Subgroups — D₄×Dic₄₂ #13061626

Order 1344.

560 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	C₂²×Dic₄₂	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	D₄×Dic₂₁	C₂	1	2	yes	maximal
4	#10621407	C₂	1	2	yes	maximal, complemented
5	#10621371	C₂	1	2	yes	maximal, complemented
6	#10621361	C₂	1	4	yes	maximal, complemented
7	#10621352	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₁₄	—	3	1	no	maximal, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×Dic₂₁	C₂²	1	4	yes	—
13	C₂×Dic₄₂	—	2	8	no	—
14	C₂×Dic₄₂	C₂²	1	2	yes	—
15	C₂×Dic₄₂	C₂²	1	4	yes	—
16	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
20	#66636	C₂²	1	4	yes	—
21	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₂₁	C₂²	1	2	yes	—
23	C₄⋊Dic₂₁	C₂²	1	2	yes	—
24	#66629	C₂²	1	2	yes	—
25	#66629	C₂²	1	4	yes	—
26	C₄×Dic₂₁	C₂²	1	2	yes	—
27	C₂²×Dic₁₄	—	3	2	no	—
28	C₄×C₇×D₄	D₃	1	1	yes	characteristic
29	D₄×Dic₇	—	3	2	no	—
30	#6487	—	3	2	no	—
31	#6451	—	3	2	no	—
32	#6441	—	3	4	no	—
33	#6432	—	3	1	no	—
34	C₄×Dic₁₄	—	3	1	no	—
35	D₄×Dic₆	—	7	1	no	maximal, Hall
36	C₂³×C₃×C₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄×C₇	—	2	4	no	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₂₁	—	2	8	no	—
44	C₂×Dic₂₁	C₂³	1	2	yes	—
45	C₂×Dic₂₁	C₂³	1	4	yes	—
46	C₂×Dic₂₁	C₂³	1	2	yes	—
47	Dic₄₂	—	2	4	no	—
48	Dic₄₂	—	2	8	no	—
49	Dic₄₂	D₄	1	4	yes	direct product factor
50	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
52	C₂²×Dic₇	—	3	4	no	—
53	C₂×Dic₁₄	—	3	1	no	—
54	C₂×Dic₁₄	—	3	2	no	—
55	C₂×Dic₁₄	—	3	4	no	—
56	C₂×Dic₁₄	—	6	8	no	—
57	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	—
59	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
60	C₂²⋊Dic₇	—	3	4	no	—
61	C₄⋊Dic₇	—	3	1	no	—
62	C₄⋊Dic₇	—	3	2	no	—
63	C₄⋊Dic₇	—	3	2	no	—
64	Dic₇⋊C₄	—	3	2	no	—
65	Dic₇⋊C₄	—	3	4	no	—
66	C₄×Dic₇	—	3	2	no	—
67	C₂²×Dic₆	—	7	2	no	—
68	C₃×C₄×D₄	D₇	1	1	yes	characteristic
69	D₄×Dic₃	—	7	2	no	—
70	#924	—	7	2	no	—
71	#888	—	7	2	no	—
72	#878	—	7	4	no	—
73	#869	—	7	1	no	—
74	C₄×Dic₆	—	7	1	no	—
75	C₂²×C₃×C₇	—	2	2	no	—
76	C₂²×C₃×C₇	—	2	2	no	—
77	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	complemented
78	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
79	C₃×C₄×C₇	—	2	2	no	—
80	C₃×C₄×C₇	—	2	1	no	—
81	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
83	Dic₂₁	—	2	4	no	—
84	Dic₂₁	—	2	2	no	—
85	Dic₂₁	C₂×D₄	1	4	yes	—
86	D₄×Q₈	—	21	1	no	Sylow
87	C₂³×C₇	C₂²×D₃	1	2	yes	—
88	C₇×D₄	Dic₆	1	4	yes	complemented
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
92	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₂×Dic₇	—	3	2	no	—
95	C₂×Dic₇	—	3	2	no	—
96	C₂×Dic₇	—	3	4	no	—
97	C₂×Dic₇	—	6	8	no	—
98	Dic₁₄	—	3	4	no	—
99	Dic₁₄	—	6	8	no	—
100	Dic₁₄	—	6	4	no	—
101	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
103	C₂²×Dic₃	—	7	4	no	—
104	C₂×Dic₆	—	7	1	no	—
105	C₂×Dic₆	—	7	2	no	—
106	C₂×Dic₆	—	7	4	no	—
107	C₂×Dic₆	—	14	8	no	—
108	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
109	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
110	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
111	#217	—	7	4	no	—
112	C₄⋊Dic₃	—	7	1	no	—
113	C₄⋊Dic₃	—	7	2	no	—
114	C₄⋊Dic₃	—	7	2	no	—
115	Dic₃⋊C₄	—	7	2	no	—
116	Dic₃⋊C₄	—	7	4	no	—
117	C₄×Dic₃	—	7	2	no	—
118	C₂×C₃×C₇	—	2	4	no	—
119	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂²×Q₈	—	21	2	no	—
123	C₄⋊Q₈	—	21	2	no	—
124	C₄⋊Q₈	—	21	1	no	—
125	C₂²⋊Q₈	—	21	2	no	—
126	C₂²⋊Q₈	—	21	4	no	—
127	C₄×Q₈	—	21	1	no	—
128	C₄×D₄	—	21	2	no	—
129	C₄×D₄	D₂₁	1	1	yes	characteristic
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	C₂×Dic₆	1	4	yes	complemented
133	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	2	1	no	—
136	C₄×C₇	C₂×Dic₆	1	2	yes	complemented
137	C₄×C₇	D₃×D₄	1	2	yes	—
138	Dic₇	—	3	4	no	—
139	Dic₇	—	6	4	no	—
140	Dic₇	—	6	2	no	—
141	C₂³×C₃	C₂²×D₇	1	2	yes	—
142	C₃×D₄	Dic₁₄	1	4	yes	complemented
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
146	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
148	C₂×Dic₃	—	7	2	no	—
149	C₂×Dic₃	—	7	2	no	—
150	C₂×Dic₃	—	7	4	no	—
151	C₂×Dic₃	—	14	8	no	—
152	Dic₆	—	7	4	no	—
153	Dic₆	—	14	8	no	—
154	Dic₆	—	14	4	no	—
155	C₃×C₇	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
156	C₂×Q₈	—	21	1	no	—
157	C₂×Q₈	—	21	4	no	—
158	C₂×Q₈	—	21	2	no	—
159	C₂×Q₈	—	42	8	no	—
160	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
161	C₂²×C₄	—	21	4	no	—
162	C₂²×C₄	C₂×D₂₁	1	2	yes	—
163	C₄⋊C₄	—	21	2	no	—
164	C₄⋊C₄	—	21	4	no	—
165	C₄⋊C₄	—	21	2	no	—
166	C₄⋊C₄	—	21	1	no	—
167	C₄⋊C₄	—	21	2	no	—
168	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic
169	C₂²⋊C₄	—	21	4	no	—
170	C₂²⋊C₄	C₂×D₂₁	1	2	yes	—
171	C₄²	—	21	2	no	—
172	C₄²	C₂×D₂₁	1	1	yes	characteristic
173	C₂×C₇	—	2	4	no	—
174	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
175	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
176	C₂×C₇	#1010	1	1	yes	characteristic
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	C₂×Dic₁₄	1	4	yes	complemented
180	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	C₂×Dic₁₄	1	2	yes	complemented
184	C₃×C₄	D₄×D₇	1	2	yes	—
185	Dic₃	—	7	4	no	—
186	Dic₃	—	14	4	no	—
187	Dic₃	—	14	2	no	—
188	C₂³	C₂²×D₂₁	1	2	yes	—
189	Q₈	—	21	4	no	—
190	Q₈	—	42	8	no	—
191	Q₈	—	42	4	no	—
192	D₄	Dic₄₂	1	4	yes	direct product factor
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	21	2	no	—
195	C₂×C₄	—	21	2	no	—
196	C₂×C₄	—	21	4	no	—
197	C₂×C₄	—	42	8	no	—
198	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
199	C₂×C₄	C₂²×D₂₁	1	2	yes	—
200	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
201	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
202	C₇	D₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	C₂²×Dic₁₄	1	1	yes	characteristic
205	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
206	C₂×C₃	#6554	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	C₂×Dic₄₂	1	4	yes	complemented
210	C₂²	C₂³×D₂₁	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	21	4	no	—
214	C₄	—	42	4	no	—
215	C₄	—	42	2	no	—
216	C₄	C₂×Dic₄₂	1	2	yes	complemented
217	C₄	D₄×D₂₁	1	2	yes	—
218	C₃	D₄×Dic₁₄	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	2	4	no	—
220	C₂	C₂²×Dic₄₂	1	1	yes	characteristic
221	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
222	C₂	#10621663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₄×C₇×D₄	C₂
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₂	C₂	semidirect
4. #10621407	C₂	semidirect
5. #10621371	C₂	semidirect
6. #10621361	C₂	semidirect
7. #10621352	C₂	semidirect
8. C₄×Dic₄₂	C₂	semidirect
37. C₃×C₇×D₄	Q₈	semidirect
49. Dic₄₂	D₄	direct (N × H)
77. C₂²×C₃×C₇	C₂×Q₈	semidirect
82. C₃×C₄×C₇	C₂×Q₈	semidirect
88. C₇×D₄	Dic₆	semidirect
132. C₂²×C₇	C₂×Dic₆	semidirect
136. C₄×C₇	C₂×Dic₆	semidirect
142. C₃×D₄	Dic₁₄	semidirect
155. C₃×C₇	D₄×Q₈	semidirect
179. C₂²×C₃	C₂×Dic₁₄	semidirect
183. C₃×C₄	C₂×Dic₁₄	semidirect
192. D₄	Dic₄₂	direct (N × H)
202. C₇	D₄×Dic₆	semidirect
209. C₂²	C₂×Dic₄₂	semidirect
216. C₄	C₂×Dic₄₂	semidirect
218. C₃	D₄×Dic₁₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₄₂	2
C₃×C₄×C₇×D₄	2
D₄×Dic₂₁	2
#10621407	2
#10621371	2
#10621361	2
#10621352	2
C₄×Dic₄₂	2
D₄×Dic₁₄	3
D₄×Dic₆	7

Subgroups — D4×Dic42 #13061626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₄₂ #13061626