Subgroups — C₂×#22612296 #33244208

Order 18688.

3654 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₇₃	C₂	1	2	yes	maximal
2	#22612296	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×Dic₇₃	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₇₃	C₂	1	3	yes	maximal
5	C₂×#17988433	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₇₃	C₄	1	32	yes	complemented
8	C₂×Q₈×Dic₇₃	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₇₃	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₇₃	C₂²	1	6	yes	—
11	C₂²×C₄×Dic₇₃	C₂²	1	6	yes	—
12	C₄⋊C₄×Dic₇₃	C₂²	1	48	yes	—
13	C₄×C₄⋊Dic₇₃	C₂²	1	24	yes	—
14	#17988433	C₂²	1	24	yes	—
15	C₂×#14556498	C₂²	1	1	yes	characteristic
16	C₂×#14556498	C₂²	1	3	yes	—
17	C₂×#14556498	C₂²	1	6	yes	—
18	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₇₃	C₄	1	16	yes	complemented
20	C₂×C₄×Q₈×C₇₃	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
22	C₂²×C₄²×C₇₃	C₂²	1	3	yes	—
23	C₂³×Dic₇₃	C₂³	1	2	yes	—
24	C₂³×Dic₇₃	C₂³	1	6	yes	—
25	Q₈×Dic₇₃	C₂×C₄	1	128	yes	complemented
26	C₂×C₄⋊Dic₇₃	C₂×C₄	1	48	yes	—
27	C₂×C₄⋊Dic₇₃	C₂³	1	24	yes	—
28	C₂×C₄×Dic₇₃	—	4	12	no	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	48	yes	—
30	C₂×C₄×Dic₇₃	C₂³	1	24	yes	—
31	C₂×C₄×Dic₇₃	C₂³	1	24	yes	—
32	#14556498	C₂³	1	4	yes	—
33	#14556498	C₂³	1	12	yes	—
34	#14556498	C₂³	1	24	yes	—
35	C₂²×Q₈×C₇₃	C₂×C₄	1	16	yes	—
36	C₂²×Q₈×C₇₃	C₂×C₄	1	8	yes	—
37	C₂²×Q₈×C₇₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
40	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
41	C₄×Q₈×C₇₃	C₂×C₄	1	64	yes	complemented
42	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	24	yes	—
43	C₂×C₄⋊C₄×C₇₃	C₂³	1	12	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	24	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	12	yes	—
46	C₂²×Dic₇₃	—	2	12	no	—
47	C₂²×Dic₇₃	—	2	4	no	—
48	C₂²×Dic₇₃	—	2	4	no	—
49	C₂²×Dic₇₃	—	2	12	no	—
50	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
51	C₂²×Dic₇₃	C₂²×C₄	1	24	yes	—
52	C₂²×Dic₇₃	C₂⁴	1	4	yes	—
53	C₂²×Dic₇₃	C₂⁴	1	12	yes	—
54	C₄⋊Dic₇₃	C₂²×C₄	1	96	yes	—
55	C₄×Dic₇₃	—	4	24	no	—
56	C₄×Dic₇₃	C₂²×C₄	1	96	yes	—
57	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₇₃	C₄²	1	64	yes	complemented
59	C₂×Q₈×C₇₃	C₂²×C₄	1	32	yes	—
60	C₂×Q₈×C₇₃	C₂²×C₄	1	16	yes	—
61	C₂²×C₄×C₇₃	—	2	6	no	—
62	C₂²×C₄×C₇₃	—	2	6	no	—
63	C₂²×C₄×C₇₃	—	2	4	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
68	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
71	C₄⋊C₄×C₇₃	C₂²×C₄	1	48	yes	—
72	C₄²×C₇₃	C₂²×C₄	1	48	yes	—
73	C₂³×C₇₃	C₂³×C₄	1	4	yes	—
74	C₂³×C₇₃	C₂³×C₄	1	2	yes	—
75	C₂³×C₇₃	H(5,𝔽₂)	1	2	yes	—
76	C₂³×C₇₃	2₋¹⁺⁴	1	2	yes	—
77	C₂³×C₇₃	2₋¹⁺⁴	1	4	yes	—
78	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
79	Q₈×C₇₃	C₂×C₄²	1	64	yes	complemented
80	C₂×C₄×C₇₃	—	2	8	no	—
81	C₂×C₄×C₇₃	—	2	24	no	—
82	C₂×C₄×C₇₃	—	2	6	no	—
83	C₂×C₄×C₇₃	—	2	6	no	—
84	C₂×C₄×C₇₃	—	2	4	no	—
85	C₂×C₄×C₇₃	C₂×C₄²	1	48	yes	—
86	C₂×C₄×C₇₃	C₂³×C₄	1	24	yes	—
87	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
90	C₂×Dic₇₃	—	2	16	no	—
91	C₂×Dic₇₃	—	2	48	no	—
92	C₂×Dic₇₃	—	2	12	no	—
93	C₂×Dic₇₃	—	2	4	no	—
94	C₂×Dic₇₃	—	2	4	no	—
95	C₂×Dic₇₃	—	2	12	no	—
96	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
97	C₂×Dic₇₃	C₂³×C₄	1	24	yes	—
98	C₂²×C₇₃	C₂²×C₄²	1	4	yes	—
99	C₂²×C₇₃	#519	1	8	yes	—
100	C₂²×C₇₃	#519	1	4	yes	—
101	C₂²×C₇₃	#520	1	8	yes	—
102	C₂²×C₇₃	#520	1	4	yes	—
103	C₂²×C₇₃	C₂⁴×C₄	1	2	yes	—
104	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
105	C₂²×C₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
107	C₂²×C₇₃	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₄×C₇₃	—	2	4	no	—
109	C₄×C₇₃	—	2	12	no	—
110	C₄×C₇₃	C₂²×C₄²	1	24	yes	—
111	Dic₇₃	—	2	8	no	—
112	Dic₇₃	—	2	24	no	—
113	C₂×#2276	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	#2276	1	8	yes	complemented
115	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#519	1	2	yes	—
117	C₂×C₇₃	C₂×#519	1	1	yes	characteristic
118	C₂×C₇₃	C₂×#520	1	2	yes	—
119	C₂×C₇₃	C₂×#520	1	1	yes	characteristic
120	C₂²×C₄×Q₈	—	73	2	no	—
121	C₂²×C₄×Q₈	D₇₃	1	1	yes	characteristic
122	#2276	—	73	16	no	—
123	C₂×C₄×C₄⋊C₄	—	73	3	no	—
124	C₂×C₄×C₄⋊C₄	—	73	6	no	—
125	C₂×#376	—	73	3	no	—
126	C₇₃	C₂×#2276	1	1	yes	characteristic, complemented, Sylow
127	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
128	C₂×C₄×Q₈	—	73	32	no	—
129	C₂×C₄×Q₈	—	73	16	no	—
130	C₂×C₄×Q₈	Dic₇₃	1	16	yes	complemented
131	C₂×C₄×Q₈	C₂×D₇₃	1	8	yes	—
132	C₂²×C₄⋊C₄	—	73	6	no	—
133	C₂²×C₄⋊C₄	C₂×D₇₃	1	3	yes	—
134	C₂²×C₄²	—	73	6	no	—
135	C₂²×C₄²	—	73	6	no	—
136	C₂²×C₄²	C₂×D₇₃	1	3	yes	—
137	C₄×C₄⋊C₄	—	73	48	no	—
138	C₄×C₄⋊C₄	—	73	24	no	—
139	#376	—	73	24	no	—
140	C₂×#95	—	73	1	no	—
141	C₂×#95	—	73	3	no	—
142	C₂×#95	—	73	6	no	—
143	C₂²×Q₈	C₄×D₇₃	1	16	yes	—
144	C₂²×Q₈	C₂×Dic₇₃	1	8	yes	—
145	C₂²×Q₈	C₂²×D₇₃	1	4	yes	—
146	C₂³×C₄	—	73	6	no	—
147	C₂³×C₄	—	73	2	no	—
148	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
149	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
151	C₄×Q₈	—	73	128	no	—
152	C₄×Q₈	C₂×Dic₇₃	1	64	yes	complemented
153	C₂×C₄⋊C₄	—	73	48	no	—
154	C₂×C₄⋊C₄	—	73	24	no	—
155	C₂×C₄⋊C₄	C₂×Dic₇₃	1	24	yes	—
156	C₂×C₄⋊C₄	C₂²×D₇₃	1	12	yes	—
157	C₂×C₄²	—	292	12	no	—
158	C₂×C₄²	—	73	24	no	—
159	C₂×C₄²	—	73	48	no	—
160	C₂×C₄²	—	73	24	no	—
161	C₂×C₄²	C₂×Dic₇₃	1	24	yes	—
162	C₂×C₄²	C₂²×D₇₃	1	12	yes	—
163	#95	—	73	4	no	—
164	#95	—	73	24	no	—
165	#95	—	73	12	no	—
166	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
167	C₂×Q₈	C₄×Dic₇₃	1	64	yes	complemented
168	C₂×Q₈	C₂×C₄×D₇₃	1	32	yes	—
169	C₂×Q₈	C₂²×Dic₇₃	1	16	yes	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	73	8	no	—
174	C₂²×C₄	—	73	24	no	—
175	C₂²×C₄	—	73	4	no	—
176	C₂²×C₄	—	73	12	no	—
177	C₂²×C₄	—	146	12	no	—
178	C₂²×C₄	—	146	4	no	—
179	C₂²×C₄	—	146	4	no	—
180	C₂²×C₄	—	146	12	no	—
181	C₂²×C₄	C₂×C₄×D₇₃	1	24	yes	—
182	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
183	C₂²×C₄	C₂²×Dic₇₃	1	12	yes	—
184	C₂²×C₄	C₂²×Dic₇₃	1	12	yes	—
185	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
186	C₂²×C₄	C₂³×D₇₃	1	6	yes	—
187	C₂²×C₄	C₂³×D₇₃	1	6	yes	—
188	C₄⋊C₄	—	73	96	no	—
189	C₄⋊C₄	C₂²×Dic₇₃	1	48	yes	—
190	C₄²	—	292	24	no	—
191	C₄²	—	73	96	no	—
192	C₄²	C₂²×Dic₇₃	1	48	yes	—
193	C₂³	C₂²×C₄×D₇₃	1	4	yes	—
194	C₂³	C₂³×Dic₇₃	1	2	yes	—
195	C₂³	#14556637	1	2	yes	—
196	C₂³	#14556638	1	2	yes	—
197	C₂³	#14556639	1	4	yes	—
198	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
199	Q₈	C₂×C₄×Dic₇₃	1	64	yes	complemented
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	73	8	no	—
206	C₂×C₄	—	73	24	no	—
207	C₂×C₄	—	146	12	no	—
208	C₂×C₄	—	146	4	no	—
209	C₂×C₄	—	146	4	no	—
210	C₂×C₄	—	146	12	no	—
211	C₂×C₄	—	146	16	no	—
212	C₂×C₄	—	146	48	no	—
213	C₂×C₄	C₂×C₄×Dic₇₃	1	48	yes	—
214	C₂×C₄	C₂²×C₄×D₇₃	1	24	yes	—
215	C₂×C₄	C₂³×Dic₇₃	1	4	yes	—
216	C₂×C₄	C₂³×Dic₇₃	1	12	yes	—
217	C₂×C₄	C₂³×Dic₇₃	1	12	yes	—
218	C₂²	C₂²×C₄×Dic₇₃	1	4	yes	—
219	C₂²	#17989063	1	8	yes	—
220	C₂²	#17989065	1	4	yes	—
221	C₂²	#17989069	1	8	yes	—
222	C₂²	#17989070	1	4	yes	—
223	C₂²	C₂³×C₄×D₇₃	1	2	yes	—
224	C₂²	C₂⁴×Dic₇₃	1	1	yes	characteristic
225	C₂²	C₂×#14556637	1	1	yes	characteristic
226	C₂²	C₂×#14556638	1	1	yes	characteristic
227	C₂²	C₂×#14556639	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	12	no	—
230	C₄	—	146	8	no	—
231	C₄	—	146	24	no	—
232	C₄	C₂²×C₄×Dic₇₃	1	24	yes	—
233	C₂	#22612296	1	8	yes	direct product factor
234	C₂	C₂³×C₄×Dic₇₃	1	1	yes	characteristic
235	C₂	C₂×#17989063	1	2	yes	—
236	C₂	C₂×#17989065	1	1	yes	characteristic
237	C₂	C₂×#17989069	1	2	yes	—
238	C₂	C₂×#17989070	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇₃
commutator	C₂×C₇₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₃
Fitting	C₂²×C₄×Q₈×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂⁴
cosocle	C₂³×C₇₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₇₃
hypocenter	C₇₃	C₂×#2276
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22612296	C₂	direct (N × H)
7. C₂×Q₈×Dic₇₃	C₄	semidirect
19. C₂×C₄×Q₈×C₇₃	C₄	semidirect
25. Q₈×Dic₇₃	C₂×C₄	semidirect
41. C₄×Q₈×C₇₃	C₂×C₄	semidirect
58. C₂×Q₈×C₇₃	C₄²	semidirect
79. Q₈×C₇₃	C₂×C₄²	semidirect
114. C₂×C₇₃	#2276	semidirect
126. C₇₃	C₂×#2276	semidirect
130. C₂×C₄×Q₈	Dic₇₃	semidirect
152. C₄×Q₈	C₂×Dic₇₃	semidirect
167. C₂×Q₈	C₄×Dic₇₃	semidirect
199. Q₈	C₂×C₄×Dic₇₃	semidirect
233. C₂	#22612296	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×Q₈×Dic₇₃	2
#22612296	2
C₂×C₄⋊C₄×Dic₇₃	2
C₂×C₄×C₄⋊Dic₇₃	2
C₂×#17988433	2
C₂²×C₄×Q₈×C₇₃	2
C₂×#2276	73

Subgroups — C2×#22612296 #33244208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612296 #33244208