Subgroups — D₄×#16864638 #46845101

Order 34688.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864638	C₂	1	2	yes	maximal, complemented
2	#31773975	C₂	1	2	yes	maximal, complemented
3	#31773969	C₂	1	4	yes	maximal, complemented
4	D₄×C₂₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	#31773921	C₂	1	1	yes	maximal, characteristic, complemented
6	#31773861	C₂	1	2	yes	maximal, complemented
7	C₄×#16864638	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#16864638	—	2	8	no	—
10	C₂×#16864638	C₂²	1	1	yes	characteristic
11	C₂×#16864638	C₂²	1	2	yes	—
12	C₂×#16864638	C₂²	1	1	yes	characteristic
13	C₂×#16864638	C₂²	1	4	yes	complemented
14	C₂²×C₂₇₁⋊C₈	C₂²	1	4	yes	complemented
15	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	complemented
16	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	complemented
17	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	complemented
18	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	complemented
19	#22576071	C₂²	1	2	yes	complemented
20	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	complemented
21	C₂²×D₄×C₂₇₁	C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₇₁	C₂²	1	2	yes	—
23	C₄×D₄×C₂₇₁	C₄	1	2	yes	—
24	C₄×D₄×C₂₇₁	C₄	1	1	yes	characteristic
25	C₄×D₄×C₂₇₁	C₄	1	1	yes	characteristic
26	C₄×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₇₁	C₂²	1	2	yes	—
28	C₄×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₇₁	C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₂₇₁	C₄	1	2	yes	—
31	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
32	#16864638	—	2	8	no	—
33	#16864638	—	2	4	no	—
34	#16864638	—	2	2	no	—
35	#16864638	D₄	1	4	yes	direct product factor
36	C₂×C₂₇₁⋊C₈	—	2	8	no	—
37	C₂×C₂₇₁⋊C₈	C₂³	1	4	yes	—
38	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
39	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
40	C₂⁴×C₂₇₁	C₂×C₄	1	2	yes	—
41	C₂×D₄×C₂₇₁	—	2	4	no	—
42	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₇₁	—	2	4	no	—
46	C₂²×C₄×C₂₇₁	—	2	4	no	—
47	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
50	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
62	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
65	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
66	C₂₇₁⋊C₈	—	2	4	no	—
67	C₂₇₁⋊C₈	—	2	2	no	—
68	C₂₇₁⋊C₈	C₂×D₄	1	4	yes	complemented
69	C₂³×C₂₇₁	—	2	4	no	—
70	C₂³×C₂₇₁	—	2	4	no	—
71	C₂³×C₂₇₁	—	4	2	no	—
72	C₂³×C₂₇₁	C₂²×C₄	1	2	yes	—
73	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₂₇₁	C₂²×C₄	1	2	yes	—
75	D₄×C₂₇₁	—	2	4	no	—
76	D₄×C₂₇₁	—	2	2	no	—
77	D₄×C₂₇₁	M₁₆	1	4	yes	complemented
78	C₂×C₄×C₂₇₁	—	2	4	no	—
79	C₂×C₄×C₂₇₁	—	2	4	no	—
80	C₂×C₄×C₂₇₁	—	2	4	no	—
81	C₂×C₄×C₂₇₁	—	2	2	no	—
82	C₂×C₄×C₂₇₁	—	2	2	no	—
83	C₂×C₄×C₂₇₁	—	2	2	no	—
84	C₂×C₄×C₂₇₁	—	2	2	no	—
85	C₂×C₄×C₂₇₁	—	2	2	no	—
86	C₂×C₄×C₂₇₁	—	4	1	no	—
87	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₇₁	#41	1	2	yes	—
96	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
97	C₂²×C₂₇₁	—	2	2	no	—
98	C₂²×C₂₇₁	—	2	1	no	—
99	C₂²×C₂₇₁	—	2	1	no	—
100	C₂²×C₂₇₁	—	2	2	no	—
101	C₂²×C₂₇₁	—	2	2	no	—
102	C₂²×C₂₇₁	—	2	2	no	—
103	C₂²×C₂₇₁	—	2	2	no	—
104	C₂²×C₂₇₁	—	4	8	no	—
105	C₂²×C₂₇₁	C₄×D₄	1	2	yes	—
106	C₂²×C₂₇₁	C₂×M₁₆	1	4	yes	complemented
107	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₂₇₁	—	2	2	no	—
109	C₄×C₂₇₁	—	2	2	no	—
110	C₄×C₂₇₁	—	2	1	no	—
111	C₄×C₂₇₁	—	2	1	no	—
112	C₄×C₂₇₁	—	2	1	no	—
113	C₄×C₂₇₁	C₄×D₄	1	2	yes	—
114	C₄×C₂₇₁	C₂×M₁₆	1	2	yes	complemented
115	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₂₇₁	—	2	2	no	—
118	C₂×C₂₇₁	—	2	4	no	—
119	C₂×C₂₇₁	—	4	2	no	—
120	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₂₇₁	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₂₇₁	#568	1	1	yes	characteristic
123	C₂₇₁	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
124	D₄×M₁₆	—	271	1	no	maximal, Sylow
125	C₂²×M₁₆	—	271	2	no	—
126	C₂×C₄×D₄	D₂₇₁	1	1	yes	characteristic
127	#436	—	271	2	no	—
128	#435	—	271	4	no	—
129	C₈×D₄	—	271	2	no	—
130	#423	—	271	1	no	—
131	#407	—	271	2	no	—
132	C₄×M₁₆	—	271	1	no	—
133	C₂²×D₄	Dic₂₇₁	1	1	yes	characteristic
134	C₂³×C₄	C₂×D₂₇₁	1	2	yes	—
135	C₂×M₁₆	—	271	1	no	—
136	C₂×M₁₆	—	271	2	no	—
137	C₂×M₁₆	—	271	1	no	—
138	C₂×M₁₆	—	271	4	no	—
139	C₂×M₁₆	—	542	8	no	—
140	C₂²×C₈	—	271	4	no	—
141	C₄×D₄	Dic₂₇₁	1	2	yes	—
142	C₄×D₄	Dic₂₇₁	1	1	yes	characteristic
143	C₄×D₄	Dic₂₇₁	1	1	yes	characteristic
144	C₄×D₄	C₂×D₂₇₁	1	1	yes	characteristic
145	C₄×D₄	C₂×D₂₇₁	1	2	yes	—
146	C₄×D₄	C₂×D₂₇₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	Dic₂₇₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	Dic₂₇₁	1	2	yes	—
149	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
150	C₄⋊C₈	—	271	2	no	—
151	C₄⋊C₈	—	271	2	no	—
152	C₂²⋊C₈	—	271	4	no	—
153	C₂²⋊C₈	—	271	4	no	—
154	#97	—	271	2	no	—
155	C₄×C₈	—	271	2	no	—
156	C₂⁴	C₂×Dic₂₇₁	1	2	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	C₂×Dic₂₇₁	1	2	yes	—
159	C₂×D₄	C₂×Dic₂₇₁	1	1	yes	characteristic
160	C₂×D₄	C₂×Dic₂₇₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
164	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
165	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
166	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
167	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₇₁	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₇₁	1	2	yes	—
170	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
171	M₁₆	—	271	4	no	—
172	M₁₆	—	542	8	no	—
173	M₁₆	—	542	4	no	—
174	M₁₆	—	542	2	no	—
175	C₂×C₈	—	271	2	no	—
176	C₂×C₈	—	271	2	no	—
177	C₂×C₈	—	271	4	no	—
178	C₂×C₈	—	542	8	no	—
179	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
181	C₄⋊C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
182	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
183	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
184	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
186	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
187	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
188	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
189	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×Dic₂₇₁	1	2	yes	—
194	C₂³	C₂²×Dic₂₇₁	1	2	yes	—
195	C₂³	C₂²×Dic₂₇₁	1	1	yes	characteristic
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	#16864638	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₂₇₁	1	2	yes	—
212	C₂×C₄	C₂²×Dic₂₇₁	1	2	yes	—
213	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
214	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
215	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
216	C₂×C₄	#16864653	1	2	yes	—
217	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
218	C₈	—	271	4	no	—
219	C₈	—	542	4	no	—
220	C₈	—	542	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	D₄×Dic₂₇₁	1	2	yes	—
230	C₂²	C₂×#16864638	1	4	yes	complemented
231	C₂²	C₂³×Dic₂₇₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	D₄×Dic₂₇₁	1	2	yes	—
238	C₄	C₂×#16864638	1	2	yes	complemented
239	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
240	C₄	C₂×#16864653	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×D₄×Dic₂₇₁	1	1	yes	characteristic
245	C₂	C₂²×#16864638	1	1	yes	characteristic
246	C₂	#31774664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₇₁	C₂
prime radical	C₂²×D₄×C₂₇₁	C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₇₁
hypocenter	C₂₇₁	D₄×M₁₆
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#16864638	C₂	semidirect
2. #31773975	C₂	semidirect
3. #31773969	C₂	semidirect
4. D₄×C₂₇₁⋊C₈	C₂	semidirect
5. #31773921	C₂	semidirect
6. #31773861	C₂	semidirect
7. C₄×#16864638	C₂	semidirect
13. C₂×#16864638	C₂²	semidirect
14. C₂²×C₂₇₁⋊C₈	C₂²	semidirect
15. C₄⋊(C₂₇₁⋊C₈)	C₂²	semidirect
16. C₄⋊(C₂₇₁⋊C₈)	C₂²	semidirect
17. C₂²⋊(C₂₇₁⋊C₈)	C₂²	semidirect
18. C₂²⋊(C₂₇₁⋊C₈)	C₂²	semidirect
19. #22576071	C₂²	semidirect
20. C₄×C₂₇₁⋊C₈	C₂²	semidirect
35. #16864638	D₄	direct (N × H)
68. C₂₇₁⋊C₈	C₂×D₄	semidirect
77. D₄×C₂₇₁	M₁₆	semidirect
106. C₂²×C₂₇₁	C₂×M₁₆	semidirect
114. C₄×C₂₇₁	C₂×M₁₆	semidirect
123. C₂₇₁	D₄×M₁₆	semidirect
198. D₄	#16864638	direct (N × H)
230. C₂²	C₂×#16864638	semidirect
238. C₄	C₂×#16864638	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#16864638	2
#31773975	2
#31773969	2
D₄×C₂₇₁⋊C₈	2
#31773921	2
#31773861	2
C₄×#16864638	2
C₂×C₄×D₄×C₂₇₁	2
D₄×M₁₆	271

Subgroups — D4×#16864638 #46845101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#16864638 #46845101