Subgroups — C₂²×C₈×D₂₈ #14217629

Order 1792.

3694 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×(C₂×D₇)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₂₈	C₂	1	24	yes	maximal, direct product factor
7	C₂³×D₂₈	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₇	C₂²	1	2	yes	—
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₂²×(C₂×D₇)⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄⋊Dic₇	C₄	1	1	yes	characteristic
14	C₂³×C₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₈×D₇	—	2	16	no	—
16	C₂²×C₈×D₇	C₂²	1	24	yes	complemented
17	C₂×C₄×D₂₈	C₄	1	12	yes	—
18	C₂×C₄×D₂₈	C₂²	1	12	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	12	yes	complemented
20	C₂×(C₂×D₇)⋊C₈	C₂²	1	24	yes	complemented
21	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	12	yes	complemented
22	C₈×D₂₈	C₂²	1	64	yes	direct product factor
23	C₂²×C₈×D₄	—	7	1	no	maximal, Sylow
24	C₂⁴×D₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂³×Dic₇	C₂×C₄	1	2	yes	—
29	C₂²×D₂₈	C₈	1	16	yes	direct product factor
30	C₂²×D₂₈	C₂×C₄	1	12	yes	—
31	C₂²×C₄×D₇	—	2	8	no	—
32	C₂²×C₄×D₇	—	2	8	no	—
33	C₂²×C₄×D₇	C₂×C₄	1	12	yes	—
34	C₂²×C₄×D₇	C₂³	1	12	yes	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	D₄	1	8	yes	complemented
37	C₂²×C₇×C₈	C₂³	1	6	yes	—
38	C₂²×C₇×C₈	C₂³	1	6	yes	—
39	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₇	C₂³	1	6	yes	—
41	C₂×(C₂×D₇)⋊C₄	C₂×C₄	1	24	yes	—
42	C₂×C₄⋊Dic₇	C₂×C₄	1	12	yes	—
43	C₂²×C₇⋊C₈	—	2	8	no	—
44	C₂²×C₇⋊C₈	C₂³	1	12	yes	—
45	C₂×C₈×D₇	—	2	96	no	—
46	C₂×C₈×D₇	C₂³	1	32	yes	complemented
47	C₄×D₂₈	C₂×C₄	1	48	yes	—
48	C₄×D₂₈	C₂³	1	16	yes	—
49	C₄×C₇×C₈	C₂³	1	16	yes	complemented
50	(C₂×D₇)⋊C₈	C₂³	1	32	yes	complemented
51	C₄⋊(C₇⋊C₈)	C₂³	1	16	yes	complemented
52	C₂⁴×C₈	—	7	2	no	—
53	C₂²×C₄×D₄	—	7	1	no	—
54	C₂×C₈×D₄	—	7	24	no	—
55	C₂²×C₄⋊C₈	—	7	1	no	—
56	C₂²×C₂²⋊C₈	—	7	2	no	—
57	C₂²×C₄×C₈	D₇	1	1	yes	characteristic, complemented
58	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂³×D₇	—	2	8	no	—
60	C₂³×D₇	—	2	8	no	—
61	C₂³×D₇	C₂×C₈	1	16	yes	complemented
62	C₂³×D₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	C₂×C₈	1	8	yes	complemented
65	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₇	#41	1	4	yes	—
70	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
71	C₂²×Dic₇	—	2	8	no	—
72	C₂²×Dic₇	C₂²×C₄	1	12	yes	—
73	C₂×D₂₈	C₂×C₈	1	96	yes	direct product factor
74	C₂×D₂₈	C₂²×C₄	1	16	yes	—
75	C₂×C₄×D₇	—	2	48	no	—
76	C₂×C₄×D₇	—	2	24	no	—
77	C₂×C₄×D₇	—	2	24	no	—
78	C₂×C₄×D₇	C₂²×C₄	1	24	yes	—
79	C₂×C₄×D₇	C₂⁴	1	8	yes	—
80	C₂×C₇×C₈	—	2	12	no	—
81	C₂×C₇×C₈	C₂×D₄	1	24	yes	complemented
82	C₂×C₇×C₈	C₂⁴	1	4	yes	—
83	C₂×C₇×C₈	C₂⁴	1	4	yes	—
84	C₄²×C₇	C₂²×C₄	1	12	yes	—
85	C₄²×C₇	C₂⁴	1	4	yes	—
86	(C₂×D₇)⋊C₄	C₂²×C₄	1	32	yes	—
87	C₄⋊Dic₇	C₂²×C₄	1	16	yes	—
88	C₂×C₇⋊C₈	—	2	24	no	—
89	C₂×C₇⋊C₈	C₂⁴	1	8	yes	—
90	C₈×D₇	—	2	64	no	—
91	C₂³×D₄	—	7	1	no	—
92	C₂⁴×C₄	—	7	2	no	—
93	C₂³×C₈	—	7	2	no	—
94	C₂³×C₈	—	7	24	no	—
95	C₂³×C₈	—	14	16	no	—
96	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
97	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
98	C₂×C₄×D₄	—	7	12	no	—
99	C₂×C₄×D₄	—	7	12	no	—
100	C₂²×C₄⋊C₄	—	7	1	no	—
101	C₂²×C₂²⋊C₄	—	7	2	no	—
102	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
103	C₈×D₄	—	7	64	no	—
104	C₂×C₄⋊C₈	—	7	12	no	—
105	C₂×C₂²⋊C₈	—	7	24	no	—
106	C₂×C₄×C₈	C₂×D₇	1	12	yes	complemented
107	C₂³×C₇	C₄×D₄	1	4	yes	—
108	C₂³×C₇	C₂²×C₈	1	4	yes	—
109	C₂³×C₇	#131	1	4	yes	—
110	C₂³×C₇	C₂³×C₄	1	3	yes	—
111	C₂²×D₇	—	2	24	no	—
112	C₂²×D₇	—	2	24	no	—
113	C₂²×D₇	—	2	64	no	—
114	C₂²×D₇	C₂²×C₈	1	48	yes	complemented
115	C₂²×D₇	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₇	—	2	12	no	—
117	C₂×C₄×C₇	C₄×D₄	1	12	yes	—
118	C₂×C₄×C₇	C₂²×C₈	1	24	yes	complemented
119	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
120	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₇	C₂×#41	1	6	yes	—
124	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
125	C₂×Dic₇	—	2	24	no	—
126	C₂×Dic₇	C₂³×C₄	1	8	yes	—
127	D₂₈	C₂²×C₈	1	64	yes	direct product factor
128	C₄×D₇	—	2	48	no	—
129	C₄×D₇	—	2	8	no	—
130	C₄×D₇	—	2	8	no	—
131	C₇×C₈	—	2	4	no	—
132	C₇×C₈	C₂²×D₄	1	8	yes	complemented
133	C₇⋊C₈	—	2	8	no	—
134	C₂⁵	—	7	2	no	—
135	C₂²×D₄	—	7	16	no	—
136	C₂²×D₄	—	7	12	no	—
137	C₂³×C₄	—	7	2	no	—
138	C₂³×C₄	—	7	12	no	—
139	C₂³×C₄	—	7	12	no	—
140	C₂³×C₄	—	14	8	no	—
141	C₂³×C₄	—	14	8	no	—
142	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
143	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
144	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	7	12	no	—
147	C₂²×C₈	—	7	32	no	—
148	C₂²×C₈	—	14	8	no	—
149	C₂²×C₈	—	14	96	no	—
150	C₂²×C₈	D₂₈	1	8	yes	direct product factor
151	C₂²×C₈	C₂²×D₇	1	6	yes	—
152	C₂²×C₈	C₂²×D₇	1	6	yes	—
153	C₄×D₄	—	7	48	no	—
154	C₄×D₄	—	7	16	no	—
155	C₂×C₄⋊C₄	—	7	12	no	—
156	C₂×C₂²⋊C₄	—	7	24	no	—
157	C₂×C₄²	C₄×D₇	1	6	yes	—
158	C₂×C₄²	C₂²×D₇	1	6	yes	—
159	C₄⋊C₈	—	7	16	no	—
160	C₂²⋊C₈	—	7	32	no	—
161	C₄×C₈	C₂²×D₇	1	16	yes	complemented
162	C₂²×C₇	C₈×D₄	1	16	yes	complemented
163	C₂²×C₇	C₂×C₄×D₄	1	6	yes	—
164	C₂²×C₇	C₂³×C₈	1	6	yes	—
165	C₂²×C₇	C₂×#131	1	6	yes	—
166	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
167	C₂×D₇	—	2	8	no	—
168	C₂×D₇	—	2	96	no	—
169	C₂×D₇	—	2	8	no	—
170	C₂×D₇	C₂³×C₈	1	16	yes	complemented
171	C₄×C₇	—	2	4	no	—
172	C₄×C₇	C₂×C₄×D₄	1	6	yes	—
173	C₄×C₇	C₂³×C₈	1	8	yes	complemented
174	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
175	C₄×C₇	C₂²×#41	1	1	yes	characteristic
176	Dic₇	—	2	8	no	—
177	C₂⁴	—	7	16	no	—
178	C₂⁴	—	7	12	no	—
179	C₂⁴	—	14	8	no	—
180	C₂⁴	—	14	8	no	—
181	C₂⁴	C₂×C₄×D₇	1	1	yes	characteristic
182	C₂×D₄	—	7	96	no	—
183	C₂×D₄	—	7	16	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	7	12	no	—
186	C₂²×C₄	—	7	24	no	—
187	C₂²×C₄	—	7	8	no	—
188	C₂²×C₄	—	14	8	no	—
189	C₂²×C₄	—	14	48	no	—
190	C₂²×C₄	—	14	24	no	—
191	C₂²×C₄	—	14	24	no	—
192	C₂²×C₄	C₈×D₇	1	8	yes	complemented
193	C₂²×C₄	C₂×C₄×D₇	1	6	yes	—
194	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
195	C₂²×C₄	C₂×C₄×D₇	1	6	yes	—
196	C₂²×C₄	C₂×D₂₈	1	4	yes	—
197	C₂²×C₄	#1157	1	4	yes	—
198	C₂²×C₄	C₂³×D₇	1	3	yes	—
199	C₂×C₈	—	2	12	no	—
200	C₂×C₈	—	7	8	no	—
201	C₂×C₈	—	14	24	no	—
202	C₂×C₈	—	14	64	no	—
203	C₂×C₈	C₂×D₂₈	1	24	yes	direct product factor
204	C₂×C₈	C₂³×D₇	1	4	yes	—
205	C₂×C₈	C₂³×D₇	1	4	yes	—
206	C₄⋊C₄	—	7	16	no	—
207	C₂²⋊C₄	—	7	32	no	—
208	C₄²	C₂×C₄×D₇	1	12	yes	—
209	C₄²	C₂³×D₇	1	4	yes	—
210	C₂×C₇	C₂×C₈×D₄	1	12	yes	complemented
211	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
212	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
213	C₂×C₇	C₂²×#131	1	1	yes	characteristic
214	D₇	—	2	16	no	—
215	C₂³	—	7	48	no	—
216	C₂³	—	7	8	no	—
217	C₂³	—	14	24	no	—
218	C₂³	—	14	24	no	—
219	C₂³	—	14	64	no	—
220	C₂³	C₄×D₂₈	1	4	yes	—
221	C₂³	C₂×C₈×D₇	1	4	yes	—
222	C₂³	#6464	1	4	yes	—
223	C₂³	C₂²×C₄×D₇	1	3	yes	—
224	D₄	—	7	64	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	7	8	no	—
227	C₂×C₄	—	14	24	no	—
228	C₂×C₄	—	14	48	no	—
229	C₂×C₄	—	14	8	no	—
230	C₂×C₄	—	14	8	no	—
231	C₂×C₄	C₄×D₂₈	1	12	yes	—
232	C₂×C₄	C₂×C₈×D₇	1	24	yes	complemented
233	C₂×C₄	C₂²×C₄×D₇	1	4	yes	—
234	C₂×C₄	C₂²×C₄×D₇	1	3	yes	—
235	C₂×C₄	C₂²×C₄×D₇	1	4	yes	—
236	C₂×C₄	C₂²×D₂₈	1	6	yes	—
237	C₂×C₄	C₂×#1157	1	6	yes	—
238	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	14	8	no	—
241	C₈	C₂²×D₂₈	1	8	yes	direct product factor
242	C₇	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	7	16	no	—
244	C₂²	—	14	8	no	—
245	C₂²	—	14	96	no	—
246	C₂²	—	14	8	no	—
247	C₂²	C₈×D₂₈	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₂₈	1	6	yes	—
249	C₂²	C₂²×C₈×D₇	1	6	yes	—
250	C₂²	C₂×#6464	1	6	yes	—
251	C₂²	C₂³×C₄×D₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	14	8	no	—
254	C₄	C₂×C₄×D₂₈	1	6	yes	—
255	C₄	C₂²×C₈×D₇	1	8	yes	complemented
256	C₄	C₂³×D₂₈	1	1	yes	characteristic
257	C₄	C₂²×#1157	1	1	yes	characteristic
258	C₂	—	14	16	no	—
259	C₂	C₂×C₈×D₂₈	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₂₈	1	1	yes	characteristic
261	C₂	C₂³×C₈×D₇	1	1	yes	characteristic
262	C₂	C₂²×#6464	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇
commutator	C₂×C₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂²×C₄×C₇×C₈	C₂
socle	C₂⁴×C₇	C₂²×C₄
prime radical	C₂³×D₂₈	C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	C₂²×C₈×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×D₇	C₂	semidirect
3. C₂²×C₄×C₇×C₈	C₂	semidirect
4. C₂²×(C₂×D₇)⋊C₈	C₂	semidirect
5. C₂²×C₄⋊(C₇⋊C₈)	C₂	semidirect
6. C₂×C₈×D₂₈	C₂	direct (N × H)
16. C₂²×C₈×D₇	C₂²	semidirect
19. C₂×C₄×C₇×C₈	C₂²	semidirect
20. C₂×(C₂×D₇)⋊C₈	C₂²	semidirect
21. C₂×C₄⋊(C₇⋊C₈)	C₂²	semidirect
22. C₈×D₂₈	C₂²	direct (N × H)
29. C₂²×D₂₈	C₈	direct (N × H)
36. C₂²×C₇×C₈	D₄	semidirect
46. C₂×C₈×D₇	C₂³	semidirect
49. C₄×C₇×C₈	C₂³	semidirect
50. (C₂×D₇)⋊C₈	C₂³	semidirect
51. C₄⋊(C₇⋊C₈)	C₂³	semidirect
57. C₂²×C₄×C₈	D₇	semidirect
61. C₂³×D₇	C₂×C₈	semidirect
64. C₂²×C₄×C₇	C₂×C₈	semidirect
73. C₂×D₂₈	C₂×C₈	direct (N × H)
81. C₂×C₇×C₈	C₂×D₄	semidirect
106. C₂×C₄×C₈	C₂×D₇	semidirect
114. C₂²×D₇	C₂²×C₈	semidirect
118. C₂×C₄×C₇	C₂²×C₈	semidirect
127. D₂₈	C₂²×C₈	direct (N × H)
132. C₇×C₈	C₂²×D₄	semidirect
150. C₂²×C₈	D₂₈	direct (N × H)
161. C₄×C₈	C₂²×D₇	semidirect
162. C₂²×C₇	C₈×D₄	semidirect
170. C₂×D₇	C₂³×C₈	semidirect
173. C₄×C₇	C₂³×C₈	semidirect
192. C₂²×C₄	C₈×D₇	semidirect
203. C₂×C₈	C₂×D₂₈	direct (N × H)
210. C₂×C₇	C₂×C₈×D₄	semidirect
232. C₂×C₄	C₂×C₈×D₇	semidirect
241. C₈	C₂²×D₂₈	direct (N × H)
242. C₇	C₂²×C₈×D₄	semidirect
247. C₂²	C₈×D₂₈	direct (N × H)
255. C₄	C₂²×C₈×D₇	semidirect
259. C₂	C₂×C₈×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
C₂³×C₈×D₇	2
C₂²×C₄×D₂₈	2
C₂²×C₄×C₇×C₈	2
C₂²×(C₂×D₇)⋊C₈	2
C₂²×C₄⋊(C₇⋊C₈)	2
C₂×C₈×D₂₈	2
C₂²×C₈×D₄	7

Subgroups — C22×C8×D28 #14217629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×D₂₈ #14217629