Subgroups — C₂²×D₄×#10596831 #33949816

Order 18688.

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	C₂⁴×#10596831	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×#10596831	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊#10596831	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10596831	—	2	16	no	—
8	C₂³×#10596831	C₂²	1	2	yes	—
9	C₂³×#10596831	C₂²	1	1	yes	characteristic
10	C₂³×#10596831	C₂²	1	1	yes	characteristic
11	C₂³×#10596831	C₂²	1	24	yes	complemented
12	D₄×#10596831	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#10596831	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊#10596831	C₂²	1	24	yes	complemented
15	C₂×C₄×#10596831	C₂²	1	12	yes	complemented
16	C₂³×D₄×C₇₃	C₄	1	1	yes	characteristic
17	C₂⁴×C₄×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₇₃	C₄	1	12	yes	—
19	C₂×C₄×D₄×C₇₃	C₂²	1	12	yes	—
20	C₂²×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
22	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×#10596831	—	2	96	no	—
24	C₂²×#10596831	—	2	8	no	—
25	C₂²×#10596831	—	2	4	no	—
26	C₂²×#10596831	D₄	1	8	yes	direct product factor
27	C₂²×#10596831	C₂³	1	12	yes	—
28	C₂²×#10596831	C₂³	1	6	yes	—
29	C₂²×#10596831	C₂³	1	6	yes	—
30	C₂²×#10596831	C₂³	1	32	yes	complemented
31	C₄⋊#10596831	C₂³	1	16	yes	complemented
32	C₂²⋊#10596831	C₂³	1	32	yes	complemented
33	C₄×#10596831	C₂³	1	16	yes	complemented
34	C₂⁵×C₇₃	C₂×C₄	1	2	yes	—
35	C₂²×D₄×C₇₃	C₈	1	16	yes	complemented
36	C₂²×D₄×C₇₃	C₂×C₄	1	12	yes	—
37	C₂³×C₄×C₇₃	—	2	8	no	—
38	C₂³×C₄×C₇₃	—	2	8	no	—
39	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₇₃	C₂×C₄	1	12	yes	—
42	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₃	C₂³	1	12	yes	—
44	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₄×D₄×C₇₃	C₂×C₄	1	48	yes	—
46	C₄×D₄×C₇₃	C₂³	1	16	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	12	yes	—
48	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	24	yes	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
50	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
51	C₂×#10596831	—	2	24	no	—
52	C₂×#10596831	—	2	12	no	—
53	C₂×#10596831	—	2	64	no	—
54	C₂×#10596831	C₂×D₄	1	24	yes	direct product factor
55	C₂×#10596831	C₂⁴	1	4	yes	—
56	C₂×#10596831	C₂⁴	1	8	yes	—
57	C₂×#10596831	C₂⁴	1	4	yes	—
58	C₂⁴×C₇₃	—	2	8	no	—
59	C₂⁴×C₇₃	—	2	8	no	—
60	C₂⁴×C₇₃	C₂×C₈	1	16	yes	complemented
61	C₂⁴×C₇₃	C₂²×C₄	1	12	yes	—
62	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₄×C₇₃	C₂×C₈	1	96	yes	complemented
64	C₂×D₄×C₇₃	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₇₃	—	2	8	no	—
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	—	2	48	no	—
68	C₂²×C₄×C₇₃	—	2	24	no	—
69	C₂²×C₄×C₇₃	—	2	24	no	—
70	C₂²×C₄×C₇₃	C₂×C₈	1	8	yes	complemented
71	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₇₃	#41	1	4	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
80	C₄⋊C₄×C₇₃	C₂²×C₄	1	16	yes	—
81	C₂²⋊C₄×C₇₃	C₂²×C₄	1	32	yes	—
82	C₄²×C₇₃	C₂²×C₄	1	12	yes	—
83	C₄²×C₇₃	C₂⁴	1	4	yes	—
84	C₂³×C₇₃	—	2	24	no	—
85	C₂³×C₇₃	—	2	24	no	—
86	C₂³×C₇₃	—	2	64	no	—
87	C₂³×C₇₃	C₄×D₄	1	4	yes	—
88	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
89	C₂³×C₇₃	C₂²×C₈	1	48	yes	complemented
90	C₂³×C₇₃	#131	1	4	yes	—
91	C₂³×C₇₃	C₂³×C₄	1	8	yes	—
92	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
93	D₄×C₇₃	C₂²×C₈	1	64	yes	complemented
94	C₂×C₄×C₇₃	—	2	24	no	—
95	C₂×C₄×C₇₃	—	2	12	no	—
96	C₂×C₄×C₇₃	—	2	48	no	—
97	C₂×C₄×C₇₃	—	2	8	no	—
98	C₂×C₄×C₇₃	—	2	8	no	—
99	C₂×C₄×C₇₃	C₄×D₄	1	12	yes	—
100	C₂×C₄×C₇₃	C₂²×C₈	1	24	yes	complemented
101	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
104	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
106	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
107	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
108	#10596831	—	2	8	no	—
109	#10596831	—	2	4	no	—
110	#10596831	C₂²×D₄	1	8	yes	direct product factor
111	C₂²×C₇₃	—	2	8	no	—
112	C₂²×C₇₃	—	2	96	no	—
113	C₂²×C₇₃	—	2	8	no	—
114	C₂²×C₇₃	C₈×D₄	1	16	yes	complemented
115	C₂²×C₇₃	C₂×C₄×D₄	1	6	yes	—
116	C₂²×C₇₃	C₂³×C₈	1	6	yes	—
117	C₂²×C₇₃	C₂³×C₈	1	16	yes	complemented
118	C₂²×C₇₃	C₂×#131	1	6	yes	—
119	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₇₃	—	2	8	no	—
121	C₄×C₇₃	—	2	4	no	—
122	C₄×C₇₃	C₂×C₄×D₄	1	6	yes	—
123	C₄×C₇₃	C₂³×C₈	1	8	yes	complemented
124	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
126	C₂²×C₈×D₄	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	16	no	—
128	C₂×C₇₃	C₂×C₈×D₄	1	12	yes	complemented
129	C₂×C₇₃	C₂²×C₄×D₄	1	1	yes	characteristic
130	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
131	C₂×C₇₃	C₂²×#131	1	1	yes	characteristic
132	C₂⁴×C₈	—	73	2	no	—
133	C₂²×C₄×D₄	D₇₃	1	1	yes	characteristic
134	C₂×C₈×D₄	—	73	24	no	—
135	C₂²×C₄⋊C₈	—	73	1	no	—
136	C₂²×C₂²⋊C₈	—	73	2	no	—
137	C₂²×C₄×C₈	—	73	1	no	—
138	C₇₃	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
139	C₂³×D₄	Dic₇₃	1	1	yes	characteristic
140	C₂⁴×C₄	C₂×D₇₃	1	2	yes	—
141	C₂³×C₈	—	73	1	no	—
142	C₂³×C₈	—	73	24	no	—
143	C₂³×C₈	—	73	1	no	—
144	C₂³×C₈	—	73	2	no	—
145	C₂³×C₈	—	146	16	no	—
146	C₂×C₄×D₄	Dic₇₃	1	12	yes	—
147	C₂×C₄×D₄	C₂×D₇₃	1	12	yes	—
148	C₂²×C₄⋊C₄	Dic₇₃	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	Dic₇₃	1	2	yes	—
150	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
151	C₈×D₄	—	73	64	no	—
152	C₂×C₄⋊C₈	—	73	12	no	—
153	C₂×C₂²⋊C₈	—	73	24	no	—
154	C₂×C₄×C₈	—	73	12	no	—
155	C₂⁵	C₂×Dic₇₃	1	2	yes	—
156	C₂²×D₄	#10596831	1	16	yes	direct product factor
157	C₂²×D₄	C₂×Dic₇₃	1	12	yes	—
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₇₃	1	2	yes	—
162	C₂³×C₄	C₂×Dic₇₃	1	12	yes	—
163	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₇₃	1	12	yes	—
165	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
166	C₂²×C₈	—	73	8	no	—
167	C₂²×C₈	—	73	6	no	—
168	C₂²×C₈	—	73	12	no	—
169	C₂²×C₈	—	73	32	no	—
170	C₂²×C₈	—	73	6	no	—
171	C₂²×C₈	—	146	96	no	—
172	C₂²×C₈	—	146	8	no	—
173	C₂²×C₈	—	146	4	no	—
174	C₄×D₄	C₂×Dic₇₃	1	48	yes	—
175	C₄×D₄	C₂²×D₇₃	1	16	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₇₃	1	12	yes	—
177	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	24	yes	—
178	C₂×C₄²	C₂×Dic₇₃	1	6	yes	—
179	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
180	C₄⋊C₈	—	73	16	no	—
181	C₂²⋊C₈	—	73	32	no	—
182	C₄×C₈	—	73	16	no	—
183	C₂⁴	—	2	8	no	—
184	C₂⁴	—	2	8	no	—
185	C₂⁴	C₂×#10596831	1	16	yes	complemented
186	C₂⁴	C₂²×Dic₇₃	1	12	yes	—
187	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
188	C₂×D₄	C₂×#10596831	1	96	yes	direct product factor
189	C₂×D₄	C₂²×Dic₇₃	1	16	yes	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	48	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	C₂×#10596831	1	8	yes	complemented
196	C₂²×C₄	C₂²×Dic₇₃	1	6	yes	—
197	C₂²×C₄	C₂²×Dic₇₃	1	12	yes	—
198	C₂²×C₄	C₂²×Dic₇₃	1	24	yes	—
199	C₂²×C₄	C₂²×Dic₇₃	1	3	yes	—
200	C₂²×C₄	C₂²×Dic₇₃	1	6	yes	—
201	C₂²×C₄	D₄×D₇₃	1	4	yes	—
202	C₂²×C₄	#11924027	1	4	yes	—
203	C₂²×C₄	C₂³×D₇₃	1	8	yes	—
204	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
205	C₂×C₈	—	73	24	no	—
206	C₂×C₈	—	73	4	no	—
207	C₂×C₈	—	73	4	no	—
208	C₂×C₈	—	73	8	no	—
209	C₂×C₈	—	146	64	no	—
210	C₂×C₈	—	146	24	no	—
211	C₂×C₈	—	146	12	no	—
212	C₄⋊C₄	C₂²×Dic₇₃	1	16	yes	—
213	C₂²⋊C₄	C₂²×Dic₇₃	1	32	yes	—
214	C₄²	C₂²×Dic₇₃	1	12	yes	—
215	C₄²	C₂³×D₇₃	1	4	yes	—
216	C₂³	—	2	64	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	D₄×Dic₇₃	1	4	yes	—
220	C₂³	C₂²×#10596831	1	4	yes	—
221	C₂³	C₂²×#10596831	1	48	yes	complemented
222	C₂³	#14556598	1	4	yes	—
223	C₂³	C₂³×Dic₇₃	1	8	yes	—
224	C₂³	C₂³×Dic₇₃	1	3	yes	—
225	D₄	C₂²×#10596831	1	64	yes	direct product factor
226	C₂×C₄	—	2	24	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	48	no	—
231	C₂×C₄	D₄×Dic₇₃	1	12	yes	—
232	C₂×C₄	C₂²×#10596831	1	24	yes	complemented
233	C₂×C₄	C₂³×Dic₇₃	1	4	yes	—
234	C₂×C₄	C₂³×Dic₇₃	1	8	yes	—
235	C₂×C₄	C₂³×Dic₇₃	1	3	yes	—
236	C₂×C₄	C₂³×Dic₇₃	1	4	yes	—
237	C₂×C₄	C₂×D₄×D₇₃	1	6	yes	—
238	C₂×C₄	C₂×#11924027	1	6	yes	—
239	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
240	C₈	—	73	8	no	—
241	C₈	—	146	8	no	—
242	C₈	—	146	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	96	no	—
246	C₂²	D₄×#10596831	1	16	yes	direct product factor
247	C₂²	C₂×D₄×Dic₇₃	1	6	yes	—
248	C₂²	C₂³×#10596831	1	6	yes	—
249	C₂²	C₂³×#10596831	1	16	yes	complemented
250	C₂²	C₂×#14556598	1	6	yes	—
251	C₂²	C₂⁴×Dic₇₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	2	8	no	—
254	C₄	C₂×D₄×Dic₇₃	1	6	yes	—
255	C₄	C₂³×#10596831	1	8	yes	complemented
256	C₄	C₂²×D₄×D₇₃	1	1	yes	characteristic
257	C₄	C₂²×#11924027	1	1	yes	characteristic
258	C₂	—	2	16	no	—
259	C₂	C₂×D₄×#10596831	1	12	yes	direct product factor
260	C₂	C₂²×D₄×Dic₇₃	1	1	yes	characteristic
261	C₂	C₂⁴×#10596831	1	1	yes	characteristic
262	C₂	C₂²×#14556598	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄×D₄×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂³×D₄×C₇₃	C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×C₄×D₄	D₇₃
hypocenter	C₇₃	C₂²×C₈×D₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#10596831	C₂	semidirect
2. C₂×D₄×#10596831	C₂	direct (N × H)
3. C₂²×C₄⋊#10596831	C₂	semidirect
4. C₂²×C₂²⋊#10596831	C₂	semidirect
5. C₂²×C₄×#10596831	C₂	semidirect
11. C₂³×#10596831	C₂²	semidirect
12. D₄×#10596831	C₂²	direct (N × H)
13. C₂×C₄⋊#10596831	C₂²	semidirect
14. C₂×C₂²⋊#10596831	C₂²	semidirect
15. C₂×C₄×#10596831	C₂²	semidirect
26. C₂²×#10596831	D₄	direct (N × H)
30. C₂²×#10596831	C₂³	semidirect
31. C₄⋊#10596831	C₂³	semidirect
32. C₂²⋊#10596831	C₂³	semidirect
33. C₄×#10596831	C₂³	semidirect
35. C₂²×D₄×C₇₃	C₈	semidirect
54. C₂×#10596831	C₂×D₄	direct (N × H)
60. C₂⁴×C₇₃	C₂×C₈	semidirect
63. C₂×D₄×C₇₃	C₂×C₈	semidirect
70. C₂²×C₄×C₇₃	C₂×C₈	semidirect
89. C₂³×C₇₃	C₂²×C₈	semidirect
93. D₄×C₇₃	C₂²×C₈	semidirect
100. C₂×C₄×C₇₃	C₂²×C₈	semidirect
110. #10596831	C₂²×D₄	direct (N × H)
114. C₂²×C₇₃	C₈×D₄	semidirect
117. C₂²×C₇₃	C₂³×C₈	semidirect
123. C₄×C₇₃	C₂³×C₈	semidirect
128. C₂×C₇₃	C₂×C₈×D₄	semidirect
138. C₇₃	C₂²×C₈×D₄	semidirect
156. C₂²×D₄	#10596831	direct (N × H)
185. C₂⁴	C₂×#10596831	semidirect
188. C₂×D₄	C₂×#10596831	direct (N × H)
195. C₂²×C₄	C₂×#10596831	semidirect
221. C₂³	C₂²×#10596831	semidirect
225. D₄	C₂²×#10596831	direct (N × H)
232. C₂×C₄	C₂²×#10596831	semidirect
246. C₂²	D₄×#10596831	direct (N × H)
249. C₂²	C₂³×#10596831	semidirect
255. C₄	C₂³×#10596831	semidirect
259. C₂	C₂×D₄×#10596831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#10596831	2
C₂×D₄×#10596831	2
C₂²×C₄⋊#10596831	2
C₂²×C₂²⋊#10596831	2
C₂²×C₄×#10596831	2
C₂²×C₄×D₄×C₇₃	2
C₂²×C₈×D₄	73

Subgroups — C22×D4×#10596831 #33949816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×#10596831 #33949816