Subgroups — C₄×#17988711 #33084588

Order 18688.

1070 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988711	C₂	1	2	yes	maximal
2	C₄²×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
3	#22609796	C₂	1	2	yes	maximal
4	#22609792	C₂	1	4	yes	maximal
5	C₄×#14556520	C₂	1	2	yes	maximal
6	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
7	C₄×C₈×Dic₇₃	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₄₆	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	#17988711	C₄	1	16	yes	direct product factor
12	C₂×#14556520	C₂²	1	4	yes	—
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	2	yes	—
15	C₂×C₄×#10596831	C₂²	1	2	yes	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
19	C₄×#11924007	C₄	1	4	yes	—
20	C₄×#11924007	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₇₃	C₄	1	2	yes	—
22	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₇₃	C₂²	1	2	yes	—
24	#17988341	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
27	C₄³×C₇₃	C₂²	1	1	yes	characteristic
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂³	1	2	yes	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	C₄×Dic₁₄₆	C₈	1	16	yes	complemented
33	C₄×Dic₁₄₆	C₂×C₄	1	8	yes	—
34	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
36	C₂×#11924007	C₂×C₄	1	2	yes	—
37	C₂×#11924007	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
41	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
42	#14556520	C₂×C₄	1	16	yes	—
43	C₄×#10596831	—	2	4	no	—
44	C₄×#10596831	—	2	2	no	—
45	C₄×#10596831	C₂×C₄	1	8	yes	—
46	C₄×#10596831	Q₈	1	4	yes	complemented
47	C₈×Dic₇₃	—	2	8	no	—
48	C₈×Dic₇₃	C₂×C₄	1	16	yes	—
49	#14556498	C₂×C₄	1	4	yes	—
50	#14556498	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₇₃	—	2	4	no	—
59	C₂×Dic₁₄₆	C₄²	1	4	yes	—
60	C₂×Dic₁₄₆	C₂×C₈	1	8	yes	—
61	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
62	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
63	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
64	C₂×#10596831	—	2	2	no	—
65	C₂×#10596831	—	2	2	no	—
66	C₂×#10596831	C₂²×C₄	1	4	yes	—
67	C₂×#10596831	C₂²×C₄	1	4	yes	—
68	C₂×#10596831	C₂×Q₈	1	4	yes	—
69	C₂×#10596831	#41	1	4	yes	—
70	C₄⋊Dic₇₃	C₄²	1	4	yes	—
71	C₄⋊Dic₇₃	C₂×C₈	1	8	yes	—
72	#11924007	C₄²	1	8	yes	—
73	#11924007	C₂×C₈	1	16	yes	—
74	C₄×Dic₇₃	—	2	8	no	—
75	C₄×Dic₇₃	—	2	4	no	—
76	C₄×Dic₇₃	C₂×C₈	1	16	yes	—
77	C₄×Dic₇₃	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
86	C₄²×C₇₃	—	2	4	no	—
87	C₄²×C₇₃	—	2	2	no	—
88	C₄²×C₇₃	C₂×C₈	1	8	yes	—
89	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₇₃	#41	1	2	yes	—
92	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₇₃	#131	1	4	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
107	C₂×Dic₇₃	—	2	4	no	—
108	C₂×Dic₇₃	—	2	4	no	—
109	C₂×Dic₇₃	—	2	4	no	—
110	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
111	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
112	C₂×Dic₇₃	C₂²×C₈	1	8	yes	—
113	C₂×Dic₇₃	C₂²×C₈	1	8	yes	—
114	Dic₁₄₆	C₄×C₈	1	16	yes	complemented
115	C₈×C₇₃	—	2	8	no	—
116	#10596831	—	2	4	no	—
117	#10596831	C₄×Q₈	1	8	yes	complemented
118	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
119	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
121	C₂²×C₇₃	C₂³×C₈	1	2	yes	—
122	C₂²×C₇₃	C₂×#131	1	2	yes	—
123	C₄×C₇₃	—	2	2	no	—
124	C₄×C₇₃	—	2	4	no	—
125	C₄×C₇₃	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₇₃	C₈×Q₈	1	8	yes	complemented
127	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
128	C₄×C₇₃	C₄×#41	1	2	yes	—
129	Dic₇₃	—	2	4	no	—
130	Dic₇₃	—	2	8	no	—
131	Dic₇₃	C₂×C₄×C₈	1	8	yes	—
132	C₄×C₈×Q₈	—	73	1	no	maximal, Sylow
133	C₂×C₇₃	C₄²×Q₈	1	1	yes	characteristic
134	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₇₃	C₄×#131	1	1	yes	characteristic
136	C₂×C₇₃	C₂×C₈×Q₈	1	2	yes	—
137	C₂×C₇₃	C₈×#41	1	2	yes	—
138	C₂×C₈×Q₈	—	73	2	no	—
139	C₄²×Q₈	—	73	1	no	—
140	C₈×C₄⋊C₄	—	73	2	no	—
141	C₈×C₄⋊C₄	—	73	4	no	—
142	C₄×C₄⋊C₈	—	73	2	no	—
143	C₄×C₄⋊C₈	D₇₃	1	1	yes	characteristic
144	C₄²×C₈	—	73	2	no	—
145	C₄²×C₈	—	73	1	no	—
146	C₇₃	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄×Q₈	—	73	2	no	—
148	C₂×C₄×Q₈	—	73	1	no	—
149	C₈×Q₈	—	73	16	no	—
150	C₂×C₄⋊C₈	—	73	4	no	—
151	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
152	C₂×C₄×C₈	—	73	4	no	—
153	C₂×C₄×C₈	—	73	2	no	—
154	C₂×C₄×C₈	—	73	1	no	—
155	C₂×C₄×C₈	—	73	4	no	—
156	C₂×C₄×C₈	—	73	2	no	—
157	C₂×C₄×C₈	—	73	1	no	—
158	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
159	C₄×C₄⋊C₄	—	73	2	no	—
160	C₄×C₄⋊C₄	—	73	4	no	—
161	C₄×C₄⋊C₄	—	73	1	no	—
162	C₄×C₄⋊C₄	—	73	2	no	—
163	C₄³	—	73	2	no	—
164	C₄³	C₂×D₇₃	1	1	yes	characteristic
165	#336	—	73	4	no	—
166	#336	C₂×D₇₃	1	2	yes	—
167	C₂²×Q₈	—	73	1	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
171	C₄×Q₈	—	73	16	no	—
172	C₄×Q₈	—	73	8	no	—
173	C₂×C₄⋊C₄	—	73	2	no	—
174	C₂×C₄⋊C₄	—	73	4	no	—
175	C₂×C₄⋊C₄	—	73	1	no	—
176	C₂×C₄⋊C₄	—	73	2	no	—
177	C₂×C₄²	—	73	4	no	—
178	C₂×C₄²	—	73	2	no	—
179	C₂×C₄²	—	73	4	no	—
180	C₂×C₄²	—	73	2	no	—
181	C₂×C₄²	C₄×D₇₃	1	2	yes	—
182	C₂×C₄²	C₄×D₇₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
186	C₄⋊C₈	—	73	16	no	—
187	C₄⋊C₈	C₄×D₇₃	1	8	yes	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	73	16	no	—
190	C₄×C₈	—	73	8	no	—
191	C₄×C₈	—	73	4	no	—
192	C₄×C₈	—	146	4	no	—
193	C₄×C₈	—	146	8	no	—
194	C₄×C₈	—	146	2	no	—
195	#95	—	73	2	no	—
196	#95	—	73	4	no	—
197	C₂×Q₈	—	73	4	no	—
198	C₂×Q₈	—	73	8	no	—
199	C₂²×C₄	—	73	2	no	—
200	C₂²×C₄	—	73	2	no	—
201	C₂²×C₄	—	73	4	no	—
202	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
206	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	73	4	no	—
210	C₂×C₈	—	73	4	no	—
211	C₂×C₈	—	73	4	no	—
212	C₂×C₈	—	73	4	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	—	146	2	no	—
215	C₂×C₈	C₂×C₄×D₇₃	1	8	yes	—
216	C₄⋊C₄	—	73	8	no	—
217	C₄⋊C₄	—	73	4	no	—
218	C₄⋊C₄	—	73	16	no	—
219	C₄⋊C₄	—	73	8	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	73	16	no	—
223	C₄²	—	73	8	no	—
224	C₄²	—	146	8	no	—
225	C₄²	—	146	4	no	—
226	C₄²	C₈×D₇₃	1	8	yes	—
227	C₄²	C₂×C₄×D₇₃	1	4	yes	—
228	C₄²	Q₈×D₇₃	1	2	yes	—
229	C₄²	#11924027	1	2	yes	—
230	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
231	Q₈	—	73	16	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	73	4	no	—
236	C₂×C₄	—	73	8	no	—
237	C₂×C₄	—	73	4	no	—
238	C₂×C₄	—	73	8	no	—
239	C₂×C₄	—	146	4	no	—
240	C₂×C₄	—	146	4	no	—
241	C₂×C₄	—	146	4	no	—
242	C₂×C₄	C₄²×D₇₃	1	2	yes	—
243	C₂×C₄	C₄²×D₇₃	1	2	yes	—
244	C₂×C₄	#14556556	1	4	yes	—
245	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
247	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
248	C₂×C₄	#14556599	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
251	C₂×C₄	#41×D₇₃	1	2	yes	—
252	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
253	C₈	—	2	8	no	—
254	C₈	—	73	8	no	—
255	C₈	—	146	4	no	—
256	C₂²	C₂×C₄²×D₇₃	1	1	yes	characteristic
257	C₂²	C₂×#14556556	1	1	yes	characteristic
258	C₂²	#17989055	1	1	yes	characteristic
259	C₂²	C₂²×C₈×D₇₃	1	2	yes	—
260	C₂²	C₂×#14556599	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	73	8	no	—
264	C₄	—	146	8	no	—
265	C₄	—	146	4	no	—
266	C₄	C₄×C₈×D₇₃	1	4	yes	—
267	C₄	#17988711	1	8	yes	direct product factor
268	C₄	C₄×Q₈×D₇₃	1	2	yes	—
269	C₄	C₄×#11924027	1	2	yes	—
270	C₂	C₄×#14556556	1	1	yes	characteristic
271	C₂	C₂×C₄×C₈×D₇₃	1	1	yes	characteristic
272	C₂	C₄×#14556599	1	1	yes	characteristic
273	C₂	C₂×#17988711	1	2	yes	—
274	C₂	#22618887	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₃
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×C₄⋊C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂³×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₇₃
hypocenter	C₇₃	C₄×C₈×Q₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. #17988711	C₄	direct (N × H)
32. C₄×Dic₁₄₆	C₈	semidirect
46. C₄×#10596831	Q₈	semidirect
114. Dic₁₄₆	C₄×C₈	semidirect
117. #10596831	C₄×Q₈	semidirect
126. C₄×C₇₃	C₈×Q₈	semidirect
146. C₇₃	C₄×C₈×Q₈	semidirect
267. C₄	#17988711	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17988711	2
C₄²×Dic₁₄₆	2
#22609796	2
#22609792	2
C₄×#14556520	2
C₄²×#10596831	2
C₄×C₈×Dic₇₃	2
C₄×C₄⋊C₈×C₇₃	2
C₄×C₈×Q₈	73

Subgroups — C4×#17988711 #33084588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988711 #33084588