Subgroups — C₂×C₇×Q₈×Q₁₆ #13238571

Order 1792.

1830 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₈²	C₂	1	2	yes	maximal
2	C₇×Q₈×Q₁₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₇×#500	C₂	1	3	yes	maximal
4	C₂×C₇×#477	C₂	1	6	yes	maximal
5	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₇×Q₁₆	C₂	1	3	yes	maximal
7	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
8	C₇×Q₈²	—	2	8	no	—
9	C₇×Q₈²	C₂²	1	16	yes	—
10	C₂×C₇×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	6	yes	—
12	C₂×C₄×C₇×Q₈	—	2	2	no	—
13	C₂×C₄×C₇×Q₈	C₂²	1	6	yes	—
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₇×#500	C₂²	1	24	yes	—
16	C₇×#477	C₂²	1	48	yes	—
17	C₇×C₈×Q₈	C₂²	1	8	yes	—
18	C₄×C₇×Q₁₆	C₂²	1	24	yes	—
19	C₂×C₇×#107	C₂²	1	3	yes	—
20	C₂×C₇×#107	C₂²	1	6	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
22	C₂×C₇×#103	C₂²	1	6	yes	—
23	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
24	C₂×Q₈×Q₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×Q₈	—	2	6	no	—
26	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
27	C₂²×C₇×Q₈	C₂³	1	2	yes	—
28	C₂×C₇×Q₁₆	Q₈	1	8	yes	direct product factor
29	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
30	C₂²×C₇×C₈	C₂³	1	3	yes	—
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊Q₈	—	2	48	no	—
33	C₇×C₄⋊Q₈	—	2	12	no	—
34	C₇×C₄⋊Q₈	C₂³	1	24	yes	—
35	C₄×C₇×Q₈	—	2	16	no	—
36	C₄×C₇×Q₈	—	2	12	no	—
37	C₄×C₇×Q₈	D₄	1	4	yes	—
38	C₄×C₇×Q₈	C₂³	1	24	yes	—
39	C₄×C₇×Q₈	C₂³	1	4	yes	—
40	C₂×C₇×C₄⋊C₄	—	2	6	no	—
41	C₂×C₇×C₄⋊C₄	—	2	6	no	—
42	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
44	C₂×C₄²×C₇	—	2	6	no	—
45	C₂×C₄²×C₇	C₂³	1	3	yes	—
46	C₇×#107	C₂³	1	24	yes	—
47	C₇×#107	C₂³	1	12	yes	—
48	C₇×C₄⋊C₈	C₂³	1	12	yes	—
49	C₇×#103	C₂³	1	24	yes	—
50	C₄×C₇×C₈	C₂³	1	12	yes	—
51	C₂×Q₈²	C₂×C₇	1	2	yes	—
52	Q₈×Q₁₆	C₂×C₇	1	16	yes	direct product factor
53	C₂×#500	C₂×C₇	1	3	yes	—
54	C₂×#477	C₂×C₇	1	6	yes	—
55	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
56	C₂×C₄×Q₁₆	C₂×C₇	1	3	yes	—
57	C₂×C₇×Q₈	—	2	2	no	—
58	C₂×C₇×Q₈	—	2	24	no	—
59	C₂×C₇×Q₈	—	2	4	no	—
60	C₂×C₇×Q₈	—	8	12	no	—
61	C₂×C₇×Q₈	Q₁₆	1	8	yes	direct product factor
62	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
63	C₂×C₇×Q₈	C₂×Q₈	1	8	yes	—
64	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₇	—	2	2	no	—
66	C₂²×C₄×C₇	—	2	6	no	—
67	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
70	C₇×Q₁₆	C₂×Q₈	1	16	yes	direct product factor
71	C₂×C₇×C₈	—	2	6	no	—
72	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
73	C₂×C₇×C₈	C₂⁴	1	6	yes	—
74	C₂×C₇×C₈	C₂⁴	1	2	yes	—
75	C₇×C₄⋊C₄	—	2	24	no	—
76	C₇×C₄⋊C₄	—	2	24	no	—
77	C₇×C₄⋊C₄	—	2	6	no	—
78	C₇×C₄⋊C₄	C₂×D₄	1	12	yes	—
79	C₇×C₄⋊C₄	C₂⁴	1	12	yes	—
80	C₄²×C₇	—	2	24	no	—
81	C₄²×C₇	C₂×D₄	1	6	yes	—
82	C₄²×C₇	C₂⁴	1	6	yes	—
83	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
84	Q₈²	—	2	8	no	—
85	Q₈²	C₂²×C₇	1	16	yes	—
86	C₂×C₄⋊Q₈	—	2	6	no	—
87	C₂×C₄⋊Q₈	C₂²×C₇	1	6	yes	—
88	C₂×C₄×Q₈	—	2	2	no	—
89	C₂×C₄×Q₈	C₂²×C₇	1	6	yes	—
90	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
91	#500	C₂²×C₇	1	24	yes	—
92	#477	C₂²×C₇	1	48	yes	—
93	C₈×Q₈	C₂²×C₇	1	8	yes	—
94	C₄×Q₁₆	C₂²×C₇	1	24	yes	—
95	C₂×#107	C₂²×C₇	1	6	yes	—
96	C₂×#107	C₂²×C₇	1	3	yes	—
97	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
98	C₂×#103	C₂²×C₇	1	6	yes	—
99	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
100	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
101	C₇×Q₈	—	2	8	no	—
102	C₇×Q₈	—	2	4	no	—
103	C₇×Q₈	—	8	24	no	—
104	C₇×Q₈	C₂×Q₁₆	1	16	yes	direct product factor
105	C₇×Q₈	C₂²×Q₈	1	8	yes	—
106	C₂×C₄×C₇	—	2	8	no	—
107	C₂×C₄×C₇	—	2	6	no	—
108	C₂×C₄×C₇	—	2	4	no	—
109	C₂×C₄×C₇	—	2	12	no	—
110	C₂×C₄×C₇	—	4	12	no	—
111	C₂×C₄×C₇	C₂×Q₁₆	1	12	yes	—
112	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
113	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
115	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
117	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
118	C₇×C₈	—	2	6	no	—
119	C₇×C₈	C₂²×Q₈	1	4	yes	—
120	C₂²×Q₈	—	2	6	no	—
121	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
122	C₂²×Q₈	C₂³×C₇	1	2	yes	—
123	C₂×Q₁₆	C₇×Q₈	1	8	yes	direct product factor
124	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
125	C₂²×C₈	C₂³×C₇	1	3	yes	—
126	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	48	no	—
128	C₄⋊Q₈	—	2	12	no	—
129	C₄⋊Q₈	C₂³×C₇	1	24	yes	—
130	C₄×Q₈	—	2	16	no	—
131	C₄×Q₈	—	2	12	no	—
132	C₄×Q₈	C₇×D₄	1	4	yes	—
133	C₄×Q₈	C₂³×C₇	1	24	yes	—
134	C₄×Q₈	C₂³×C₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	6	no	—
136	C₂×C₄⋊C₄	—	2	6	no	—
137	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
138	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
139	C₂×C₄²	—	2	6	no	—
140	C₂×C₄²	C₂³×C₇	1	3	yes	—
141	#107	C₂³×C₇	1	12	yes	—
142	#107	C₂³×C₇	1	24	yes	—
143	C₄⋊C₈	C₂³×C₇	1	12	yes	—
144	#103	C₂³×C₇	1	24	yes	—
145	C₄×C₈	C₂³×C₇	1	12	yes	—
146	C₂²×C₇	D₄×Q₈	1	2	yes	—
147	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
148	C₂²×C₇	#576	1	2	yes	—
149	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
150	C₄×C₇	—	2	8	no	—
151	C₄×C₇	—	2	6	no	—
152	C₄×C₇	—	4	12	no	—
153	C₄×C₇	D₄×Q₈	1	2	yes	—
154	C₄×C₇	C₂²×Q₁₆	1	12	yes	—
155	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
156	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	24	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	8	12	no	—
161	C₂×Q₈	C₇×Q₁₆	1	8	yes	direct product factor
162	C₂×Q₈	C₂×C₇×D₄	1	4	yes	—
163	C₂×Q₈	C₂×C₇×Q₈	1	8	yes	—
164	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
168	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
169	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
170	Q₁₆	C₂×C₇×Q₈	1	16	yes	direct product factor
171	C₂×C₈	—	2	6	no	—
172	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
173	C₂×C₈	C₂⁴×C₇	1	6	yes	—
174	C₂×C₈	C₂⁴×C₇	1	2	yes	—
175	C₄⋊C₄	—	2	24	no	—
176	C₄⋊C₄	—	2	24	no	—
177	C₄⋊C₄	—	2	6	no	—
178	C₄⋊C₄	C₂×C₇×D₄	1	12	yes	—
179	C₄⋊C₄	C₂⁴×C₇	1	12	yes	—
180	C₄²	—	2	24	no	—
181	C₄²	C₂×C₇×D₄	1	6	yes	—
182	C₄²	C₂⁴×C₇	1	6	yes	—
183	C₂×C₇	Q₈×Q₁₆	1	4	yes	direct product factor
184	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
185	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
186	C₂×C₇	C₂×#576	1	1	yes	characteristic
187	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	2	8	no	—
190	Q₈	—	8	24	no	—
191	Q₈	C₂×C₇×Q₁₆	1	16	yes	direct product factor
192	Q₈	C₂²×C₇×Q₈	1	8	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	4	12	no	—
198	C₂×C₄	C₂×C₇×Q₁₆	1	12	yes	—
199	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
200	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
202	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
203	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
204	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
205	C₈	—	2	6	no	—
206	C₈	C₂²×C₇×Q₈	1	4	yes	—
207	C₇	C₂×Q₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	C₇×D₄×Q₈	1	2	yes	—
209	C₂²	C₂²×C₇×Q₁₆	1	2	yes	—
210	C₂²	C₇×#576	1	2	yes	—
211	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	12	no	—
215	C₄	C₇×D₄×Q₈	1	2	yes	—
216	C₄	C₂²×C₇×Q₁₆	1	12	yes	—
217	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
218	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
219	C₂	C₇×Q₈×Q₁₆	1	4	yes	direct product factor
220	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
221	C₂	C₂³×C₇×Q₁₆	1	1	yes	characteristic
222	C₂	C₂×C₇×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×Q₁₆	C₂	direct (N × H)
24. C₂×Q₈×Q₁₆	C₇	direct (N × H)
28. C₂×C₇×Q₁₆	Q₈	direct (N × H)
52. Q₈×Q₁₆	C₂×C₇	direct (N × H)
61. C₂×C₇×Q₈	Q₁₆	direct (N × H)
70. C₇×Q₁₆	C₂×Q₈	direct (N × H)
104. C₇×Q₈	C₂×Q₁₆	direct (N × H)
123. C₂×Q₁₆	C₇×Q₈	direct (N × H)
161. C₂×Q₈	C₇×Q₁₆	direct (N × H)
170. Q₁₆	C₂×C₇×Q₈	direct (N × H)
183. C₂×C₇	Q₈×Q₁₆	direct (N × H)
191. Q₈	C₂×C₇×Q₁₆	direct (N × H)
207. C₇	C₂×Q₈×Q₁₆	direct (N × H)
219. C₂	C₇×Q₈×Q₁₆	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×Q₈²	2
C₇×Q₈×Q₁₆	2
C₂×C₇×#500	2
C₂×C₇×#477	2
C₂×C₇×C₈×Q₈	2
C₂×C₄×C₇×Q₁₆	2
C₂×Q₈×Q₁₆	7

Subgroups — C2×C7×Q8×Q16 #13238571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×Q₈×Q₁₆ #13238571