Subgroups — C₂²×C₄×#11924038 #34031067

Order 18688.

3950 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924038	C₂	1	2	yes	maximal
2	C₂×C₄×#11924038	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556646	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#10596833	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924038	C₄	1	32	yes	direct product factor
7	C₂²×#11924038	C₂²	1	24	yes	—
8	C₂³×#10596833	C₂²	1	4	yes	—
9	C₄×#11924038	C₂²	1	64	yes	direct product factor
10	C₂×#14556646	C₂²	1	24	yes	complemented
11	C₂×C₄×#10596833	C₂²	1	24	yes	complemented
12	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
13	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
14	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×D₇₃	C₄	1	12	yes	—
19	C₂×C₄²×D₇₃	C₂²	1	12	yes	—
20	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#11924038	C₂×C₄	1	192	yes	direct product factor
22	C₂×#11924038	C₂³	1	32	yes	—
23	C₂²×#10596833	C₂×C₄	1	32	yes	complemented
24	C₂²×#10596833	C₂³	1	24	yes	—
25	#14556646	C₂³	1	32	yes	complemented
26	C₄×#10596833	C₂³	1	32	yes	complemented
27	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	—	2	8	no	—
32	C₂²×C₄×D₇₃	—	2	8	no	—
33	C₂²×C₄×D₇₃	C₂×C₄	1	16	yes	—
34	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
35	C₂²×C₄×D₇₃	C₂×C₄	1	12	yes	—
36	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
37	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
39	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
41	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
43	C₄²×D₇₃	C₂×C₄	1	48	yes	—
44	C₄²×D₇₃	C₂³	1	16	yes	—
45	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂×C₄	1	12	yes	—
49	#11924038	C₂²×C₄	1	128	yes	direct product factor
50	C₂×#10596833	C₂²×C₄	1	96	yes	complemented
51	C₂×#10596833	C₂⁴	1	16	yes	—
52	C₂³×D₇₃	—	2	4	no	—
53	C₂³×D₇₃	—	2	4	no	—
54	C₂³×D₇₃	C₄²	1	8	yes	—
55	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
56	C₂²×Dic₇₃	—	2	4	no	—
57	C₂²×Dic₇₃	M₁₆	1	8	yes	complemented
58	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
59	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
60	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
61	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
62	C₂²×Dic₇₃	C₂⁴	1	3	yes	—
63	C₂×C₄×D₇₃	—	2	48	no	—
64	C₂×C₄×D₇₃	—	2	48	no	—
65	C₂×C₄×D₇₃	C₄²	1	48	yes	—
66	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	16	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	48	yes	—
70	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
71	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
72	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
73	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
74	C₄×Dic₇₃	C₂⁴	1	4	yes	—
75	C₄×Dic₇₃	C₂⁴	1	4	yes	—
76	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	—	2	4	no	—
78	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
79	C₂²×C₄×C₇₃	M₁₆	1	8	yes	complemented
80	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
83	C₄²×C₇₃	C₂²×C₄	1	16	yes	—
84	C₂³×C₇₃	C₂×C₄²	1	4	yes	—
85	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
86	C₂³×C₇₃	C₂×M₁₆	1	4	yes	—
87	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
88	C₂²×D₇₃	—	2	12	no	—
89	C₂²×D₇₃	—	2	12	no	—
90	C₂²×D₇₃	—	2	32	no	—
91	C₂²×D₇₃	C₂×C₄²	1	24	yes	—
92	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
93	C₂×C₄×C₇₃	—	2	12	no	—
94	C₂×C₄×C₇₃	C₂×C₄²	1	24	yes	—
95	C₂×C₄×C₇₃	C₂×M₁₆	1	24	yes	complemented
96	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
99	C₂×Dic₇₃	—	2	12	no	—
100	C₂×Dic₇₃	C₂×C₄²	1	12	yes	—
101	C₂×Dic₇₃	C₂×M₁₆	1	24	yes	complemented
102	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
103	C₂×Dic₇₃	C₂³×C₄	1	4	yes	—
104	C₂×Dic₇₃	C₂³×C₄	1	12	yes	—
105	C₂×Dic₇₃	C₂³×C₄	1	3	yes	—
106	C₂×Dic₇₃	C₂⁵	1	1	yes	characteristic
107	C₄×D₇₃	—	2	32	no	—
108	C₄×D₇₃	—	2	32	no	—
109	C₄×D₇₃	C₂×C₄²	1	48	yes	—
110	C₄×D₇₃	C₂³×C₄	1	16	yes	—
111	#10596833	C₂³×C₄	1	32	yes	complemented
112	C₂²×C₇₃	C₄×M₁₆	1	16	yes	complemented
113	C₂²×C₇₃	C₂²×C₄²	1	6	yes	—
114	C₂²×C₇₃	C₂²×M₁₆	1	6	yes	—
115	C₂²×C₇₃	C₂²×M₁₆	1	6	yes	—
116	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
117	C₂×D₇₃	—	2	48	no	—
118	C₂×D₇₃	—	2	4	no	—
119	C₂×D₇₃	—	2	4	no	—
120	C₂×D₇₃	C₂²×C₄²	1	8	yes	—
121	C₄×C₇₃	—	2	4	no	—
122	C₄×C₇₃	C₂²×C₄²	1	8	yes	—
123	C₄×C₇₃	C₂²×M₁₆	1	8	yes	complemented
124	Dic₇₃	—	2	4	no	—
125	Dic₇₃	C₂²×C₄²	1	6	yes	—
126	Dic₇₃	C₂²×M₁₆	1	8	yes	complemented
127	Dic₇₃	C₂⁴×C₄	1	2	yes	—
128	C₂²×C₄×M₁₆	—	73	1	no	maximal, Sylow
129	C₂×C₇₃	C₂×C₄×M₁₆	1	12	yes	complemented
130	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
131	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
132	C₂×C₇₃	C₂³×M₁₆	1	1	yes	characteristic
133	D₇₃	—	2	8	no	—
134	C₂³×M₁₆	—	73	2	no	—
135	C₂³×C₄²	—	73	1	no	—
136	C₂×C₄×M₁₆	—	73	24	no	—
137	C₂²×#97	—	73	2	no	—
138	C₂²×C₄×C₈	—	73	2	no	—
139	C₇₃	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
140	C₂⁴×C₄	—	73	1	no	—
141	C₂⁴×C₄	—	73	1	no	—
142	C₂⁴×C₄	—	73	1	no	—
143	C₂²×M₁₆	—	73	32	no	—
144	C₂²×M₁₆	—	73	24	no	—
145	C₂³×C₈	—	73	4	no	—
146	C₂²×C₄²	—	73	1	no	—
147	C₂²×C₄²	—	73	1	no	—
148	C₂²×C₄²	—	73	1	no	—
149	C₂²×C₄²	—	73	12	no	—
150	C₂²×C₄²	—	73	12	no	—
151	C₂²×C₄²	#64408	1	1	yes	characteristic
152	C₄×M₁₆	—	73	64	no	—
153	C₂×#97	—	73	24	no	—
154	C₂×C₄×C₈	—	73	24	no	—
155	C₂⁵	—	73	1	no	—
156	C₂³×C₄	—	73	1	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂³×C₄	—	73	16	no	—
159	C₂³×C₄	—	73	6	no	—
160	C₂³×C₄	—	73	6	no	—
161	C₂³×C₄	—	73	1	no	—
162	C₂³×C₄	—	73	12	no	—
163	C₂³×C₄	—	73	12	no	—
164	C₂³×C₄	—	146	8	no	—
165	C₂³×C₄	—	146	8	no	—
166	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
167	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
168	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
169	C₂×M₁₆	—	73	192	no	—
170	C₂×M₁₆	—	73	32	no	—
171	C₂²×C₈	—	73	32	no	—
172	C₂²×C₈	—	73	24	no	—
173	C₂×C₄²	—	73	6	no	—
174	C₂×C₄²	—	73	6	no	—
175	C₂×C₄²	—	73	12	no	—
176	C₂×C₄²	—	73	6	no	—
177	C₂×C₄²	—	73	6	no	—
178	C₂×C₄²	—	73	48	no	—
179	C₂×C₄²	—	73	16	no	—
180	C₂×C₄²	C₂×#64408	1	12	yes	—
181	#97	—	73	32	no	—
182	C₄×C₈	—	73	32	no	—
183	C₂⁴	—	73	6	no	—
184	C₂⁴	—	73	8	no	—
185	C₂⁴	—	146	4	no	—
186	C₂⁴	—	146	4	no	—
187	C₂⁴	C₂²×#64408	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	73	8	no	—
190	C₂²×C₄	—	73	3	no	—
191	C₂²×C₄	—	73	6	no	—
192	C₂²×C₄	—	73	3	no	—
193	C₂²×C₄	—	73	8	no	—
194	C₂²×C₄	—	73	12	no	—
195	C₂²×C₄	—	73	4	no	—
196	C₂²×C₄	—	73	48	no	—
197	C₂²×C₄	—	73	48	no	—
198	C₂²×C₄	—	73	6	no	—
199	C₂²×C₄	—	73	16	no	—
200	C₂²×C₄	—	73	16	no	—
201	C₂²×C₄	—	146	48	no	—
202	C₂²×C₄	—	146	4	no	—
203	C₂²×C₄	—	146	48	no	—
204	C₂²×C₄	C₄×#64408	1	8	yes	—
205	C₂²×C₄	#11924038	1	8	yes	direct product factor
206	C₂²×C₄	C₂²×#64408	1	6	yes	—
207	C₂²×C₄	C₂²×#64408	1	6	yes	—
208	C₂²×C₄	C₂²×#64408	1	6	yes	—
209	M₁₆	—	73	128	no	—
210	C₂×C₈	—	73	96	no	—
211	C₂×C₈	—	73	16	no	—
212	C₄²	—	73	12	no	—
213	C₄²	—	73	12	no	—
214	C₄²	—	73	16	no	—
215	C₄²	—	73	4	no	—
216	C₄²	—	73	4	no	—
217	C₄²	C₂²×#64408	1	16	yes	—
218	C₂³	—	73	24	no	—
219	C₂³	—	73	4	no	—
220	C₂³	—	146	12	no	—
221	C₂³	—	146	12	no	—
222	C₂³	—	146	32	no	—
223	C₂³	C₂×C₄×#64408	1	4	yes	—
224	C₂³	D₇₃⋊M₁₆	1	4	yes	—
225	C₂³	C₂×#11924038	1	4	yes	—
226	C₂³	C₂³×#64408	1	3	yes	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	73	12	no	—
229	C₂×C₄	—	73	12	no	—
230	C₂×C₄	—	73	24	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	3	no	—
233	C₂×C₄	—	73	4	no	—
234	C₂×C₄	—	73	16	no	—
235	C₂×C₄	—	73	48	no	—
236	C₂×C₄	—	73	4	no	—
237	C₂×C₄	—	146	32	no	—
238	C₂×C₄	—	146	12	no	—
239	C₂×C₄	—	146	32	no	—
240	C₂×C₄	C₂×C₄×#64408	1	24	yes	—
241	C₂×C₄	C₂×#11924038	1	24	yes	direct product factor
242	C₂×C₄	C₂³×#64408	1	4	yes	—
243	C₂×C₄	C₂³×#64408	1	4	yes	—
244	C₂×C₄	C₂³×#64408	1	4	yes	—
245	C₈	—	73	32	no	—
246	C₂²	—	73	8	no	—
247	C₂²	—	146	4	no	—
248	C₂²	—	146	4	no	—
249	C₂²	—	146	48	no	—
250	C₂²	C₄×#11924038	1	16	yes	direct product factor
251	C₂²	C₂²×C₄×#64408	1	6	yes	—
252	C₂²	C₂×D₇₃⋊M₁₆	1	6	yes	—
253	C₂²	C₂²×#11924038	1	6	yes	—
254	C₂²	C₂⁴×#64408	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	73	2	no	—
257	C₄	—	73	6	no	—
258	C₄	—	73	8	no	—
259	C₄	—	146	4	no	—
260	C₄	C₂²×C₄×#64408	1	8	yes	—
261	C₄	C₂²×#11924038	1	8	yes	direct product factor
262	C₂	—	146	8	no	—
263	C₂	C₂×C₄×#11924038	1	12	yes	direct product factor
264	C₂	C₂³×C₄×#64408	1	1	yes	characteristic
265	C₂	C₂²×D₇₃⋊M₁₆	1	1	yes	characteristic
266	C₂	C₂³×#11924038	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#64408
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×Dic₇₃	C₂⁵
hypercenter	C₂²×C₄²	#64408
hypocenter	C₇₃	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11924038	C₂	direct (N × H)
3. C₂²×#14556646	C₂	semidirect
4. C₂²×C₄×#10596833	C₂	semidirect
6. C₂²×#11924038	C₄	direct (N × H)
9. C₄×#11924038	C₂²	direct (N × H)
10. C₂×#14556646	C₂²	semidirect
11. C₂×C₄×#10596833	C₂²	semidirect
21. C₂×#11924038	C₂×C₄	direct (N × H)
23. C₂²×#10596833	C₂×C₄	semidirect
25. #14556646	C₂³	semidirect
26. C₄×#10596833	C₂³	semidirect
49. #11924038	C₂²×C₄	direct (N × H)
50. C₂×#10596833	C₂²×C₄	semidirect
57. C₂²×Dic₇₃	M₁₆	semidirect
79. C₂²×C₄×C₇₃	M₁₆	semidirect
95. C₂×C₄×C₇₃	C₂×M₁₆	semidirect
101. C₂×Dic₇₃	C₂×M₁₆	semidirect
111. #10596833	C₂³×C₄	semidirect
112. C₂²×C₇₃	C₄×M₁₆	semidirect
123. C₄×C₇₃	C₂²×M₁₆	semidirect
126. Dic₇₃	C₂²×M₁₆	semidirect
129. C₂×C₇₃	C₂×C₄×M₁₆	semidirect
139. C₇₃	C₂²×C₄×M₁₆	semidirect
205. C₂²×C₄	#11924038	direct (N × H)
241. C₂×C₄	C₂×#11924038	direct (N × H)
250. C₂²	C₄×#11924038	direct (N × H)
261. C₄	C₂²×#11924038	direct (N × H)
263. C₂	C₂×C₄×#11924038	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×#11924038	2
C₂×C₄×#11924038	2
C₂²×#14556646	2
C₂²×C₄×#10596833	2
C₂²×C₄²×D₇₃	2
C₂²×C₄×M₁₆	73

Subgroups — C22×C4×#11924038 #34031067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11924038 #34031067