Subgroups — C₂×C₄×#14556666 #34029784

Order 18688.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556666	C₂	1	2	yes	maximal
2	C₄×#14556666	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989476	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11924031	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	C₄	1	16	yes	direct product factor
7	C₂×#14556666	C₂²	1	8	yes	—
8	C₂²×#11924031	C₂²	1	4	yes	—
9	#17989476	C₂²	1	8	yes	complemented
10	C₄×#11924031	C₂²	1	8	yes	complemented
11	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₇₃	C₄	1	4	yes	—
18	C₄×C₈×D₇₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
20	#14556666	C₂×C₄	1	32	yes	direct product factor
21	C₂×#11924031	C₂×C₄	1	16	yes	complemented
22	C₂×#11924031	C₂³	1	8	yes	—
23	C₂²×C₄×D₇₃	C₈	1	2	yes	—
24	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂³	1	1	yes	characteristic
27	C₂×C₈×D₇₃	C₈	1	8	yes	—
28	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
29	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₇₃	C₂³	1	2	yes	—
32	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₇₃	C₈	1	4	yes	—
35	C₄²×D₇₃	C₂×C₄	1	4	yes	—
36	C₄×#10596831	C₂×C₄	1	2	yes	—
37	C₄×#10596831	C₂³	1	2	yes	—
38	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
40	C₈×Dic₇₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
45	#11924031	C₂²×C₄	1	16	yes	complemented
46	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
48	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₇₃	—	2	8	no	—
50	C₂×C₄×D₇₃	C₄²	1	4	yes	—
51	C₂×C₄×D₇₃	C₂×C₈	1	8	yes	—
52	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
55	C₂×#10596831	C₂×C₈	1	4	yes	—
56	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
57	C₂×#10596831	C₂²×C₄	1	4	yes	—
58	C₂×#10596831	C₂²×C₄	1	2	yes	—
59	C₂×#10596831	C₂⁴	1	1	yes	characteristic
60	C₈×D₇₃	C₄²	1	8	yes	—
61	C₈×D₇₃	C₂×C₈	1	16	yes	—
62	C₈×D₇₃	C₂²×C₄	1	8	yes	—
63	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
64	C₄×Dic₇₃	C₂×C₈	1	8	yes	—
65	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₄²	1	4	yes	—
69	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₇₃	C₂×C₈	1	2	yes	—
73	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂²×D₇₃	—	2	4	no	—
76	C₂²×D₇₃	C₄×C₈	1	4	yes	—
77	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₇₃	M₃₂	1	8	yes	complemented
79	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
80	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×Dic₇₃	—	2	4	no	—
85	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
86	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
87	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
88	C₄×D₇₃	—	2	16	no	—
89	C₄×D₇₃	C₄×C₈	1	8	yes	—
90	C₄×D₇₃	C₂×C₄²	1	4	yes	—
91	C₄×D₇₃	C₂²×C₈	1	4	yes	—
92	C₈×C₇₃	C₂×C₄²	1	4	yes	—
93	C₈×C₇₃	C₂²×C₈	1	4	yes	—
94	#10596831	C₂×C₄²	1	2	yes	—
95	#10596831	C₂²×C₈	1	4	yes	—
96	#10596831	C₂³×C₄	1	2	yes	—
97	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
98	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
99	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
100	C₂×D₇₃	—	2	8	no	—
101	C₂×D₇₃	—	2	4	no	—
102	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
103	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₇₃	C₂×M₃₂	1	8	yes	complemented
105	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
107	Dic₇₃	—	2	4	no	—
108	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
109	C₂×C₄×M₃₂	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	C₄×M₃₂	1	4	yes	complemented
111	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
112	C₂×C₇₃	C₂²×M₃₂	1	2	yes	—
113	D₇₃	—	2	4	no	—
114	C₂²×M₃₂	—	73	2	no	—
115	C₂²×C₄×C₈	—	73	1	no	—
116	C₄×M₃₂	—	73	8	no	—
117	C₂×#346	—	73	2	no	—
118	C₂×C₄×C₁₆	—	73	2	no	—
119	C₇₃	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₈	—	73	1	no	—
121	C₂³×C₈	—	73	1	no	—
122	C₂²×C₄²	—	73	1	no	—
123	C₂×M₃₂	—	73	16	no	—
124	C₂×M₃₂	—	73	8	no	—
125	C₂²×C₁₆	—	73	4	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	4	no	—
130	C₂×C₄×C₈	—	73	4	no	—
131	C₂×C₄×C₈	#64408	1	1	yes	characteristic
132	#346	—	73	8	no	—
133	C₄×C₁₆	—	73	8	no	—
134	C₂³×C₄	—	73	1	no	—
135	C₂³×C₄	—	73	2	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	8	no	—
139	C₂²×C₈	—	73	8	no	—
140	C₂²×C₈	—	73	4	no	—
141	C₂²×C₈	—	73	2	no	—
142	C₂²×C₈	—	73	2	no	—
143	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
144	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
145	C₂×C₄²	—	73	2	no	—
146	C₂×C₄²	—	73	1	no	—
147	C₂×C₄²	—	73	4	no	—
148	C₂×C₄²	—	73	4	no	—
149	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
150	M₃₂	—	73	32	no	—
151	C₂×C₁₆	—	73	8	no	—
152	C₂×C₁₆	—	73	16	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	4	no	—
155	C₄×C₈	—	73	2	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	—	73	2	no	—
158	C₄×C₈	C₂×#64408	1	4	yes	—
159	C₂⁴	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	2	no	—
162	C₂²×C₄	—	73	4	no	—
163	C₂²×C₄	—	73	2	no	—
164	C₂²×C₄	—	73	4	no	—
165	C₂²×C₄	—	73	2	no	—
166	C₂²×C₄	—	73	8	no	—
167	C₂²×C₄	—	146	8	no	—
168	C₂²×C₄	#11924036	1	2	yes	—
169	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
170	C₂×C₈	—	73	4	no	—
171	C₂×C₈	—	73	1	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	4	no	—
175	C₂×C₈	—	73	16	no	—
176	C₂×C₈	—	73	8	no	—
177	C₂×C₈	—	73	8	no	—
178	C₂×C₈	C₄×#64408	1	4	yes	—
179	C₂×C₈	#11924036	1	4	yes	—
180	C₂×C₈	C₂²×#64408	1	2	yes	—
181	C₂×C₈	C₂²×#64408	1	2	yes	—
182	C₄²	—	73	8	no	—
183	C₄²	—	73	2	no	—
184	C₄²	—	73	2	no	—
185	C₄²	#11924036	1	2	yes	—
186	C₄²	C₂²×#64408	1	2	yes	—
187	C₁₆	—	73	16	no	—
188	C₂³	—	73	2	no	—
189	C₂³	—	73	4	no	—
190	C₂³	—	146	4	no	—
191	C₂³	C₂×#11924036	1	1	yes	characteristic
192	C₂×C₄	—	73	2	no	—
193	C₂×C₄	—	73	4	no	—
194	C₂×C₄	—	73	4	no	—
195	C₂×C₄	—	73	4	no	—
196	C₂×C₄	—	73	8	no	—
197	C₂×C₄	—	73	4	no	—
198	C₂×C₄	—	146	4	no	—
199	C₂×C₄	—	146	16	no	—
200	C₂×C₄	#14556666	1	8	yes	direct product factor
201	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
202	C₂×C₄	C₂×#11924036	1	4	yes	—
203	C₂×C₄	C₂×#11924036	1	2	yes	—
204	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
205	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
206	C₈	—	73	2	no	—
207	C₈	—	73	2	no	—
208	C₈	—	73	4	no	—
209	C₈	C₂×C₄×#64408	1	4	yes	—
210	C₈	C₂×#11924036	1	4	yes	—
211	C₂²	—	73	4	no	—
212	C₂²	—	146	4	no	—
213	C₂²	—	146	8	no	—
214	C₂²	C₄×#11924036	1	2	yes	—
215	C₂²	C₂×#14556666	1	4	yes	—
216	C₂²	C₂²×#11924036	1	1	yes	characteristic
217	C₄	—	73	4	no	—
218	C₄	—	146	4	no	—
219	C₄	C₄×#11924036	1	2	yes	—
220	C₄	C₂×#14556666	1	8	yes	direct product factor
221	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
222	C₄	C₂²×#11924036	1	1	yes	characteristic
223	C₂	—	146	4	no	—
224	C₂	C₄×#14556666	1	4	yes	direct product factor
225	C₂	C₂×C₄×#11924036	1	1	yes	characteristic
226	C₂	C₂²×#14556666	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×#14556666	C₂	direct (N × H)
3. C₂×#17989476	C₂	semidirect
4. C₂×C₄×#11924031	C₂	semidirect
6. C₂×#14556666	C₄	direct (N × H)
9. #17989476	C₂²	semidirect
10. C₄×#11924031	C₂²	semidirect
20. #14556666	C₂×C₄	direct (N × H)
21. C₂×#11924031	C₂×C₄	semidirect
45. #11924031	C₂²×C₄	semidirect
78. C₂×C₄×C₇₃	M₃₂	semidirect
104. C₄×C₇₃	C₂×M₃₂	semidirect
110. C₂×C₇₃	C₄×M₃₂	semidirect
119. C₇₃	C₂×C₄×M₃₂	semidirect
200. C₂×C₄	#14556666	direct (N × H)
220. C₄	C₂×#14556666	direct (N × H)
224. C₂	C₄×#14556666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556666	2
C₄×#14556666	2
C₂×#17989476	2
C₂×C₄×#11924031	2
C₂×C₄×C₈×D₇₃	2
C₂×C₄×M₃₂	73

Subgroups — C2×C4×#14556666 #34029784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14556666 #34029784