Subgroups — C₂×Q₈×Dic₂₈ #14234246

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×Q₈×Dic₁₄	C₂	1	2	yes	maximal
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₂₈	C₂	1	16	yes	maximal, direct product factor
4	C₂×#89622	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₂₈	C₂	1	3	yes	maximal
6	C₂×#89454	C₂	1	3	yes	maximal
7	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₂×Q₈×Dic₇	—	2	2	no	—
9	C₂²×Dic₂₈	C₂²	1	1	yes	characteristic
10	Q₈×Dic₁₄	—	2	8	no	—
11	Q₈×Dic₁₄	C₂²	1	16	yes	—
12	C₂×#6451	—	2	6	no	—
13	C₂×#6432	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₄	C₂²	1	6	yes	—
15	C₇×C₈×Q₈	C₂²	1	8	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
18	C₂×#6392	C₂²	1	3	yes	—
19	C₂×#6392	C₂²	1	6	yes	—
20	C₂×#6390	C₂²	1	6	yes	—
21	#89622	C₂²	1	48	yes	—
22	C₄×Dic₂₈	C₂²	1	24	yes	—
23	#89454	C₂²	1	24	yes	—
24	C₂×Q₈×Q₁₆	—	7	1	no	maximal, Sylow
25	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
26	C₂²×Dic₁₄	—	2	6	no	—
27	C₂²×Dic₁₄	C₂³	1	2	yes	—
28	C₂²×C₇×C₈	C₂³	1	3	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₇×Q₈	D₄	1	4	yes	—
31	C₄×C₇×Q₈	C₂³	1	4	yes	—
32	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
33	C₂×C₄²×C₇	C₂³	1	3	yes	—
34	Q₈×Dic₇	—	2	16	no	—
35	C₂×C₄⋊Dic₇	—	2	6	no	—
36	C₂×C₄⋊Dic₇	C₂³	1	6	yes	—
37	C₂×Dic₇⋊C₄	—	2	6	no	—
38	C₂×C₄×Dic₇	—	2	6	no	—
39	C₂×Dic₂₈	Q₈	1	8	yes	direct product factor
40	C₂×Dic₂₈	C₂³	1	4	yes	—
41	#6451	—	2	48	no	—
42	#6432	—	2	12	no	—
43	#6432	C₂³	1	24	yes	—
44	C₄×Dic₁₄	—	2	12	no	—
45	C₄×Dic₁₄	C₂³	1	24	yes	—
46	C₇×C₄⋊C₈	C₂³	1	12	yes	—
47	C₄×C₇×C₈	C₂³	1	12	yes	—
48	#6392	C₂³	1	24	yes	—
49	#6392	C₂³	1	12	yes	—
50	#6390	C₂³	1	24	yes	—
51	C₂×Q₈²	—	7	2	no	—
52	Q₈×Q₁₆	—	7	16	no	—
53	C₂×#500	—	7	3	no	—
54	C₂×#477	—	7	6	no	—
55	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
56	C₂×C₄×Q₁₆	—	7	3	no	—
57	C₂×C₇×Q₈	Q₁₆	1	8	yes	complemented
58	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₇	—	2	2	no	—
63	C₂²×Dic₇	—	2	6	no	—
64	C₂×Dic₁₄	—	2	2	no	—
65	C₂×Dic₁₄	—	2	4	no	—
66	C₂×Dic₁₄	—	2	24	no	—
67	C₂×Dic₁₄	—	8	12	no	—
68	C₂×Dic₁₄	C₂×Q₈	1	8	yes	—
69	C₂×Dic₁₄	C₂⁴	1	4	yes	—
70	C₂×C₇×C₈	—	2	6	no	—
71	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
72	C₂×C₇×C₈	C₂⁴	1	6	yes	—
73	C₂×C₇×C₈	C₂⁴	1	2	yes	—
74	C₇×C₄⋊C₄	C₂×D₄	1	12	yes	—
75	C₄²×C₇	C₂×D₄	1	6	yes	—
76	C₄²×C₇	C₂⁴	1	6	yes	—
77	C₄⋊Dic₇	—	2	6	no	—
78	C₄⋊Dic₇	—	2	24	no	—
79	C₄⋊Dic₇	C₂⁴	1	12	yes	—
80	Dic₇⋊C₄	—	2	24	no	—
81	C₄×Dic₇	—	2	24	no	—
82	Dic₂₈	C₂×Q₈	1	16	yes	direct product factor
83	C₂²×Q₁₆	—	7	1	no	—
84	Q₈²	—	7	16	no	—
85	Q₈²	—	14	8	no	—
86	C₂×C₄⋊Q₈	—	7	6	no	—
87	C₂×C₄⋊Q₈	—	14	6	no	—
88	C₂×C₄×Q₈	—	7	6	no	—
89	C₂×C₄×Q₈	—	14	2	no	—
90	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
91	#500	—	7	24	no	—
92	#477	—	7	48	no	—
93	C₈×Q₈	C₂×D₇	1	8	yes	—
94	C₄×Q₁₆	—	7	24	no	—
95	C₂×#107	—	7	6	no	—
96	C₂×#107	—	7	3	no	—
97	C₂×C₄⋊C₈	C₂×D₇	1	3	yes	—
98	C₂×#103	—	7	6	no	—
99	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
100	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
101	C₇×Q₈	C₂×Q₁₆	1	16	yes	complemented
102	C₂×C₄×C₇	—	2	6	no	—
103	C₂×C₄×C₇	C₂×Q₁₆	1	12	yes	—
104	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
107	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
108	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
109	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
110	C₂×Dic₇	—	2	8	no	—
111	C₂×Dic₇	—	2	4	no	—
112	C₂×Dic₇	—	2	12	no	—
113	C₂×Dic₇	—	4	12	no	—
114	Dic₁₄	—	2	8	no	—
115	Dic₁₄	—	2	4	no	—
116	Dic₁₄	—	8	24	no	—
117	Dic₁₄	C₂²×Q₈	1	8	yes	—
118	C₇×C₈	—	2	6	no	—
119	C₇×C₈	C₂²×Q₈	1	4	yes	—
120	C₂²×Q₈	—	7	2	no	—
121	C₂²×Q₈	—	14	6	no	—
122	C₂²×Q₈	D₂₈	1	1	yes	characteristic
123	C₂×Q₁₆	—	7	8	no	—
124	C₂×Q₁₆	—	7	4	no	—
125	C₂²×C₈	C₂²×D₇	1	3	yes	—
126	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	7	24	no	—
128	C₄⋊Q₈	—	14	48	no	—
129	C₄⋊Q₈	—	14	12	no	—
130	C₄×Q₈	—	7	24	no	—
131	C₄×Q₈	—	14	16	no	—
132	C₄×Q₈	—	14	12	no	—
133	C₄×Q₈	D₂₈	1	4	yes	—
134	C₄×Q₈	C₂²×D₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	7	6	no	—
136	C₂×C₄⋊C₄	—	14	6	no	—
137	C₂×C₄⋊C₄	—	14	6	no	—
138	C₂×C₄⋊C₄	D₂₈	1	3	yes	—
139	C₂×C₄²	—	14	6	no	—
140	C₂×C₄²	C₂²×D₇	1	3	yes	—
141	#107	—	7	12	no	—
142	#107	—	7	24	no	—
143	C₄⋊C₈	C₂²×D₇	1	12	yes	—
144	#103	—	7	24	no	—
145	C₄×C₈	C₂²×D₇	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	6	no	—
151	C₄×C₇	D₄×Q₈	1	2	yes	—
152	C₄×C₇	C₂²×Q₁₆	1	12	yes	—
153	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
154	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₇	—	2	8	no	—
156	Dic₇	—	4	12	no	—
157	C₂×Q₈	—	7	8	no	—
158	C₂×Q₈	—	7	4	no	—
159	C₂×Q₈	—	14	2	no	—
160	C₂×Q₈	—	14	24	no	—
161	C₂×Q₈	—	14	4	no	—
162	C₂×Q₈	—	56	12	no	—
163	C₂×Q₈	Dic₂₈	1	8	yes	direct product factor
164	C₂×Q₈	C₂×D₂₈	1	4	yes	—
165	C₂²×C₄	—	14	2	no	—
166	C₂²×C₄	—	14	6	no	—
167	C₂²×C₄	C₂×D₂₈	1	3	yes	—
168	C₂²×C₄	C₂×D₂₈	1	3	yes	—
169	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
170	Q₁₆	—	7	16	no	—
171	C₂×C₈	—	2	6	no	—
172	C₂×C₈	Q₈×D₇	1	4	yes	—
173	C₂×C₈	C₂³×D₇	1	6	yes	—
174	C₂×C₈	C₂³×D₇	1	2	yes	—
175	C₄⋊C₄	—	7	12	no	—
176	C₄⋊C₄	—	14	24	no	—
177	C₄⋊C₄	—	14	6	no	—
178	C₄⋊C₄	—	14	24	no	—
179	C₄⋊C₄	C₂×D₂₈	1	12	yes	—
180	C₄²	—	14	24	no	—
181	C₄²	C₂×D₂₈	1	6	yes	—
182	C₄²	C₂³×D₇	1	6	yes	—
183	C₂×C₇	Q₈×Q₁₆	1	4	yes	complemented
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×D₂₈	1	1	yes	characteristic
188	Q₈	—	7	8	no	—
189	Q₈	—	14	8	no	—
190	Q₈	—	14	4	no	—
191	Q₈	—	56	24	no	—
192	Q₈	C₂×Dic₂₈	1	16	yes	direct product factor
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	14	8	no	—
195	C₂×C₄	—	14	4	no	—
196	C₂×C₄	—	14	12	no	—
197	C₂×C₄	—	28	12	no	—
198	C₂×C₄	C₂×Dic₂₈	1	12	yes	—
199	C₂×C₄	C₂²×D₂₈	1	6	yes	—
200	C₂×C₄	C₂²×D₂₈	1	6	yes	—
201	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
202	C₂×C₄	C₂×Q₈×D₇	1	2	yes	—
203	C₂×C₄	#6554	1	2	yes	—
204	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
205	C₈	—	2	6	no	—
206	C₈	C₂×Q₈×D₇	1	4	yes	—
207	C₇	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂²	Q₈×D₂₈	1	2	yes	—
209	C₂²	C₂²×Dic₂₈	1	2	yes	—
210	C₂²	#90443	1	2	yes	—
211	C₂²	C₂³×D₂₈	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	14	8	no	—
214	C₄	—	28	12	no	—
215	C₄	Q₈×D₂₈	1	2	yes	—
216	C₄	C₂²×Dic₂₈	1	12	yes	—
217	C₄	C₂²×Q₈×D₇	1	1	yes	characteristic
218	C₄	C₂×#6554	1	1	yes	characteristic
219	C₂	Q₈×Dic₂₈	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₂₈	1	1	yes	characteristic
221	C₂	C₂³×Dic₂₈	1	1	yes	characteristic
222	C₂	C₂×#90443	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
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
3. Q₈×Dic₂₈	C₂	direct (N × H)
39. C₂×Dic₂₈	Q₈	direct (N × H)
57. C₂×C₇×Q₈	Q₁₆	semidirect
82. Dic₂₈	C₂×Q₈	direct (N × H)
101. C₇×Q₈	C₂×Q₁₆	semidirect
163. C₂×Q₈	Dic₂₈	direct (N × H)
183. C₂×C₇	Q₈×Q₁₆	semidirect
192. Q₈	C₂×Dic₂₈	direct (N × H)
207. C₇	C₂×Q₈×Q₁₆	semidirect
219. C₂	Q₈×Dic₂₈	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×Dic₁₄	2
C₂×C₇×C₈×Q₈	2
Q₈×Dic₂₈	2
C₂×#89622	2
C₂×C₄×Dic₂₈	2
C₂×#89454	2
C₂×Q₈×Q₁₆	7

Subgroups — C2×Q8×Dic28 #14234246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₂₈ #14234246