Subgroups — C₂×C₄×#14556665 #34029782

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×#14556665	C₂	1	2	yes	maximal
2	C₄×#14556665	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989475	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11924031	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₄	1	16	yes	direct product factor
7	C₂×#14556665	C₂²	1	8	yes	—
8	C₂²×#11924031	C₂²	1	4	yes	—
9	#17989475	C₂²	1	8	yes	complemented
10	C₄×#11924031	C₂²	1	8	yes	complemented
11	C₂³×#10596831	C₄	1	1	yes	characteristic
12	C₂³×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₄	1	4	yes	—
15	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×C₄×#10596831	C₂²	1	4	yes	—
19	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
20	#14556665	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×#10596831	C₈	1	8	yes	—
24	C₂²×#10596831	C₂×C₄	1	8	yes	—
25	C₂²×#10596831	C₂×C₄	1	2	yes	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₂×C₄	1	4	yes	—
30	C₂²×#10596831	C₂³	1	1	yes	characteristic
31	C₂²×#10596831	C₂³	1	2	yes	—
32	C₄×#10596831	C₂×C₄	1	2	yes	—
33	C₄×#10596831	C₂×C₄	1	2	yes	—
34	C₄×#10596831	C₂×C₄	1	4	yes	—
35	C₄×#10596831	C₂×C₄	1	4	yes	—
36	C₄×#10596831	C₂³	1	2	yes	—
37	C₄×#10596831	C₂³	1	2	yes	—
38	C₂³×C₄×C₇₃	C₈	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	C₈	1	4	yes	—
41	C₂×C₄²×C₇₃	C₈	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	#11924031	C₂²×C₄	1	16	yes	complemented
46	C₂×#10596831	C₄²	1	8	yes	—
47	C₂×#10596831	C₄²	1	4	yes	—
48	C₂×#10596831	C₂×C₈	1	16	yes	—
49	C₂×#10596831	C₂×C₈	1	4	yes	—
50	C₂×#10596831	C₂×C₈	1	4	yes	—
51	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
52	C₂×#10596831	C₂²×C₄	1	2	yes	—
53	C₂×#10596831	C₂²×C₄	1	4	yes	—
54	C₂×#10596831	C₂²×C₄	1	8	yes	—
55	C₂×#10596831	C₂²×C₄	1	2	yes	—
56	C₂×#10596831	C₂²×C₄	1	2	yes	—
57	C₂×#10596831	C₂⁴	1	1	yes	characteristic
58	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	—	2	8	no	—
60	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂×C₈	1	8	yes	—
63	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₇₃	C₂×C₈	1	2	yes	—
70	C₄²×C₇₃	C₂×C₈	1	8	yes	—
71	C₄²×C₇₃	C₂×C₈	1	2	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₇₃	—	2	4	no	—
75	C₂³×C₇₃	C₄×C₈	1	4	yes	—
76	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
77	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	4	no	—
79	C₂×C₄×C₇₃	—	2	16	no	—
80	C₂×C₄×C₇₃	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₇₃	M₃₂	1	8	yes	complemented
83	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
92	#10596831	C₂×C₄²	1	2	yes	—
93	#10596831	C₂×C₄²	1	4	yes	—
94	#10596831	C₂²×C₈	1	4	yes	—
95	#10596831	C₂²×C₈	1	4	yes	—
96	#10596831	C₂³×C₄	1	2	yes	—
97	C₂²×C₇₃	—	2	8	no	—
98	C₂²×C₇₃	—	2	4	no	—
99	C₂²×C₇₃	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
101	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
102	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₇₃	—	2	4	no	—
104	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
105	C₄×C₇₃	C₂×C₄×C₈	1	4	yes	—
106	C₄×C₇₃	C₂×M₃₂	1	8	yes	complemented
107	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
108	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
109	C₂×C₄×M₃₂	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	—	2	4	no	—
111	C₂×C₇₃	C₄×M₃₂	1	4	yes	complemented
112	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
113	C₂×C₇₃	C₂²×M₃₂	1	2	yes	—
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	#64408	1	1	yes	characteristic
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	4	no	—
128	C₂×C₄×C₈	—	73	4	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	#346	—	73	8	no	—
133	C₄×C₁₆	—	73	8	no	—
134	C₂³×C₄	#10596833	1	2	yes	—
135	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
136	C₂²×C₈	—	73	8	no	—
137	C₂²×C₈	—	73	8	no	—
138	C₂²×C₈	—	73	4	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	73	2	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	73	1	no	—
144	C₂²×C₈	—	73	2	no	—
145	C₂×C₄²	#10596833	1	2	yes	—
146	C₂×C₄²	#10596833	1	4	yes	—
147	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
148	C₂×C₄²	C₂×#64408	1	4	yes	—
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	4	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	—	73	2	no	—
158	C₄×C₈	—	73	2	no	—
159	C₂⁴	C₂×#10596833	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	C₄×#64408	1	4	yes	—
162	C₂²×C₄	C₂×#10596833	1	2	yes	—
163	C₂²×C₄	C₂×#10596833	1	8	yes	—
164	C₂²×C₄	C₂×#10596833	1	2	yes	—
165	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
166	C₂²×C₄	C₂×#10596833	1	2	yes	—
167	C₂²×C₄	#11924036	1	4	yes	—
168	C₂²×C₄	C₂²×#64408	1	2	yes	—
169	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
170	C₂×C₈	—	73	16	no	—
171	C₂×C₈	—	73	4	no	—
172	C₂×C₈	—	73	4	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	2	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	2	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	73	8	no	—
179	C₂×C₈	—	73	4	no	—
180	C₂×C₈	—	73	4	no	—
181	C₂×C₈	—	73	8	no	—
182	C₄²	C₂×#10596833	1	2	yes	—
183	C₄²	C₂×#10596833	1	8	yes	—
184	C₄²	C₂×#10596833	1	2	yes	—
185	C₄²	C₂²×#64408	1	2	yes	—
186	C₄²	C₂²×#64408	1	2	yes	—
187	C₁₆	—	73	16	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#10596833	1	4	yes	—
190	C₂³	C₂²×#10596833	1	2	yes	—
191	C₂³	C₂²×#10596833	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	C₄×#10596833	1	8	yes	—
195	C₂×C₄	C₄×#10596833	1	4	yes	—
196	C₂×C₄	#14556665	1	8	yes	direct product factor
197	C₂×C₄	C₂×C₄×#64408	1	4	yes	—
198	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
199	C₂×C₄	C₂²×#10596833	1	4	yes	—
200	C₂×C₄	C₂²×#10596833	1	4	yes	—
201	C₂×C₄	C₂²×#10596833	1	2	yes	—
202	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
203	C₂×C₄	C₂×#11924036	1	4	yes	—
204	C₂×C₄	C₂×#11924036	1	2	yes	—
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	—	73	4	no	—
210	C₈	—	73	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	8	no	—
213	C₂²	C₂×C₄×#10596833	1	4	yes	—
214	C₂²	C₂×C₄×#10596833	1	2	yes	—
215	C₂²	C₂×#14556665	1	4	yes	—
216	C₂²	C₂³×#10596833	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	C₂×C₄×#10596833	1	4	yes	—
219	C₄	C₂×C₄×#10596833	1	2	yes	—
220	C₄	C₂×#14556665	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	—	2	4	no	—
224	C₂	C₄×#14556665	1	4	yes	direct product factor
225	C₂	C₂²×C₄×#10596833	1	1	yes	characteristic
226	C₂	C₂²×#14556665	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄×#14556665	C₂	direct (N × H)
3. C₂×#17989475	C₂	semidirect
4. C₂×C₄×#11924031	C₂	semidirect
6. C₂×#14556665	C₄	direct (N × H)
9. #17989475	C₂²	semidirect
10. C₄×#11924031	C₂²	semidirect
20. #14556665	C₂×C₄	direct (N × H)
21. C₂×#11924031	C₂×C₄	semidirect
45. #11924031	C₂²×C₄	semidirect
82. C₂×C₄×C₇₃	M₃₂	semidirect
106. C₄×C₇₃	C₂×M₃₂	semidirect
111. C₂×C₇₃	C₄×M₃₂	semidirect
119. C₇₃	C₂×C₄×M₃₂	semidirect
196. C₂×C₄	#14556665	direct (N × H)
220. C₄	C₂×#14556665	direct (N × H)
224. C₂	C₄×#14556665	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×#14556665	2
C₄×#14556665	2
C₂×#17989475	2
C₂×C₄×#11924031	2
C₂²×C₄×#10596831	2
C₂×C₄×M₃₂	73

Subgroups — C2×C4×#14556665 #34029782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14556665 #34029782