Subgroups — C₃×D₄×D₂₈ #13060894

Order 1344.

856 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₄×D₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×D₂₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6493	C₂	1	2	yes	maximal, complemented
5	C₃×#6458	C₂	1	2	yes	maximal, complemented
6	C₃×#6445	C₂	1	4	yes	maximal, complemented
7	C₃×#6437	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×D₂₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₃×D₇	C₂²	1	4	yes	complemented
11	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	complemented
13	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	complemented
14	C₃×D₄×D₇	—	2	8	no	—
15	C₂×C₃×D₂₈	—	2	8	no	—
16	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×D₂₈	C₂²	1	2	yes	complemented
18	C₂×C₃×D₂₈	C₂²	1	2	yes	complemented
19	C₂×C₃×D₂₈	C₂²	1	4	yes	complemented
20	C₂×C₃×D₂₈	C₂²	1	2	yes	complemented
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	complemented
22	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	complemented
24	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic, complemented
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	complemented
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×D₄×D₇	C₂×C₃	1	2	yes	complemented
29	C₂²×D₂₈	C₂×C₃	1	2	yes	complemented
30	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
31	#6493	C₂×C₃	1	2	yes	complemented
32	#6458	C₂×C₃	1	2	yes	complemented
33	#6445	C₂×C₃	1	4	yes	complemented
34	#6437	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄×D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
36	C₃×D₄²	—	7	1	no	maximal, Hall
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₂²×C₃×D₇	—	2	8	no	—
39	C₂²×C₃×D₇	—	2	8	no	—
40	C₂²×C₃×D₇	—	4	4	no	—
41	C₂²×C₃×D₇	C₂³	1	4	yes	—
42	C₂²×C₃×D₇	C₂³	1	2	yes	—
43	C₃×C₇×D₄	D₄	1	4	yes	complemented
44	C₂×C₃×C₄×C₇	—	2	4	no	—
45	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₃×C₂²⋊D₇	—	2	8	no	—
50	C₂×C₃×Dic₇	C₂³	1	2	yes	—
51	C₃×D₂₈	—	2	8	no	—
52	C₃×D₂₈	—	2	4	no	—
53	C₃×D₂₈	—	2	4	no	—
54	C₃×D₂₈	—	4	2	no	—
55	C₃×D₂₈	D₄	1	4	yes	direct product factor
56	C₃×C₄×D₇	—	2	4	no	—
57	C₂³×D₇	C₂²×C₃	1	4	yes	complemented
58	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	complemented
60	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	complemented
61	D₄×D₇	—	2	8	no	—
62	C₂×D₂₈	—	2	8	no	—
63	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×D₂₈	C₂²×C₃	1	2	yes	complemented
65	C₂×D₂₈	C₂²×C₃	1	2	yes	complemented
66	C₂×D₂₈	C₂²×C₃	1	4	yes	complemented
67	C₂×D₂₈	C₂²×C₃	1	2	yes	complemented
68	C₂×C₄×D₇	C₂²×C₃	1	2	yes	complemented
69	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
71	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	complemented
73	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
74	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
75	C₂²×C₃×D₄	—	7	2	no	—
76	C₂²×C₃×D₄	—	7	2	no	—
77	C₃×#127	—	7	1	no	—
78	C₃×#121	—	7	2	no	—
79	C₃×#121	—	7	2	no	—
80	C₃×#120	—	7	4	no	—
81	C₃×C₄×D₄	—	7	1	no	—
82	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
83	C₂²×C₃×C₇	—	2	2	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	C₂×D₄	1	4	yes	complemented
86	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₃×D₇	—	2	2	no	—
88	C₂×C₃×D₇	—	2	2	no	—
89	C₂×C₃×D₇	—	2	4	no	—
90	C₂×C₃×D₇	—	2	4	no	—
91	C₂×C₃×D₇	—	2	4	no	—
92	C₂×C₃×D₇	—	4	16	no	—
93	C₂×C₃×D₇	C₂×D₄	1	4	yes	complemented
94	C₃×C₄×C₇	—	2	2	no	—
95	C₃×C₄×C₇	—	2	1	no	—
96	C₃×C₄×C₇	C₂×D₄	1	2	yes	complemented
97	C₃×C₄×C₇	C₂×D₄	1	2	yes	complemented
98	C₃×Dic₇	—	2	2	no	—
99	D₄²	—	7	1	no	Sylow
100	C₂³×C₇	C₂³×C₃	1	2	yes	—
101	C₂²×D₇	—	2	8	no	—
102	C₂²×D₇	—	2	8	no	—
103	C₂²×D₇	—	4	4	no	—
104	C₂²×D₇	C₂³×C₃	1	4	yes	—
105	C₂²×D₇	C₂³×C₃	1	2	yes	—
106	C₇×D₄	C₃×D₄	1	4	yes	complemented
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
110	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
112	C₂²⋊D₇	—	2	8	no	—
113	C₂×Dic₇	C₂³×C₃	1	2	yes	—
114	D₂₈	—	2	8	no	—
115	D₂₈	—	2	4	no	—
116	D₂₈	—	2	4	no	—
117	D₂₈	—	4	2	no	—
118	D₂₈	C₃×D₄	1	4	yes	direct product factor
119	C₄×D₇	—	2	4	no	—
120	C₂⁴×C₃	—	7	4	no	—
121	C₂×C₃×D₄	—	7	4	no	—
122	C₂×C₃×D₄	—	7	1	no	—
123	C₂×C₃×D₄	—	7	4	no	—
124	C₂×C₃×D₄	—	7	2	no	—
125	C₂×C₃×D₄	—	7	2	no	—
126	C₂×C₃×D₄	—	7	2	no	—
127	C₂×C₃×D₄	—	14	8	no	—
128	C₂×C₃×D₄	—	14	8	no	—
129	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	7	2	no	—
131	C₂²×C₃×C₄	C₂×D₇	1	2	yes	complemented
132	C₃×C₄⋊C₄	—	7	1	no	—
133	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	4	no	—
136	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
137	C₃×C₄²	C₂×D₇	1	1	yes	characteristic, complemented
138	C₂×C₃×C₇	—	2	4	no	—
139	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₃×D₇	—	2	4	no	—
143	C₃×D₇	—	4	4	no	—
144	C₂²×D₄	—	7	2	no	—
145	C₂²×D₄	—	7	2	no	—
146	#127	—	7	1	no	—
147	#121	—	7	2	no	—
148	#121	—	7	2	no	—
149	#120	—	7	4	no	—
150	C₄×D₄	—	7	1	no	—
151	C₄×D₄	C₃×D₇	1	1	yes	characteristic, complemented
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	C₂×C₃×D₄	1	4	yes	complemented
155	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
156	C₂×D₇	—	2	2	no	—
157	C₂×D₇	—	2	2	no	—
158	C₂×D₇	—	2	4	no	—
159	C₂×D₇	—	2	4	no	—
160	C₂×D₇	—	2	4	no	—
161	C₂×D₇	—	4	16	no	—
162	C₂×D₇	C₂×C₃×D₄	1	4	yes	complemented
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	C₂×C₃×D₄	1	2	yes	complemented
166	C₄×C₇	C₂×C₃×D₄	1	2	yes	complemented
167	Dic₇	—	2	2	no	—
168	C₂³×C₃	—	7	2	no	—
169	C₂³×C₃	—	7	4	no	—
170	C₂³×C₃	—	14	8	no	—
171	C₂³×C₃	—	14	8	no	—
172	C₂³×C₃	—	28	4	no	—
173	C₂³×C₃	C₂²×D₇	1	2	yes	—
174	C₃×D₄	—	7	4	no	—
175	C₃×D₄	—	14	8	no	—
176	C₃×D₄	—	14	8	no	—
177	C₃×D₄	—	14	4	no	—
178	C₃×D₄	—	14	4	no	—
179	C₃×D₄	—	28	2	no	—
180	C₃×D₄	D₂₈	1	4	yes	direct product factor
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	7	2	no	—
183	C₂×C₃×C₄	—	14	4	no	—
184	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₃×C₇	D₄²	1	1	yes	characteristic, complemented, Hall
189	C₂⁴	—	7	4	no	—
190	C₂×D₄	—	7	4	no	—
191	C₂×D₄	—	7	1	no	—
192	C₂×D₄	—	7	4	no	—
193	C₂×D₄	—	7	2	no	—
194	C₂×D₄	—	7	2	no	—
195	C₂×D₄	—	7	2	no	—
196	C₂×D₄	—	14	8	no	—
197	C₂×D₄	—	14	8	no	—
198	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
199	C₂²×C₄	—	7	2	no	—
200	C₂²×C₄	C₂×C₃×D₇	1	2	yes	complemented
201	C₄⋊C₄	—	7	1	no	—
202	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
203	C₂²⋊C₄	—	7	2	no	—
204	C₂²⋊C₄	—	7	4	no	—
205	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	complemented
206	C₄²	C₂×C₃×D₇	1	1	yes	characteristic, complemented
207	C₂×C₇	—	2	4	no	—
208	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
209	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
211	D₇	—	2	4	no	—
212	D₇	—	4	4	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	7	4	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₂²×C₃	—	14	4	no	—
219	C₂²×C₃	—	14	4	no	—
220	C₂²×C₃	—	14	4	no	—
221	C₂²×C₃	—	28	16	no	—
222	C₂²×C₃	C₂×D₂₈	1	4	yes	complemented
223	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	14	2	no	—
227	C₃×C₄	C₂×D₂₈	1	2	yes	complemented
228	C₃×C₄	D₄×D₇	1	2	yes	complemented
229	C₂³	—	7	2	no	—
230	C₂³	—	7	4	no	—
231	C₂³	—	14	8	no	—
232	C₂³	—	14	8	no	—
233	C₂³	—	28	4	no	—
234	C₂³	C₂²×C₃×D₇	1	2	yes	—
235	D₄	—	7	4	no	—
236	D₄	—	14	8	no	—
237	D₄	—	14	8	no	—
238	D₄	—	14	4	no	—
239	D₄	—	14	4	no	—
240	D₄	—	28	2	no	—
241	D₄	C₃×D₂₈	1	4	yes	direct product factor
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	7	2	no	—
244	C₂×C₄	—	14	4	no	—
245	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
247	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
249	C₇	C₃×D₄²	1	1	yes	characteristic, complemented, Sylow
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	14	4	no	—
252	C₂×C₃	—	28	4	no	—
253	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
254	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
255	C₂×C₃	#6553	1	1	yes	characteristic
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	7	4	no	—
259	C₂²	—	14	2	no	—
260	C₂²	—	14	2	no	—
261	C₂²	—	14	4	no	—
262	C₂²	—	14	4	no	—
263	C₂²	—	14	4	no	—
264	C₂²	—	28	16	no	—
265	C₂²	C₂×C₃×D₂₈	1	4	yes	complemented
266	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	14	2	no	—
270	C₄	C₂×C₃×D₂₈	1	2	yes	complemented
271	C₄	C₃×D₄×D₇	1	2	yes	complemented
272	C₃	D₄×D₂₈	1	1	yes	characteristic, direct product factor, Sylow
273	C₂	—	2	4	no	—
274	C₂	—	14	4	no	—
275	C₂	—	28	4	no	—
276	C₂	C₂²×C₃×D₂₈	1	1	yes	characteristic
277	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
278	C₂	C₃×#6553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×C₄×D₄	D₇
hypocenter	C₇	C₃×D₄²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₄×D₇	C₂	semidirect
2. C₂²×C₃×D₂₈	C₂	semidirect
3. C₃×C₄×C₇×D₄	C₂	semidirect
4. C₃×#6493	C₂	semidirect
5. C₃×#6458	C₂	semidirect
6. C₃×#6445	C₂	semidirect
7. C₃×#6437	C₂	semidirect
8. C₃×C₄×D₂₈	C₂	semidirect
9. D₄×D₂₈	C₃	direct (N × H)
10. C₂³×C₃×D₇	C₂²	semidirect
12. C₂²×C₃×C₄×C₇	C₂²	semidirect
13. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
17. C₂×C₃×D₂₈	C₂²	semidirect
18. C₂×C₃×D₂₈	C₂²	semidirect
19. C₂×C₃×D₂₈	C₂²	semidirect
20. C₂×C₃×D₂₈	C₂²	semidirect
21. C₂×C₃×C₄×D₇	C₂²	semidirect
22. C₃×C₇×C₄⋊C₄	C₂²	semidirect
23. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
24. C₃×C₄²×C₇	C₂²	semidirect
25. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
26. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
27. C₃×C₄⋊Dic₇	C₂²	semidirect
28. C₂×D₄×D₇	C₂×C₃	semidirect
29. C₂²×D₂₈	C₂×C₃	semidirect
30. C₄×C₇×D₄	C₂×C₃	semidirect
31. #6493	C₂×C₃	semidirect
32. #6458	C₂×C₃	semidirect
33. #6445	C₂×C₃	semidirect
34. #6437	C₂×C₃	semidirect
35. C₄×D₂₈	C₂×C₃	semidirect
43. C₃×C₇×D₄	D₄	semidirect
55. C₃×D₂₈	D₄	direct (N × H)
57. C₂³×D₇	C₂²×C₃	semidirect
59. C₂²×C₄×C₇	C₂²×C₃	semidirect
60. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
64. C₂×D₂₈	C₂²×C₃	semidirect
65. C₂×D₂₈	C₂²×C₃	semidirect
66. C₂×D₂₈	C₂²×C₃	semidirect
67. C₂×D₂₈	C₂²×C₃	semidirect
68. C₂×C₄×D₇	C₂²×C₃	semidirect
69. C₇×C₄⋊C₄	C₂²×C₃	semidirect
70. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
71. C₄²×C₇	C₂²×C₃	semidirect
72. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
73. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
74. C₄⋊Dic₇	C₂²×C₃	semidirect
82. C₃×C₄×D₄	D₇	semidirect
85. C₂²×C₃×C₇	C₂×D₄	semidirect
93. C₂×C₃×D₇	C₂×D₄	semidirect
96. C₃×C₄×C₇	C₂×D₄	semidirect
97. C₃×C₄×C₇	C₂×D₄	semidirect
106. C₇×D₄	C₃×D₄	semidirect
118. D₂₈	C₃×D₄	direct (N × H)
131. C₂²×C₃×C₄	C₂×D₇	semidirect
133. C₃×C₄⋊C₄	C₂×D₇	semidirect
136. C₃×C₂²⋊C₄	C₂×D₇	semidirect
137. C₃×C₄²	C₂×D₇	semidirect
151. C₄×D₄	C₃×D₇	semidirect
154. C₂²×C₇	C₂×C₃×D₄	semidirect
162. C₂×D₇	C₂×C₃×D₄	semidirect
165. C₄×C₇	C₂×C₃×D₄	semidirect
166. C₄×C₇	C₂×C₃×D₄	semidirect
180. C₃×D₄	D₂₈	direct (N × H)
188. C₃×C₇	D₄²	semidirect
200. C₂²×C₄	C₂×C₃×D₇	semidirect
202. C₄⋊C₄	C₂×C₃×D₇	semidirect
205. C₂²⋊C₄	C₂×C₃×D₇	semidirect
206. C₄²	C₂×C₃×D₇	semidirect
222. C₂²×C₃	C₂×D₂₈	semidirect
227. C₃×C₄	C₂×D₂₈	semidirect
228. C₃×C₄	D₄×D₇	semidirect
241. D₄	C₃×D₂₈	direct (N × H)
249. C₇	C₃×D₄²	semidirect
265. C₂²	C₂×C₃×D₂₈	semidirect
270. C₄	C₂×C₃×D₂₈	semidirect
271. C₄	C₃×D₄×D₇	semidirect
272. C₃	D₄×D₂₈	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×C₃×D₄×D₇	2
C₂²×C₃×D₂₈	2
C₃×C₄×C₇×D₄	2
C₃×#6493	2
C₃×#6458	2
C₃×#6445	2
C₃×#6437	2
C₃×C₄×D₂₈	2
D₄×D₂₈	3
C₃×D₄²	7

Subgroups — C3×D4×D28 #13060894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×D₂₈ #13060894