Subgroups — C₂³×#14556509 #33949468

Order 18688.

4990 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11924012	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14556509	C₂	1	28	yes	maximal, direct product factor
4	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₄₆	—	2	1	no	—
6	C₂³×Dic₁₄₆	—	2	7	no	—
7	C₂³×Dic₁₄₆	—	2	1	no	—
8	C₂³×Dic₁₄₆	—	2	7	no	—
9	C₂³×Dic₁₄₆	C₄	1	14	yes	—
10	C₂³×Dic₁₄₆	C₂²	1	14	yes	—
11	C₂⁴×Dic₇₃	C₄	1	2	yes	—
12	C₂²×#11924012	—	2	7	no	—
13	C₂²×#11924012	C₂²	1	14	yes	—
14	C₂³×#10596831	—	2	1	no	—
15	C₂×#14556509	C₂²	1	112	yes	direct product factor
16	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₇₃	—	2	7	no	—
18	C₂²×M₁₆×C₇₃	C₂²	1	14	yes	—
19	C₂³×C₈×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	—	2	84	no	—
21	C₂²×Dic₁₄₆	—	2	14	no	—
22	C₂²×Dic₁₄₆	—	2	14	no	—
23	C₂²×Dic₁₄₆	—	2	14	no	—
24	C₂²×Dic₁₄₆	—	2	42	no	—
25	C₂²×Dic₁₄₆	—	2	14	no	—
26	C₂²×Dic₁₄₆	—	2	42	no	—
27	C₂²×Dic₁₄₆	C₂×C₄	1	84	yes	—
28	C₂²×Dic₁₄₆	C₂³	1	28	yes	—
29	C₂³×Dic₇₃	—	2	14	no	—
30	C₂³×Dic₇₃	—	2	2	no	—
31	C₂³×Dic₇₃	C₂×C₄	1	28	yes	—
32	C₂×#11924012	—	2	42	no	—
33	C₂×#11924012	C₂³	1	28	yes	—
34	C₂²×#10596831	—	2	14	no	—
35	#14556509	C₂³	1	64	yes	direct product factor
36	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₃	C₂×C₄	1	7	yes	—
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	7	yes	—
40	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₇₃	D₄	1	7	yes	—
42	C₂³×C₄×C₇₃	C₂³	1	7	yes	—
43	C₂×M₁₆×C₇₃	—	2	42	no	—
44	C₂×M₁₆×C₇₃	C₂³	1	28	yes	—
45	C₂²×C₈×C₇₃	—	2	14	no	—
46	C₂×Dic₁₄₆	—	2	168	no	—
47	C₂×Dic₁₄₆	—	2	84	no	—
48	C₂×Dic₁₄₆	—	2	84	no	—
49	C₂×Dic₁₄₆	—	2	28	no	—
50	C₂×Dic₁₄₆	—	2	28	no	—
51	C₂×Dic₁₄₆	—	2	28	no	—
52	C₂×Dic₁₄₆	—	2	28	no	—
53	C₂×Dic₁₄₆	C₂²×C₄	1	56	yes	—
54	C₂×Dic₁₄₆	C₂⁴	1	8	yes	—
55	C₂²×Dic₇₃	—	2	84	no	—
56	C₂²×Dic₇₃	—	2	28	no	—
57	C₂²×Dic₇₃	C₂²×C₄	1	56	yes	—
58	#11924012	—	2	28	no	—
59	#11924012	C₂⁴	1	8	yes	—
60	C₂×#10596831	—	2	28	no	—
61	C₂⁴×C₇₃	—	2	8	no	—
62	C₂⁴×C₇₃	C₂²⋊C₄	1	7	yes	—
63	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁴×C₇₃	C₂²×C₄	1	7	yes	—
65	C₂²×C₄×C₇₃	C₂²⋊C₄	1	42	yes	—
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	7	yes	—
67	C₂²×C₄×C₇₃	C₂²⋊C₄	1	7	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	21	yes	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	21	yes	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	21	yes	—
73	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
74	M₁₆×C₇₃	—	2	28	no	—
75	M₁₆×C₇₃	C₂⁴	1	8	yes	—
76	C₂×C₈×C₇₃	—	2	28	no	—
77	C₂³×C₇₃	—	2	28	no	—
78	C₂³×C₇₃	—	2	28	no	—
79	C₂³×C₇₃	#101	1	8	yes	complemented
80	C₂³×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
81	C₂³×C₇₃	C₂×C₂²⋊C₄	1	7	yes	—
82	C₂³×C₇₃	C₂³×C₄	1	7	yes	—
83	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	42	yes	—
84	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
85	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	21	yes	—
86	C₂×C₄×C₇₃	C₂³×C₄	1	7	yes	—
87	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
88	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
89	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
90	C₂×C₄×C₇₃	C₂²×D₄	1	7	yes	—
91	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
92	C₂×Dic₇₃	—	2	56	no	—
93	C₂×Dic₇₃	—	2	56	no	—
94	C₂×Dic₇₃	C₂³×C₄	1	16	yes	—
95	Dic₁₄₆	—	2	56	no	—
96	Dic₁₄₆	—	2	56	no	—
97	Dic₁₄₆	—	2	8	no	—
98	Dic₁₄₆	—	2	8	no	—
99	C₈×C₇₃	—	2	8	no	—
100	#10596831	—	2	8	no	—
101	C₂²×C₇₃	—	2	42	no	—
102	C₂²×C₇₃	—	2	14	no	—
103	C₂²×C₇₃	C₂×#101	1	28	yes	complemented
104	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
105	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
106	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
107	C₄×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
108	C₄×C₇₃	C₂²×C₂²⋊C₄	1	7	yes	—
109	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
111	Dic₇₃	—	2	16	no	—
112	C₂³×#101	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	—	2	7	no	—
114	C₂×C₇₃	—	2	1	no	—
115	C₂×C₇₃	C₂²×#101	1	14	yes	complemented
116	C₂×C₇₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
117	C₂⁴×Q₈	—	73	1	no	—
118	C₂³×M₁₆	—	73	1	no	—
119	C₂³×M₁₆	D₇₃	1	1	yes	characteristic
120	C₂²×#101	—	73	28	no	—
121	C₇₃	C₂³×#101	1	1	yes	characteristic, complemented, Sylow
122	C₂³×Q₈	—	73	14	no	—
123	C₂³×Q₈	—	73	14	no	—
124	C₂³×Q₈	—	146	7	no	—
125	C₂³×Q₈	—	146	1	no	—
126	C₂³×Q₈	—	146	7	no	—
127	C₂³×Q₈	—	146	1	no	—
128	C₂⁴×C₄	—	73	2	no	—
129	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
130	C₂²×M₁₆	—	2	7	no	—
131	C₂²×M₁₆	—	73	14	no	—
132	C₂²×M₁₆	—	146	7	no	—
133	C₂²×M₁₆	C₂×D₇₃	1	14	yes	—
134	C₂³×C₈	—	2	1	no	—
135	C₂³×C₈	—	146	1	no	—
136	C₂×#101	—	73	112	no	—
137	C₂⁵	C₄×D₇₃	1	1	yes	characteristic
138	C₂²×Q₈	—	73	84	no	—
139	C₂²×Q₈	—	73	28	no	—
140	C₂²×Q₈	—	146	84	no	—
141	C₂²×Q₈	—	146	14	no	—
142	C₂²×Q₈	—	146	14	no	—
143	C₂²×Q₈	—	146	14	no	—
144	C₂²×Q₈	—	146	42	no	—
145	C₂²×Q₈	—	146	14	no	—
146	C₂²×Q₈	—	146	42	no	—
147	C₂³×C₄	—	73	28	no	—
148	C₂³×C₄	—	146	14	no	—
149	C₂³×C₄	—	146	2	no	—
150	C₂³×C₄	C₄×D₇₃	1	7	yes	—
151	C₂³×C₄	D₂₉₂	1	7	yes	—
152	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
153	C₂³×C₄	C₂²⋊D₇₃	1	7	yes	—
154	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂³×C₄	C₂²×D₇₃	1	7	yes	—
156	C₂×M₁₆	—	2	42	no	—
157	C₂×M₁₆	—	73	28	no	—
158	C₂×M₁₆	—	146	42	no	—
159	C₂×M₁₆	C₂²×D₇₃	1	28	yes	—
160	C₂²×C₈	—	2	14	no	—
161	C₂²×C₈	—	146	14	no	—
162	#101	—	73	64	no	—
163	C₂⁴	—	2	8	no	—
164	C₂⁴	#11924005	1	7	yes	—
165	C₂⁴	#11924005	1	1	yes	characteristic
166	C₂⁴	C₂×C₄×D₇₃	1	7	yes	—
167	C₂×Q₈	—	73	56	no	—
168	C₂×Q₈	—	73	8	no	—
169	C₂×Q₈	—	146	168	no	—
170	C₂×Q₈	—	146	84	no	—
171	C₂×Q₈	—	146	84	no	—
172	C₂×Q₈	—	146	28	no	—
173	C₂×Q₈	—	146	28	no	—
174	C₂×Q₈	—	146	28	no	—
175	C₂×Q₈	—	146	28	no	—
176	C₂²×C₄	—	73	56	no	—
177	C₂²×C₄	—	146	84	no	—
178	C₂²×C₄	—	146	28	no	—
179	C₂²×C₄	#11924005	1	42	yes	—
180	C₂²×C₄	#11924005	1	7	yes	—
181	C₂²×C₄	#11924005	1	7	yes	—
182	C₂²×C₄	C₂×C₄×D₇₃	1	21	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₇₃	1	7	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₇₃	1	21	yes	—
185	C₂²×C₄	C₂×D₂₉₂	1	7	yes	—
186	C₂²×C₄	C₂×D₂₉₂	1	21	yes	—
187	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
188	M₁₆	—	2	28	no	—
189	M₁₆	—	73	8	no	—
190	M₁₆	—	146	28	no	—
191	M₁₆	C₂³×D₇₃	1	8	yes	—
192	C₂×C₈	—	2	28	no	—
193	C₂×C₈	—	146	28	no	—
194	C₂³	—	2	28	no	—
195	C₂³	—	2	28	no	—
196	C₂³	#14556509	1	8	yes	direct product factor
197	C₂³	C₂×#11924005	1	21	yes	—
198	C₂³	C₂×#11924005	1	7	yes	—
199	C₂³	C₂²×C₄×D₇₃	1	7	yes	—
200	Q₈	—	146	8	no	—
201	Q₈	—	146	8	no	—
202	Q₈	—	146	56	no	—
203	Q₈	—	146	56	no	—
204	C₂×C₄	—	73	16	no	—
205	C₂×C₄	—	146	56	no	—
206	C₂×C₄	—	146	56	no	—
207	C₂×C₄	C₂×#11924005	1	42	yes	—
208	C₂×C₄	C₂×#11924005	1	21	yes	—
209	C₂×C₄	C₂×#11924005	1	21	yes	—
210	C₂×C₄	C₂²×C₄×D₇₃	1	7	yes	—
211	C₂×C₄	C₂²×C₂²⋊D₇₃	1	7	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₇₃	1	7	yes	—
213	C₂×C₄	C₂²×D₂₉₂	1	7	yes	—
214	C₂×C₄	C₂²×D₂₉₂	1	7	yes	—
215	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
216	C₈	—	2	8	no	—
217	C₈	—	146	8	no	—
218	C₂²	—	2	42	no	—
219	C₂²	—	2	14	no	—
220	C₂²	C₂×#14556509	1	28	yes	direct product factor
221	C₂²	C₂²×#11924005	1	7	yes	—
222	C₂²	C₂²×#11924005	1	7	yes	—
223	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
224	C₄	—	146	16	no	—
225	C₄	C₂²×#11924005	1	7	yes	—
226	C₄	C₂²×#11924005	1	7	yes	—
227	C₄	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
228	C₄	C₂³×D₂₉₂	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	7	no	—
231	C₂	C₂²×#14556509	1	14	yes	direct product factor
232	C₂	C₂³×#11924005	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11924005
commutator	C₂²×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂³×M₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²⋊C₄
prime radical	C₂⁵×C₇₃	C₂×C₄
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂³×M₁₆	D₇₃
hypocenter	C₇₃	C₂³×#101
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#14556509	C₂	direct (N × H)
15. C₂×#14556509	C₂²	direct (N × H)
35. #14556509	C₂³	direct (N × H)
79. C₂³×C₇₃	#101	semidirect
103. C₂²×C₇₃	C₂×#101	semidirect
115. C₂×C₇₃	C₂²×#101	semidirect
121. C₇₃	C₂³×#101	semidirect
196. C₂³	#14556509	direct (N × H)
220. C₂²	C₂×#14556509	direct (N × H)
231. C₂	C₂²×#14556509	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×Dic₁₄₆	2
C₂³×#11924012	2
C₂²×#14556509	2
C₂³×M₁₆×C₇₃	2
C₂³×#101	73

Subgroups — C23×#14556509 #33949468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556509 #33949468