C₂×C₇×Q₈	—	2	2	no	—
60	C₂×C₇×Q₈	—	2	4	no	—
61	C₂×C₇×Q₈	C₂×C₈	1	8	yes	—
62	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₇×Q₁₆	C₂×C₈	1	16	yes	complemented
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	2	4	no	—
71	C₂×C₇×C₈	—	2	4	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	4	2	no	—
74	C₂×C₇×C₈	—	4	4	no	—
75	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
79	C₇×C₄⋊C₄	—	2	4	no	—
80	C₇×C₄⋊C₄	—	2	2	no	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₄²×C₇	—	2	4	no	—
84	C₄²×C₇	—	2	4	no	—
85	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇	#41	1	1	yes	characteristic
88	C₄²×C₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Q₁₆	C₄×C₇	1	1	yes	characteristic
90	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
91	C₈×Q₈	—	2	4	no	—
92	C₈×Q₈	C₂²×C₇	1	8	yes	—
93	C₄×Q₁₆	C₄×C₇	1	4	yes	—
94	C₄×Q₁₆	C₂²×C₇	1	4	yes	—
95	C₂×#107	C₄×C₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	2	2	no	—
97	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
98	C₂×#103	C₄×C₇	1	2	yes	—
99	C₂×C₄×C₈	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
103	#334	C₂²×C₇	1	4	yes	—
104	Q₈⋊C₈	C₂²×C₇	1	8	yes	—
105	#322	C₂²×C₇	1	4	yes	—
106	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
107	C₇×Q₈	—	2	4	no	—
108	C₇×Q₈	—	2	8	no	—
109	C₇×Q₈	C₂²×C₈	1	8	yes	—
110	C₂×C₄×C₇	—	2	4	no	—
111	C₂×C₄×C₇	—	2	8	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇	#131	1	2	yes	—
120	C₂×C₄×C₇	#137	1	4	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	—	2	4	no	—
126	C₇×C₈	—	4	2	no	—
127	C₇×C₈	—	4	4	no	—
128	C₇×C₈	C₂²×C₈	1	4	yes	—
129	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
130	C₂×Q₁₆	C₇×C₈	1	8	yes	complemented
131	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂²×C₈	C₇×D₄	1	2	yes	—
137	C₄×Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
140	C₄×Q₈	C₂³×C₇	1	4	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
145	#107	C₂×C₄×C₇	1	4	yes	—
146	C₄⋊C₈	—	2	8	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	C₂³×C₇	1	4	yes	—
149	#103	C₂×C₄×C₇	1	8	yes	—
150	C₄×C₈	—	2	8	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
153	C₄×C₈	C₇×D₄	1	2	yes	—
154	C₄×C₈	C₂³×C₇	1	2	yes	—
155	C₄×C₈	C₂³×C₇	1	2	yes	—
156	C₄×C₈	C₂³×C₇	1	2	yes	—
157	C₂²×C₇	C₈×D₄	1	2	yes	—
158	C₂²×C₇	#441	1	2	yes	—
159	C₂²×C₇	#444	1	2	yes	—
160	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₇	—	2	8	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	4	4	no	—
165	C₄×C₇	C₈×D₄	1	2	yes	—
166	C₄×C₇	#441	1	2	yes	—
167	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₇	C₂×#131	1	1	yes	characteristic
169	C₄×C₇	C₂×#137	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₇×C₈	1	8	yes	—
173	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₇×#41	1	1	yes	characteristic
179	Q₁₆	C₂×C₇×C₈	1	16	yes	complemented
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
187	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
188	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
189	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
197	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
198	C₄²	C₇×#41	1	1	yes	characteristic
199	C₄²	C₂⁴×C₇	1	1	yes	characteristic
200	C₂×C₇	#1579	1	4	yes	direct product factor
201	C₂×C₇	C₂×C₈×D₄	1	1	yes	characteristic
202	C₂×C₇	C₂×#441	1	1	yes	characteristic
203	C₂×C₇	C₂×#444	1	1	yes	characteristic
204	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	2	8	no	—
207	Q₈	C₂²×C₇×C₈	1	8	yes	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
214	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
215	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
217	C₂×C₄	C₇×#131	1	2	yes	—
218	C₂×C₄	C₇×#137	1	4	yes	—
219	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	4	no	—
224	C₈	—	4	2	no	—
225	C₈	—	4	4	no	—
226	C₈	C₂²×C₇×C₈	1	4	yes	—
227	C₇	C₂×#1579	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	C₇×C₈×D₄	1	2	yes	—
229	C₂²	C₇×#441	1	2	yes	—
230	C₂²	C₇×#444	1	2	yes	—
231	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
232	C₄	—	2	8	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	C₇×C₈×D₄	1	2	yes	—
237	C₄	C₇×#441	1	2	yes	—
238	C₄	C₂³×C₇×C₈	1	1	yes	characteristic
239	C₄	C₂×C₇×#131	1	1	yes	characteristic
240	C₄	C₂×C₇×#137	1	2	yes	—
241	C₂	C₇×#1579	1	4	yes	direct product factor
242	C₂	C₂×C₇×C₈×D₄	1	1	yes	characteristic
243	C₂	C₂×C₇×#441	1	1	yes	characteristic
244	C₂	C₂×C₇×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₄	C₂³×C₇×C₈
Frattini, cosocle	C₄²	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1579	C₂	direct (N × H)
24. C₂×#1579	C₇	direct (N × H)
26. C₂×C₇×Q₁₆	C₈	semidirect
55. #1579	C₂×C₇	direct (N × H)
68. C₇×Q₁₆	C₂×C₈	semidirect
130. C₂×Q₁₆	C₇×C₈	semidirect
179. Q₁₆	C₂×C₇×C₈	semidirect
200. C₂×C₇	#1579	direct (N × H)
227. C₇	C₂×#1579	direct (N × H)
241. C₂	C₇×#1579	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×C₈×Q₈	2
C₂×C₄×C₇×Q₁₆	2
C₇×#1579	2
C₂×C₇×#334	2
C₂×C₇×Q₈⋊C₈	2
C₂×C₇×#322	2
C₂×#1579	7

Subgroups — C2×C7×#1579 #13195063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1579 #13195063