Subgroups — D₄×D₈₄ #13061642

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×D₄×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₂²×D₈₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621413	C₂	1	2	yes	maximal, complemented
5	#10621378	C₂	1	2	yes	maximal, complemented
6	#10621365	C₂	1	4	yes	maximal, complemented
7	#10621357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×D₂₈	—	3	1	no	maximal, Hall
10	C₂³×D₂₁	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×#4268	C₂²	1	4	yes	complemented
14	D₄×D₂₁	—	2	8	no	—
15	C₂×D₈₄	—	2	8	no	—
16	C₂×D₈₄	C₂²	1	1	yes	characteristic
17	C₂×D₈₄	C₂²	1	2	yes	complemented
18	C₂×D₈₄	C₂²	1	2	yes	complemented
19	C₂×D₈₄	C₂²	1	4	yes	complemented
20	C₂×D₈₄	C₂²	1	2	yes	complemented
21	C₂×C₄×D₂₁	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	#66631	C₂²	1	2	yes	complemented
26	#66631	C₂²	1	4	yes	complemented
27	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic, complemented
28	C₂×D₄×D₇	—	3	2	no	—
29	C₂²×D₂₈	—	3	2	no	—
30	C₄×C₇×D₄	D₃	1	1	yes	characteristic, complemented
31	#6493	—	3	2	no	—
32	#6458	—	3	2	no	—
33	#6445	—	3	4	no	—
34	#6437	—	3	1	no	—
35	C₄×D₂₈	—	3	1	no	—
36	D₄×D₁₂	—	7	1	no	maximal, Hall
37	C₂³×C₃×C₇	C₂³	1	2	yes	—
38	C₂²×D₂₁	—	2	8	no	—
39	C₂²×D₂₁	—	2	8	no	—
40	C₂²×D₂₁	—	4	4	no	—
41	C₂²×D₂₁	C₂³	1	4	yes	—
42	C₂²×D₂₁	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	#4268	—	2	8	no	—
50	C₂×Dic₂₁	C₂³	1	2	yes	—
51	D₈₄	—	2	8	no	—
52	D₈₄	—	2	4	no	—
53	D₈₄	—	2	4	no	—
54	D₈₄	—	4	2	no	—
55	D₈₄	D₄	1	4	yes	direct product factor
56	C₄×D₂₁	—	2	4	no	—
57	C₂³×D₇	—	3	4	no	—
58	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₃	1	2	yes	complemented
60	C₂×C₂²⋊D₇	—	3	4	no	—
61	D₄×D₇	—	6	8	no	—
62	C₂×D₂₈	—	3	1	no	—
63	C₂×D₂₈	—	3	2	no	—
64	C₂×D₂₈	—	3	2	no	—
65	C₂×D₂₈	—	3	4	no	—
66	C₂×D₂₈	—	3	2	no	—
67	C₂×D₂₈	—	6	8	no	—
68	C₂×C₄×D₇	—	3	2	no	—
69	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
70	C₇×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
71	C₄²×C₇	C₂×D₃	1	1	yes	characteristic, complemented
72	(C₂×D₇)⋊C₄	—	3	2	no	—
73	(C₂×D₇)⋊C₄	—	3	4	no	—
74	C₄⋊Dic₇	—	3	1	no	—
75	C₂×D₃×D₄	—	7	2	no	—
76	C₂²×D₁₂	—	7	2	no	—
77	C₃×C₄×D₄	D₇	1	1	yes	characteristic, complemented
78	#930	—	7	2	no	—
79	#895	—	7	2	no	—
80	#882	—	7	4	no	—
81	#874	—	7	1	no	—
82	C₄×D₁₂	—	7	1	no	—
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×D₂₁	—	2	2	no	—
88	C₂×D₂₁	—	2	2	no	—
89	C₂×D₂₁	—	2	4	no	—
90	C₂×D₂₁	—	2	4	no	—
91	C₂×D₂₁	—	2	4	no	—
92	C₂×D₂₁	—	4	16	no	—
93	C₂×D₂₁	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	Dic₂₁	—	2	2	no	—
99	D₄²	—	21	1	no	Sylow
100	C₂³×C₇	C₂²×D₃	1	2	yes	—
101	C₂²×D₇	—	3	2	no	—
102	C₂²×D₇	—	3	4	no	—
103	C₂²×D₇	—	6	8	no	—
104	C₂²×D₇	—	6	8	no	—
105	C₂²×D₇	—	12	4	no	—
106	C₇×D₄	D₁₂	1	4	yes	complemented
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
110	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
112	C₂²⋊D₇	—	6	8	no	—
113	C₂×Dic₇	—	3	2	no	—
114	D₂₈	—	3	4	no	—
115	D₂₈	—	6	8	no	—
116	D₂₈	—	6	4	no	—
117	D₂₈	—	6	4	no	—
118	D₂₈	—	12	2	no	—
119	C₄×D₇	—	6	4	no	—
120	C₂³×D₃	—	7	4	no	—
121	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×D₇	1	2	yes	complemented
123	C₂×#67	—	7	4	no	—
124	D₃×D₄	—	14	8	no	—
125	C₂×D₁₂	—	7	1	no	—
126	C₂×D₁₂	—	7	2	no	—
127	C₂×D₁₂	—	7	2	no	—
128	C₂×D₁₂	—	7	4	no	—
129	C₂×D₁₂	—	7	2	no	—
130	C₂×D₁₂	—	14	8	no	—
131	C₂×C₄×D₃	—	7	2	no	—
132	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic, complemented
133	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	complemented
134	C₃×C₄²	C₂×D₇	1	1	yes	characteristic, complemented
135	(C₂×D₃)⋊C₄	—	7	2	no	—
136	(C₂×D₃)⋊C₄	—	7	4	no	—
137	C₄⋊Dic₃	—	7	1	no	—
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	D₂₁	—	2	4	no	—
143	D₂₁	—	4	4	no	—
144	C₂²×D₄	—	21	2	no	—
145	C₂²×D₄	—	21	2	no	—
146	#127	—	21	1	no	—
147	#121	—	21	2	no	—
148	#121	—	21	2	no	—
149	#120	—	21	4	no	—
150	C₄×D₄	—	21	1	no	—
151	C₄×D₄	D₂₁	1	1	yes	characteristic, complemented
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	C₂×D₁₂	1	4	yes	complemented
155	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
156	C₂×D₇	—	3	4	no	—
157	C₂×D₇	—	6	2	no	—
158	C₂×D₇	—	6	2	no	—
159	C₂×D₇	—	6	4	no	—
160	C₂×D₇	—	6	4	no	—
161	C₂×D₇	—	6	4	no	—
162	C₂×D₇	—	12	16	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	C₂×D₁₂	1	2	yes	complemented
166	C₄×C₇	D₃×D₄	1	2	yes	complemented
167	Dic₇	—	6	2	no	—
168	C₂³×C₃	C₂²×D₇	1	2	yes	—
169	C₂²×D₃	—	7	2	no	—
170	C₂²×D₃	—	7	4	no	—
171	C₂²×D₃	—	14	8	no	—
172	C₂²×D₃	—	14	8	no	—
173	C₂²×D₃	—	28	4	no	—
174	C₃×D₄	D₂₈	1	4	yes	complemented
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
178	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
180	#67	—	14	8	no	—
181	C₂×Dic₃	—	7	2	no	—
182	D₁₂	—	7	4	no	—
183	D₁₂	—	14	8	no	—
184	D₁₂	—	14	4	no	—
185	D₁₂	—	14	4	no	—
186	D₁₂	—	28	2	no	—
187	C₄×D₃	—	14	4	no	—
188	C₃×C₇	D₄²	1	1	yes	characteristic, complemented, Hall
189	C₂⁴	—	21	4	no	—
190	C₂×D₄	—	21	4	no	—
191	C₂×D₄	—	21	1	no	—
192	C₂×D₄	—	21	4	no	—
193	C₂×D₄	—	21	2	no	—
194	C₂×D₄	—	21	2	no	—
195	C₂×D₄	—	21	2	no	—
196	C₂×D₄	—	42	8	no	—
197	C₂×D₄	—	42	8	no	—
198	C₂×D₄	C₂×D₂₁	1	1	yes	characteristic
199	C₂²×C₄	—	21	2	no	—
200	C₂²×C₄	C₂×D₂₁	1	2	yes	complemented
201	C₄⋊C₄	—	21	1	no	—
202	C₄⋊C₄	C₂×D₂₁	1	1	yes	characteristic, complemented
203	C₂²⋊C₄	—	21	2	no	—
204	C₂²⋊C₄	—	21	4	no	—
205	C₂²⋊C₄	C₂×D₂₁	1	2	yes	complemented
206	C₄²	C₂×D₂₁	1	1	yes	characteristic, complemented
207	C₂×C₇	—	2	4	no	—
208	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
209	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
210	C₂×C₇	#1009	1	1	yes	characteristic
211	D₇	—	6	4	no	—
212	D₇	—	12	4	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	C₂×D₂₈	1	4	yes	complemented
216	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
217	C₂×D₃	—	7	4	no	—
218	C₂×D₃	—	14	2	no	—
219	C₂×D₃	—	14	2	no	—
220	C₂×D₃	—	14	4	no	—
221	C₂×D₃	—	14	4	no	—
222	C₂×D₃	—	14	4	no	—
223	C₂×D₃	—	28	16	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	C₂×D₂₈	1	2	yes	complemented
227	C₃×C₄	D₄×D₇	1	2	yes	complemented
228	Dic₃	—	14	2	no	—
229	C₂³	—	21	2	no	—
230	C₂³	—	21	4	no	—
231	C₂³	—	42	8	no	—
232	C₂³	—	42	8	no	—
233	C₂³	—	84	4	no	—
234	C₂³	C₂²×D₂₁	1	2	yes	—
235	D₄	—	21	4	no	—
236	D₄	—	42	8	no	—
237	D₄	—	42	8	no	—
238	D₄	—	42	4	no	—
239	D₄	—	42	4	no	—
240	D₄	—	84	2	no	—
241	D₄	D₈₄	1	4	yes	direct product factor
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	21	2	no	—
244	C₂×C₄	—	42	4	no	—
245	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₂₁	1	2	yes	—
247	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
249	C₇	D₄×D₁₂	1	1	yes	characteristic, complemented, Sylow
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
252	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
253	C₂×C₃	#6553	1	1	yes	characteristic
254	D₃	—	14	4	no	—
255	D₃	—	28	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	21	4	no	—
259	C₂²	—	42	2	no	—
260	C₂²	—	42	2	no	—
261	C₂²	—	42	4	no	—
262	C₂²	—	42	4	no	—
263	C₂²	—	42	4	no	—
264	C₂²	—	84	16	no	—
265	C₂²	C₂×D₈₄	1	4	yes	complemented
266	C₂²	C₂³×D₂₁	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	42	2	no	—
270	C₄	C₂×D₈₄	1	2	yes	complemented
271	C₄	D₄×D₂₁	1	2	yes	complemented
272	C₃	D₄×D₂₈	1	1	yes	characteristic, complemented, Sylow
273	C₂	—	2	4	no	—
274	C₂	—	42	4	no	—
275	C₂	—	84	4	no	—
276	C₂	C₂²×D₈₄	1	1	yes	characteristic
277	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
278	C₂	#10621662	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
hypercenter	C₄×D₄	D₂₁
hypocenter	C₃×C₇	D₄²
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. C₂×D₄×D₂₁	C₂	semidirect
2. C₂²×D₈₄	C₂	semidirect
3. C₃×C₄×C₇×D₄	C₂	semidirect
4. #10621413	C₂	semidirect
5. #10621378	C₂	semidirect
6. #10621365	C₂	semidirect
7. #10621357	C₂	semidirect
8. C₄×D₈₄	C₂	semidirect
10. C₂³×D₂₁	C₂²	semidirect
12. C₂²×C₃×C₄×C₇	C₂²	semidirect
13. C₂×#4268	C₂²	semidirect
17. C₂×D₈₄	C₂²	semidirect
18. C₂×D₈₄	C₂²	semidirect
19. C₂×D₈₄	C₂²	semidirect
20. C₂×D₈₄	C₂²	semidirect
21. C₂×C₄×D₂₁	C₂²	semidirect
22. C₃×C₇×C₄⋊C₄	C₂²	semidirect
23. C₃×C₇×C₂²⋊C₄	C₂²	semidirect
24. C₃×C₄²×C₇	C₂²	semidirect
25. #66631	C₂²	semidirect
26. #66631	C₂²	semidirect
27. C₄⋊Dic₂₁	C₂²	semidirect
30. C₄×C₇×D₄	D₃	semidirect
43. C₃×C₇×D₄	D₄	semidirect
55. D₈₄	D₄	direct (N × H)
59. C₂²×C₄×C₇	C₂×D₃	semidirect
69. C₇×C₄⋊C₄	C₂×D₃	semidirect
70. C₇×C₂²⋊C₄	C₂×D₃	semidirect
71. C₄²×C₇	C₂×D₃	semidirect
77. C₃×C₄×D₄	D₇	semidirect
85. C₂²×C₃×C₇	C₂×D₄	semidirect
93. C₂×D₂₁	C₂×D₄	semidirect
96. C₃×C₄×C₇	C₂×D₄	semidirect
97. C₃×C₄×C₇	C₂×D₄	semidirect
106. C₇×D₄	D₁₂	semidirect
122. C₂²×C₃×C₄	C₂×D₇	semidirect
132. C₃×C₄⋊C₄	C₂×D₇	semidirect
133. C₃×C₂²⋊C₄	C₂×D₇	semidirect
134. C₃×C₄²	C₂×D₇	semidirect
151. C₄×D₄	D₂₁	semidirect
154. C₂²×C₇	C₂×D₁₂	semidirect
165. C₄×C₇	C₂×D₁₂	semidirect
166. C₄×C₇	D₃×D₄	semidirect
174. C₃×D₄	D₂₈	semidirect
188. C₃×C₇	D₄²	semidirect
200. C₂²×C₄	C₂×D₂₁	semidirect
202. C₄⋊C₄	C₂×D₂₁	semidirect
205. C₂²⋊C₄	C₂×D₂₁	semidirect
206. C₄²	C₂×D₂₁	semidirect
215. C₂²×C₃	C₂×D₂₈	semidirect
226. C₃×C₄	C₂×D₂₈	semidirect
227. C₃×C₄	D₄×D₇	semidirect
241. D₄	D₈₄	direct (N × H)
249. C₇	D₄×D₁₂	semidirect
265. C₂²	C₂×D₈₄	semidirect
270. C₄	C₂×D₈₄	semidirect
271. C₄	D₄×D₂₁	semidirect
272. C₃	D₄×D₂₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₂₁	2
C₂²×D₈₄	2
C₃×C₄×C₇×D₄	2
#10621413	2
#10621378	2
#10621365	2
#10621357	2
C₄×D₈₄	2
D₄×D₂₈	3
D₄×D₁₂	7

Subgroups — D4×D84 #13061642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₈₄ #13061642