Subgroups — C₂²×#17988610 #33949595

Order 18688.

3438 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	C₂³×#11924012	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988610	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556520	C₂	1	4	yes	maximal, complemented
5	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₇₃	C₄	1	2	yes	—
7	C₂²×#11924012	—	2	8	no	—
8	C₂²×#11924012	C₂²	1	12	yes	—
9	C₂³×#10596831	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₇₃	C₄	1	12	yes	—
11	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
14	C₂²×C₄×Dic₇₃	C₂²	1	12	yes	—
15	#17988610	C₂²	1	64	yes	direct product factor
16	C₂×#14556520	C₂²	1	48	yes	complemented
17	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₇₃	—	2	8	no	—
19	C₂²×M₁₆×C₇₃	C₂²	1	12	yes	—
20	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
21	C₂³×Dic₇₃	—	2	16	no	—
22	C₂³×Dic₇₃	C₂×C₄	1	24	yes	—
23	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₂×#11924012	—	2	48	no	—
26	C₂×#11924012	C₂³	1	16	yes	—
27	C₂²×#10596831	—	2	8	no	—
28	C₂²×#10596831	C₂³	1	12	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	48	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₇₃	C₂³	1	16	yes	—
34	C₂×C₄×Dic₇₃	C₂³	1	12	yes	—
35	#14556520	C₂³	1	64	yes	complemented
36	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₃	C₂×C₄	1	6	yes	—
38	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	8	yes	—
40	C₂³×C₄×C₇₃	Q₈	1	8	yes	—
41	C₂³×C₄×C₇₃	C₂³	1	6	yes	—
42	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₇₃	—	2	48	no	—
44	C₂×M₁₆×C₇₃	C₂³	1	16	yes	—
45	C₂²×C₈×C₇₃	—	2	8	no	—
46	C₂²×C₈×C₇₃	C₂³	1	12	yes	—
47	C₂²×Dic₇₃	—	2	8	no	—
48	C₂²×Dic₇₃	—	2	96	no	—
49	C₂²×Dic₇₃	M₁₆	1	16	yes	complemented
50	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
51	C₂²×Dic₇₃	C₂²×C₄	1	32	yes	—
52	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
53	#11924012	—	2	32	no	—
54	C₂×#10596831	—	2	24	no	—
55	C₂×#10596831	C₂⁴	1	8	yes	—
56	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	24	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
59	C₄×Dic₇₃	C₂⁴	1	8	yes	—
60	C₂⁴×C₇₃	—	2	8	no	—
61	C₂⁴×C₇₃	C₄⋊C₄	1	8	yes	—
62	C₂⁴×C₇₃	C₂²×C₄	1	6	yes	—
63	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₄⋊C₄	1	48	yes	—
65	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	24	yes	—
71	C₂²×C₄×C₇₃	C₂×Q₈	1	24	yes	—
72	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
74	C₂²×C₄×C₇₃	C₂⁴	1	4	yes	—
75	M₁₆×C₇₃	—	2	32	no	—
76	C₂×C₈×C₇₃	—	2	24	no	—
77	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
78	C₂³×C₇₃	—	2	24	no	—
79	C₂³×C₇₃	—	2	32	no	—
80	C₂³×C₇₃	C₂×C₄⋊C₄	1	24	yes	—
81	C₂³×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
82	C₂³×C₇₃	C₂×M₁₆	1	8	yes	—
83	C₂³×C₇₃	C₂³×C₄	1	4	yes	—
84	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
85	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	48	yes	—
86	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	12	yes	—
87	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	24	yes	—
88	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
92	C₂×C₄×C₇₃	C₂²×Q₈	1	8	yes	—
93	C₂×C₄×C₇₃	C₂²×Q₈	1	6	yes	—
94	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
95	C₂×Dic₇₃	—	2	24	no	—
96	C₂×Dic₇₃	—	2	64	no	—
97	C₂×Dic₇₃	C₂×M₁₆	1	48	yes	complemented
98	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
99	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
100	C₈×C₇₃	—	2	8	no	—
101	#10596831	—	2	8	no	—
102	C₂²×C₇₃	—	2	48	no	—
103	C₂²×C₇₃	—	2	8	no	—
104	C₂²×C₇₃	#423	1	16	yes	complemented
105	C₂²×C₇₃	C₂²×C₄⋊C₄	1	8	yes	—
106	C₂²×C₇₃	C₂²×C₄⋊C₄	1	6	yes	—
107	C₂²×C₇₃	C₂²×M₁₆	1	12	yes	—
108	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₇₃	C₂²×C₄⋊C₄	1	6	yes	—
110	C₄×C₇₃	C₂²×C₄⋊C₄	1	8	yes	—
111	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
112	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
113	Dic₇₃	—	2	8	no	—
114	Dic₇₃	C₂²×M₁₆	1	16	yes	complemented
115	C₂²×#423	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	—	2	8	no	—
117	C₂×C₇₃	C₂×#423	1	12	yes	complemented
118	C₂×C₇₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₇₃	C₂³×M₁₆	1	2	yes	—
120	C₂³×M₁₆	—	73	1	no	—
121	C₂³×M₁₆	D₇₃	1	1	yes	characteristic
122	C₂³×C₄²	—	73	1	no	—
123	C₂×#423	—	73	24	no	—
124	C₂²×C₄⋊C₈	—	73	4	no	—
125	C₇₃	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴×C₄	—	73	2	no	—
127	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
128	C₂²×M₁₆	—	2	8	no	—
129	C₂²×M₁₆	—	73	12	no	—
130	C₂²×M₁₆	—	146	8	no	—
131	C₂²×M₁₆	C₂×D₇₃	1	12	yes	—
132	C₂³×C₈	—	73	2	no	—
133	C₂³×C₈	C₂×D₇₃	1	2	yes	—
134	C₂²×C₄²	—	73	1	no	—
135	C₂²×C₄²	—	73	12	no	—
136	C₂²×C₄²	—	73	1	no	—
137	C₂²×C₄²	—	73	12	no	—
138	C₂²×C₄²	—	73	2	no	—
139	#423	—	73	64	no	—
140	C₂×C₄⋊C₈	—	73	48	no	—
141	C₂⁵	C₄×D₇₃	1	1	yes	characteristic
142	C₂³×C₄	—	73	2	no	—
143	C₂³×C₄	—	73	24	no	—
144	C₂³×C₄	—	73	2	no	—
145	C₂³×C₄	—	146	16	no	—
146	C₂³×C₄	Dic₁₄₆	1	8	yes	—
147	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
148	C₂³×C₄	C₄×D₇₃	1	6	yes	—
149	C₂³×C₄	C₂²⋊D₇₃	1	8	yes	—
150	C₂³×C₄	C₂²×D₇₃	1	6	yes	—
151	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	48	no	—
153	C₂×M₁₆	—	73	16	no	—
154	C₂×M₁₆	—	146	48	no	—
155	C₂×M₁₆	C₂²×D₇₃	1	16	yes	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	73	12	no	—
158	C₂²×C₈	—	146	8	no	—
159	C₂²×C₈	C₂²×D₇₃	1	12	yes	—
160	C₂×C₄²	—	73	12	no	—
161	C₂×C₄²	—	73	12	no	—
162	C₂×C₄²	—	73	48	no	—
163	C₂×C₄²	—	73	12	no	—
164	C₂×C₄²	—	73	12	no	—
165	C₂×C₄²	—	73	16	no	—
166	C₄⋊C₈	—	73	64	no	—
167	C₂⁴	—	2	8	no	—
168	C₂⁴	#11924007	1	8	yes	—
169	C₂⁴	C₂×C₄×D₇₃	1	6	yes	—
170	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
171	C₂²×C₄	—	73	16	no	—
172	C₂²×C₄	—	73	12	no	—
173	C₂²×C₄	—	73	32	no	—
174	C₂²×C₄	—	73	12	no	—
175	C₂²×C₄	—	146	96	no	—
176	C₂²×C₄	—	146	8	no	—
177	C₂²×C₄	#11924007	1	48	yes	—
178	C₂²×C₄	#11924007	1	8	yes	—
179	C₂²×C₄	C₂×C₄×D₇₃	1	12	yes	—
180	C₂²×C₄	C₂×C₄×D₇₃	1	6	yes	—
181	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₇₃	1	24	yes	—
184	C₂²×C₄	C₂×Dic₁₄₆	1	24	yes	—
185	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
186	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
187	C₂²×C₄	C₂³×D₇₃	1	4	yes	—
188	M₁₆	—	2	32	no	—
189	M₁₆	—	146	32	no	—
190	C₂×C₈	—	2	24	no	—
191	C₂×C₈	—	73	8	no	—
192	C₂×C₈	—	146	24	no	—
193	C₂×C₈	C₂³×D₇₃	1	8	yes	—
194	C₄²	—	73	16	no	—
195	C₄²	—	73	24	no	—
196	C₄²	—	73	16	no	—
197	C₄²	—	73	8	no	—
198	C₂³	—	2	32	no	—
199	C₂³	—	2	24	no	—
200	C₂³	C₂×#11924007	1	24	yes	—
201	C₂³	C₂×#11924007	1	4	yes	—
202	C₂³	M₁₆×D₇₃	1	8	yes	—
203	C₂³	C₂²×C₄×D₇₃	1	4	yes	—
204	C₂³	C₂²×C₄×D₇₃	1	3	yes	—
205	C₂×C₄	—	73	8	no	—
206	C₂×C₄	—	73	48	no	—
207	C₂×C₄	—	73	8	no	—
208	C₂×C₄	—	146	64	no	—
209	C₂×C₄	—	146	24	no	—
210	C₂×C₄	C₂×#11924007	1	48	yes	—
211	C₂×C₄	C₂×#11924007	1	24	yes	—
212	C₂×C₄	C₂×#11924007	1	12	yes	—
213	C₂×C₄	C₂²×C₄×D₇₃	1	4	yes	—
214	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₇₃	1	6	yes	—
216	C₂×C₄	C₂²×C₂²⋊D₇₃	1	8	yes	—
217	C₂×C₄	C₂²×Dic₁₄₆	1	8	yes	—
218	C₂×C₄	C₂²×Dic₁₄₆	1	6	yes	—
219	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
220	C₈	—	2	8	no	—
221	C₈	—	146	8	no	—
222	C₂²	—	2	48	no	—
223	C₂²	—	2	8	no	—
224	C₂²	#17988610	1	16	yes	direct product factor
225	C₂²	C₂²×#11924007	1	8	yes	—
226	C₂²	C₂²×#11924007	1	6	yes	—
227	C₂²	C₂×M₁₆×D₇₃	1	12	yes	—
228	C₂²	C₂³×C₄×D₇₃	1	1	yes	characteristic
229	C₄	—	73	16	no	—
230	C₄	—	146	8	no	—
231	C₄	C₂²×#11924007	1	6	yes	—
232	C₄	C₂²×#11924007	1	8	yes	—
233	C₄	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
234	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
235	C₂	—	2	8	no	—
236	C₂	C₂×#17988610	1	12	yes	direct product factor
237	C₂	C₂³×#11924007	1	1	yes	characteristic
238	C₂	C₂²×M₁₆×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
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	C₂²×#423
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#17988610	C₂	direct (N × H)
4. C₂²×#14556520	C₂	semidirect
15. #17988610	C₂²	direct (N × H)
16. C₂×#14556520	C₂²	semidirect
35. #14556520	C₂³	semidirect
49. C₂²×Dic₇₃	M₁₆	semidirect
97. C₂×Dic₇₃	C₂×M₁₆	semidirect
104. C₂²×C₇₃	#423	semidirect
114. Dic₇₃	C₂²×M₁₆	semidirect
117. C₂×C₇₃	C₂×#423	semidirect
125. C₇₃	C₂²×#423	semidirect
224. C₂²	#17988610	direct (N × H)
236. C₂	C₂×#17988610	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×#11924012	2
C₂³×C₄×Dic₇₃	2
C₂×#17988610	2
C₂²×#14556520	2
C₂³×M₁₆×C₇₃	2
C₂²×#423	73

Subgroups — C22×#17988610 #33949595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988610 #33949595