Subgroups — C₂×Q₈×Dic₃₁₆ #35168719

Order 20224.

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×Q₈×Dic₁₅₈	C₂	1	2	yes	maximal
2	Q₈×Dic₃₁₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098968	C₂	1	3	yes	maximal
4	C₂×#19098859	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₃₁₆	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
8	Q₈×Dic₁₅₈	—	2	8	no	—
9	Q₈×Dic₁₅₈	C₂²	1	16	yes	—
10	C₂×#14579161	C₂²	1	6	yes	—
11	C₂×#14579160	—	2	6	no	—
12	C₂×Q₈×Dic₇₉	—	2	2	no	—
13	C₂×C₄×Dic₁₅₈	C₂²	1	6	yes	—
14	#19098968	C₂²	1	24	yes	—
15	#19098859	C₂²	1	48	yes	—
16	C₄×Dic₃₁₆	C₂²	1	24	yes	—
17	C₂×#14579095	C₂²	1	3	yes	—
18	C₂×#14579095	C₂²	1	6	yes	—
19	C₂×#14579086	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₇₉	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
24	C₂²×Dic₁₅₈	—	2	6	no	—
25	C₂²×Dic₁₅₈	C₂³	1	2	yes	—
26	C₂×Dic₃₁₆	Q₈	1	8	yes	direct product factor
27	C₂×Dic₃₁₆	C₂³	1	4	yes	—
28	#14579161	—	2	12	no	—
29	#14579161	C₂³	1	24	yes	—
30	#14579160	—	2	48	no	—
31	Q₈×Dic₇₉	—	2	16	no	—
32	C₄×Dic₁₅₈	—	2	12	no	—
33	C₄×Dic₁₅₈	C₂³	1	24	yes	—
34	C₂×C₄⋊Dic₇₉	—	2	6	no	—
35	C₂×C₄⋊Dic₇₉	C₂³	1	6	yes	—
36	C₂×Dic₇₉⋊C₄	—	2	6	no	—
37	C₂×C₄×Dic₇₉	—	2	6	no	—
38	#14579095	C₂³	1	12	yes	—
39	#14579095	C₂³	1	24	yes	—
40	#14579086	C₂³	1	24	yes	—
41	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₇₉	D₄	1	4	yes	—
45	C₄×Q₈×C₇₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
47	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
48	C₄⋊C₈×C₇₉	C₂³	1	12	yes	—
49	C₄×C₈×C₇₉	C₂³	1	12	yes	—
50	C₂×Dic₁₅₈	—	2	2	no	—
51	C₂×Dic₁₅₈	—	2	24	no	—
52	C₂×Dic₁₅₈	—	2	4	no	—
53	C₂×Dic₁₅₈	—	8	12	no	—
54	C₂×Dic₁₅₈	C₂×Q₈	1	8	yes	—
55	C₂×Dic₁₅₈	C₂⁴	1	4	yes	—
56	C₂²×Dic₇₉	—	2	2	no	—
57	C₂²×Dic₇₉	—	2	6	no	—
58	Dic₃₁₆	C₂×Q₈	1	16	yes	direct product factor
59	C₄⋊Dic₇₉	—	2	24	no	—
60	C₄⋊Dic₇₉	—	2	6	no	—
61	C₄⋊Dic₇₉	C₂⁴	1	12	yes	—
62	Dic₇₉⋊C₄	—	2	24	no	—
63	C₄×Dic₇₉	—	2	24	no	—
64	C₂×Q₈×C₇₉	Q₁₆	1	8	yes	complemented
65	C₂×Q₈×C₇₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₇₉	—	2	6	no	—
70	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₇₉	C₂⁴	1	6	yes	—
72	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
73	C₄⋊C₄×C₇₉	C₂×D₄	1	12	yes	—
74	C₄²×C₇₉	C₂×D₄	1	6	yes	—
75	C₄²×C₇₉	C₂⁴	1	6	yes	—
76	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₇₉	C₂×Q₁₆	1	16	yes	complemented
78	C₂×C₄×C₇₉	—	2	6	no	—
79	C₂×C₄×C₇₉	C₂×Q₁₆	1	12	yes	—
80	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
81	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂²×D₄	1	6	yes	—
83	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
84	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
85	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
86	C₂×Dic₇₉	—	2	8	no	—
87	C₂×Dic₇₉	—	2	4	no	—
88	C₂×Dic₇₉	—	2	12	no	—
89	C₂×Dic₇₉	—	4	12	no	—
90	Dic₁₅₈	—	2	8	no	—
91	Dic₁₅₈	—	2	4	no	—
92	Dic₁₅₈	—	8	24	no	—
93	Dic₁₅₈	C₂²×Q₈	1	8	yes	—
94	C₈×C₇₉	—	2	6	no	—
95	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
96	C₂²×C₇₉	D₄×Q₈	1	2	yes	—
97	C₂²×C₇₉	C₂²×Q₁₆	1	2	yes	—
98	C₂²×C₇₉	#576	1	2	yes	—
99	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
100	C₄×C₇₉	—	2	6	no	—
101	C₄×C₇₉	D₄×Q₈	1	2	yes	—
102	C₄×C₇₉	C₂²×Q₁₆	1	12	yes	—
103	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
104	C₄×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₇₉	—	2	8	no	—
106	Dic₇₉	—	4	12	no	—
107	C₂×Q₈×Q₁₆	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	Q₈×Q₁₆	1	4	yes	complemented
109	C₂×C₇₉	C₂×D₄×Q₈	1	1	yes	characteristic
110	C₂×C₇₉	C₂³×Q₁₆	1	1	yes	characteristic
111	C₂×C₇₉	C₂×#576	1	1	yes	characteristic
112	C₂×Q₈²	—	79	2	no	—
113	Q₈×Q₁₆	—	79	16	no	—
114	C₂×#500	—	79	3	no	—
115	C₂×#477	—	79	6	no	—
116	C₂×C₈×Q₈	D₇₉	1	1	yes	characteristic
117	C₂×C₄×Q₁₆	—	79	3	no	—
118	C₇₉	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₁₆	—	79	1	no	—
120	Q₈²	—	79	16	no	—
121	Q₈²	—	158	8	no	—
122	C₂×C₄⋊Q₈	—	79	6	no	—
123	C₂×C₄⋊Q₈	—	158	6	no	—
124	C₂×C₄×Q₈	—	79	6	no	—
125	C₂×C₄×Q₈	—	158	2	no	—
126	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
127	#500	—	79	24	no	—
128	#477	—	79	48	no	—
129	C₈×Q₈	C₂×D₇₉	1	8	yes	—
130	C₄×Q₁₆	—	79	24	no	—
131	C₂×#107	—	79	6	no	—
132	C₂×#107	—	79	3	no	—
133	C₂×C₄⋊C₈	C₂×D₇₉	1	3	yes	—
134	C₂×#103	—	79	6	no	—
135	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
136	C₂²×Q₈	—	79	2	no	—
137	C₂²×Q₈	—	158	6	no	—
138	C₂²×Q₈	D₃₁₆	1	1	yes	characteristic
139	C₂×Q₁₆	—	79	8	no	—
140	C₂×Q₁₆	—	79	4	no	—
141	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
142	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	79	24	no	—
144	C₄⋊Q₈	—	158	48	no	—
145	C₄⋊Q₈	—	158	12	no	—
146	C₄×Q₈	—	79	24	no	—
147	C₄×Q₈	—	158	16	no	—
148	C₄×Q₈	—	158	12	no	—
149	C₄×Q₈	D₃₁₆	1	4	yes	—
150	C₄×Q₈	C₂²×D₇₉	1	4	yes	—
151	C₂×C₄⋊C₄	—	79	6	no	—
152	C₂×C₄⋊C₄	—	158	6	no	—
153	C₂×C₄⋊C₄	—	158	6	no	—
154	C₂×C₄⋊C₄	D₃₁₆	1	3	yes	—
155	C₂×C₄²	—	158	6	no	—
156	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
157	#107	—	79	12	no	—
158	#107	—	79	24	no	—
159	C₄⋊C₈	C₂²×D₇₉	1	12	yes	—
160	#103	—	79	24	no	—
161	C₄×C₈	C₂²×D₇₉	1	12	yes	—
162	C₂×Q₈	—	79	8	no	—
163	C₂×Q₈	—	79	4	no	—
164	C₂×Q₈	—	632	12	no	—
165	C₂×Q₈	—	158	4	no	—
166	C₂×Q₈	—	158	24	no	—
167	C₂×Q₈	—	158	2	no	—
168	C₂×Q₈	Dic₃₁₆	1	8	yes	direct product factor
169	C₂×Q₈	C₂×D₃₁₆	1	4	yes	—
170	C₂²×C₄	—	158	6	no	—
171	C₂²×C₄	—	158	2	no	—
172	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
173	C₂²×C₄	C₂×D₃₁₆	1	3	yes	—
174	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
175	Q₁₆	—	79	16	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	Q₈×D₇₉	1	4	yes	—
178	C₂×C₈	C₂³×D₇₉	1	6	yes	—
179	C₂×C₈	C₂³×D₇₉	1	2	yes	—
180	C₄⋊C₄	—	79	12	no	—
181	C₄⋊C₄	—	158	24	no	—
182	C₄⋊C₄	—	158	6	no	—
183	C₄⋊C₄	—	158	24	no	—
184	C₄⋊C₄	C₂×D₃₁₆	1	12	yes	—
185	C₄²	—	158	24	no	—
186	C₄²	C₂×D₃₁₆	1	6	yes	—
187	C₄²	C₂³×D₇₉	1	6	yes	—
188	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
189	Q₈	—	79	8	no	—
190	Q₈	—	632	24	no	—
191	Q₈	—	158	4	no	—
192	Q₈	—	158	8	no	—
193	Q₈	C₂×Dic₃₁₆	1	16	yes	direct product factor
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	316	12	no	—
196	C₂×C₄	—	158	12	no	—
197	C₂×C₄	—	158	8	no	—
198	C₂×C₄	—	158	4	no	—
199	C₂×C₄	C₂×Dic₃₁₆	1	12	yes	—
200	C₂×C₄	C₂²×D₃₁₆	1	6	yes	—
201	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₃₁₆	1	6	yes	—
203	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
204	C₂×C₄	#14579208	1	2	yes	—
205	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×Q₈×D₇₉	1	4	yes	—
208	C₂²	Q₈×D₃₁₆	1	2	yes	—
209	C₂²	C₂²×Dic₃₁₆	1	2	yes	—
210	C₂²	#19099368	1	2	yes	—
211	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	316	12	no	—
214	C₄	—	158	8	no	—
215	C₄	Q₈×D₃₁₆	1	2	yes	—
216	C₄	C₂²×Dic₃₁₆	1	12	yes	—
217	C₄	C₂²×Q₈×D₇₉	1	1	yes	characteristic
218	C₄	C₂×#14579208	1	1	yes	characteristic
219	C₂	Q₈×Dic₃₁₆	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₃₁₆	1	1	yes	characteristic
221	C₂	C₂³×Dic₃₁₆	1	1	yes	characteristic
222	C₂	C₂×#19099368	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×C₈×Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₇₉
hypocenter	C₇₉	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₃₁₆	C₂	direct (N × H)
26. C₂×Dic₃₁₆	Q₈	direct (N × H)
58. Dic₃₁₆	C₂×Q₈	direct (N × H)
64. C₂×Q₈×C₇₉	Q₁₆	semidirect
77. Q₈×C₇₉	C₂×Q₁₆	semidirect
108. C₂×C₇₉	Q₈×Q₁₆	semidirect
118. C₇₉	C₂×Q₈×Q₁₆	semidirect
168. C₂×Q₈	Dic₃₁₆	direct (N × H)
193. Q₈	C₂×Dic₃₁₆	direct (N × H)
219. C₂	Q₈×Dic₃₁₆	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×Q₈×Dic₁₅₈	2
Q₈×Dic₃₁₆	2
C₂×#19098968	2
C₂×#19098859	2
C₂×C₄×Dic₃₁₆	2
C₂×C₈×Q₈×C₇₉	2
C₂×Q₈×Q₁₆	79

Subgroups — C2×Q8×Dic316 #35168719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₃₁₆ #35168719