Subgroups — D₄×#11761611 #22568940

Order 8576.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11761611	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863475	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863473	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863467	C₂	1	2	yes	maximal, complemented
6	#16863464	C₂	1	2	yes	maximal, complemented
7	D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863418	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863358	C₂	1	2	yes	maximal, complemented
10	C₄×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₆₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
14	C₂×#11761611	—	2	8	no	—
15	C₂×#11761611	C₂²	1	1	yes	characteristic
16	C₂×#11761611	C₂²	1	2	yes	—
17	C₂×#11761611	C₂²	1	1	yes	characteristic
18	C₂×#11761611	C₂²	1	2	yes	complemented
19	C₂×#11761611	C₂²	1	2	yes	complemented
20	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	complemented
21	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
22	D₄×Dic₆₇	C₄	1	1	yes	characteristic
23	#14524005	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
25	(C₂²⋊D₆₇)⋊C₄	C₄	1	2	yes	—
26	C₄⋊C₄×D₆₇	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₆₇	C₄	1	2	yes	—
28	C₄²×D₆₇	C₂²	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
30	Dic₆₇⋊C₈	C₂²	1	2	yes	complemented
31	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
32	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	complemented
33	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	complemented
34	#14523953	C₂²	1	1	yes	characteristic, complemented
35	#14523953	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₆₇	C₂²	1	2	yes	complemented
38	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₆₇	C₂×C₄	1	2	yes	—
43	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
45	D₄×D₆₇	—	2	4	no	—
46	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₆₇	—	2	4	no	—
48	C₂×C₄×D₆₇	—	2	4	no	—
49	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₆₇	C₂³	1	2	yes	—
54	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
55	#11761611	—	2	4	no	—
56	#11761611	—	2	2	no	—
57	#11761611	—	2	4	no	—
58	#11761611	—	2	4	no	—
59	#11761611	D₄	1	4	yes	direct product factor
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
63	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
65	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
67	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
68	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₆₇	—	2	4	no	—
83	C₂²×D₆₇	—	2	4	no	—
84	C₂²×D₆₇	—	4	2	no	—
85	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₆₇	M₁₆	1	4	yes	complemented
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₆₇	—	2	4	no	—
95	C₂×Dic₆₇	—	2	4	no	—
96	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
99	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
100	D₂₆₈	—	2	2	no	—
101	C₄×D₆₇	—	2	4	no	—
102	C₄×D₆₇	—	2	2	no	—
103	C₄×D₆₇	—	2	2	no	—
104	C₄×D₆₇	—	2	2	no	—
105	C₄×D₆₇	—	2	4	no	—
106	C₄×D₆₇	—	4	1	no	—
107	C₄×D₆₇	C₂×D₄	1	2	yes	—
108	C₄×D₆₇	#41	1	2	yes	—
109	C₈×C₆₇	—	2	2	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	C₂×D₄	1	2	yes	complemented
112	C₆₇⋊C₈	—	2	2	no	—
113	C₆₇⋊C₈	—	2	1	no	—
114	C₆₇⋊C₈	C₂×D₄	1	2	yes	complemented
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	C₂×M₁₆	1	4	yes	complemented
118	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₆₇	—	2	1	no	—
120	C₂×D₆₇	—	2	1	no	—
121	C₂×D₆₇	—	2	2	no	—
122	C₂×D₆₇	—	2	2	no	—
123	C₂×D₆₇	—	2	2	no	—
124	C₂×D₆₇	—	4	8	no	—
125	C₂×D₆₇	C₄×D₄	1	2	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	Dic₆₇	—	2	2	no	—
132	Dic₆₇	—	2	1	no	—
133	Dic₆₇	—	2	1	no	—
134	Dic₆₇	C₄×D₄	1	2	yes	—
135	C₂×C₆₇	—	2	4	no	—
136	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×C₆₇	#568	1	1	yes	characteristic
139	D₆₇	—	2	2	no	—
140	D₆₇	—	4	2	no	—
141	D₄×M₁₆	—	67	1	no	maximal, Sylow
142	C₆₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	67	2	no	—
144	C₂×C₄×D₄	—	67	1	no	—
145	#436	—	67	1	no	—
146	#436	—	67	1	no	—
147	#435	—	67	2	no	—
148	#435	—	67	2	no	—
149	C₈×D₄	—	67	1	no	—
150	C₈×D₄	D₆₇	1	1	yes	characteristic, complemented
151	#423	—	67	1	no	—
152	#407	—	67	2	no	—
153	C₄×M₁₆	—	67	1	no	—
154	C₂²×D₄	—	67	1	no	—
155	C₂³×C₄	—	67	2	no	—
156	C₂×M₁₆	—	67	1	no	—
157	C₂×M₁₆	—	67	2	no	—
158	C₂×M₁₆	—	67	2	no	—
159	C₂×M₁₆	—	67	2	no	—
160	C₂×M₁₆	—	67	1	no	—
161	C₂×M₁₆	—	134	8	no	—
162	C₂²×C₈	—	67	2	no	—
163	C₂²×C₈	C₂×D₆₇	1	2	yes	complemented
164	C₄×D₄	—	67	1	no	—
165	C₄×D₄	—	67	2	no	—
166	C₄×D₄	—	67	1	no	—
167	C₄×D₄	—	67	2	no	—
168	C₄×D₄	—	67	1	no	—
169	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	67	1	no	—
171	C₂×C₂²⋊C₄	—	67	2	no	—
172	C₂×C₄²	—	67	1	no	—
173	C₄⋊C₈	—	67	2	no	—
174	C₄⋊C₈	—	67	1	no	—
175	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	67	2	no	—
177	C₂²⋊C₈	—	67	2	no	—
178	C₂²⋊C₈	—	67	2	no	—
179	C₂²⋊C₈	C₂×D₆₇	1	2	yes	complemented
180	#97	—	67	1	no	—
181	#97	—	67	1	no	—
182	C₄×C₈	—	67	1	no	—
183	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
184	C₂⁴	—	67	2	no	—
185	C₂×D₄	—	67	2	no	—
186	C₂×D₄	—	67	1	no	—
187	C₂×D₄	—	134	4	no	—
188	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	—	67	1	no	—
190	C₂²×C₄	—	67	1	no	—
191	C₂²×C₄	—	67	2	no	—
192	C₂²×C₄	—	67	1	no	—
193	C₂²×C₄	—	67	2	no	—
194	C₂²×C₄	—	67	2	no	—
195	C₂²×C₄	—	67	2	no	—
196	C₂²×C₄	—	134	4	no	—
197	C₂²×C₄	—	134	4	no	—
198	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
199	M₁₆	—	67	4	no	—
200	M₁₆	—	134	4	no	—
201	M₁₆	—	134	4	no	—
202	M₁₆	—	134	2	no	—
203	M₁₆	—	134	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	67	1	no	—
206	C₂×C₈	—	67	1	no	—
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	134	4	no	—
209	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₆₇	1	2	yes	—
211	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
212	C₄⋊C₄	—	67	2	no	—
213	C₄⋊C₄	—	67	1	no	—
214	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
215	C₂²⋊C₄	—	67	2	no	—
216	C₂²⋊C₄	—	67	2	no	—
217	C₂²⋊C₄	—	67	2	no	—
218	C₂²⋊C₄	C₄×D₆₇	1	2	yes	—
219	C₄²	—	67	1	no	—
220	C₄²	—	67	1	no	—
221	C₄²	—	67	1	no	—
222	C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	C₂³	—	268	2	no	—
224	C₂³	—	67	1	no	—
225	C₂³	—	67	2	no	—
226	C₂³	—	134	4	no	—
227	C₂³	—	134	4	no	—
228	C₂³	C₂×C₄×D₆₇	1	2	yes	—
229	D₄	—	134	4	no	—
230	D₄	—	134	2	no	—
231	D₄	#11761611	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	268	1	no	—
235	C₂×C₄	—	67	1	no	—
236	C₂×C₄	—	67	2	no	—
237	C₂×C₄	—	67	2	no	—
238	C₂×C₄	—	67	1	no	—
239	C₂×C₄	—	67	1	no	—
240	C₂×C₄	—	67	2	no	—
241	C₂×C₄	—	134	4	no	—
242	C₂×C₄	—	134	4	no	—
243	C₂×C₄	—	134	4	no	—
244	C₂×C₄	—	134	2	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	67	2	no	—
254	C₈	—	134	2	no	—
255	C₈	—	134	1	no	—
256	C₈	D₄×D₆₇	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	268	8	no	—
260	C₂²	—	67	2	no	—
261	C₂²	—	134	2	no	—
262	C₂²	—	134	2	no	—
263	C₂²	—	134	2	no	—
264	C₂²	—	134	1	no	—
265	C₂²	—	134	1	no	—
266	C₂²	C₂×#11761611	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	67	2	no	—
271	C₄	—	134	2	no	—
272	C₄	—	134	1	no	—
273	C₄	—	134	1	no	—
274	C₄	C₂×#11761611	1	2	yes	complemented
275	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
276	C₄	#41×D₆₇	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	268	2	no	—
279	C₂	—	134	2	no	—
280	C₂	C₄×D₄×D₆₇	1	1	yes	characteristic
281	C₂	C₂²×#11761611	1	1	yes	characteristic
282	C₂	#16864161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₈×D₄×C₆₇	C₂
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	D₄×M₁₆
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#11761611	C₂	semidirect
3. #16863475	C₂	semidirect
4. #16863473	C₂	semidirect
5. #16863467	C₂	semidirect
6. #16863464	C₂	semidirect
7. D₄×C₆₇⋊C₈	C₂	semidirect
8. #16863418	C₂	semidirect
9. #16863358	C₂	semidirect
10. C₄×#11761611	C₂	semidirect
11. C₈×D₄×C₆₇	C₂	semidirect
18. C₂×#11761611	C₂²	semidirect
19. C₂×#11761611	C₂²	semidirect
20. C₂²×C₆₇⋊C₈	C₂²	semidirect
29. C₄⋊(C₆₇⋊C₈)	C₂²	semidirect
30. Dic₆₇⋊C₈	C₂²	semidirect
31. C₂²⋊(C₆₇⋊C₈)	C₂²	semidirect
32. (C₂×D₆₇)⋊C₈	C₂²	semidirect
33. (C₂×D₆₇)⋊C₈	C₂²	semidirect
34. #14523953	C₂²	semidirect
35. #14523953	C₂²	semidirect
36. C₄×C₆₇⋊C₈	C₂²	semidirect
37. C₂²×C₈×C₆₇	C₂²	semidirect
39. C₄⋊C₈×C₆₇	C₂²	semidirect
40. C₂²⋊C₈×C₆₇	C₂²	semidirect
41. C₄×C₈×C₆₇	C₂²	semidirect
59. #11761611	D₄	direct (N × H)
87. D₄×C₆₇	M₁₆	semidirect
111. C₈×C₆₇	C₂×D₄	semidirect
114. C₆₇⋊C₈	C₂×D₄	semidirect
117. C₂²×C₆₇	C₂×M₁₆	semidirect
128. C₄×C₆₇	C₂×M₁₆	semidirect
142. C₆₇	D₄×M₁₆	semidirect
150. C₈×D₄	D₆₇	semidirect
163. C₂²×C₈	C₂×D₆₇	semidirect
175. C₄⋊C₈	C₂×D₆₇	semidirect
179. C₂²⋊C₈	C₂×D₆₇	semidirect
183. C₄×C₈	C₂×D₆₇	semidirect
231. D₄	#11761611	direct (N × H)
256. C₈	D₄×D₆₇	semidirect
266. C₂²	C₂×#11761611	semidirect
274. C₄	C₂×#11761611	semidirect

Subgroups — D4×#11761611 #22568940

Conjugacy classes of subgroups

Complements

Subgroups — D₄×#11761611 #22568940