Subgroups — C₄×#395×C₇₃ #32923264

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#395×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₇₃	C₂	1	3	yes	maximal
4	C₇₃×#1446	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1444	C₂	1	3	yes	maximal
6	C₇₃×#1441	C₂	1	3	yes	maximal
7	C₄×#95×C₇₃	C₂	1	3	yes	maximal
8	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₇₃	—	2	12	no	—
10	C₂²×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
12	C₂²×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
13	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₇₃	C₂²	1	3	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	3	yes	—
16	#395×C₇₃	C₄	1	8	yes	direct product factor
17	C₄×C₄⋊C₄×C₇₃	—	2	12	no	—
18	C₂×#95×C₇₃	C₂²	1	6	yes	—
19	C₂×#95×C₇₃	C₂²	1	3	yes	—
20	C₂×#95×C₇₃	C₂²	1	3	yes	—
21	C₂×#95×C₇₃	C₂²	1	3	yes	—
22	C₂×#95×C₇₃	C₂²	1	3	yes	—
23	C₂²×Q₈×C₇₃	—	2	6	no	—
24	C₂²×Q₈×C₇₃	—	2	6	no	—
25	C₂²×Q₈×C₇₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
28	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
31	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
32	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₇₃	—	2	32	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
35	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	12	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
38	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
39	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
40	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
41	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	12	yes	—
42	C₂×C₄²×C₇₃	—	2	6	no	—
43	C₂×C₄²×C₇₃	—	2	12	no	—
44	C₂×C₄²×C₇₃	—	2	6	no	—
45	C₂×C₄²×C₇₃	—	2	6	no	—
46	C₂×C₄²×C₇₃	D₄	1	12	yes	—
47	C₂×C₄²×C₇₃	Q₈	1	12	yes	complemented
48	#95×C₇₃	—	2	6	no	—
49	#95×C₇₃	—	2	6	no	—
50	#95×C₇₃	—	2	6	no	—
51	#95×C₇₃	—	2	12	no	—
52	#95×C₇₃	C₂×C₄	1	12	yes	—
53	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
54	C₂×Q₈×C₇₃	—	2	8	no	—
55	C₂×Q₈×C₇₃	—	2	24	no	—
56	C₂×Q₈×C₇₃	—	2	24	no	—
57	C₂²×C₄×C₇₃	—	2	3	no	—
58	C₂²×C₄×C₇₃	—	2	6	no	—
59	C₂²×C₄×C₇₃	—	2	3	no	—
60	C₂²×C₄×C₇₃	—	2	3	no	—
61	C₂²×C₄×C₇₃	—	2	3	no	—
62	C₂²×C₄×C₇₃	—	2	6	no	—
63	C₂²×C₄×C₇₃	—	2	3	no	—
64	C₂²×C₄×C₇₃	—	2	3	no	—
65	C₂²×C₄×C₇₃	—	2	3	no	—
66	C₂²×C₄×C₇₃	—	2	6	no	—
67	C₂²×C₄×C₇₃	—	2	6	no	—
68	C₂²×C₄×C₇₃	—	2	6	no	—
69	C₂²×C₄×C₇₃	—	2	6	no	—
70	C₂²×C₄×C₇₃	—	2	3	no	—
71	C₂²×C₄×C₇₃	—	2	3	no	—
72	C₂²×C₄×C₇₃	—	2	3	no	—
73	C₂²×C₄×C₇₃	—	2	3	no	—
74	C₂²×C₄×C₇₃	—	2	3	no	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
81	C₂²×C₄×C₇₃	C₂×Q₈	1	6	yes	—
82	C₂²×C₄×C₇₃	C₂×Q₈	1	6	yes	—
83	C₂²×C₄×C₇₃	C₂×Q₈	1	6	yes	—
84	C₂²×C₄×C₇₃	#41	1	2	yes	—
85	C₂²×C₄×C₇₃	#41	1	6	yes	—
86	C₂²×C₄×C₇₃	#41	1	6	yes	—
87	C₂²×C₄×C₇₃	#41	1	2	yes	—
88	C₄⋊C₄×C₇₃	—	2	24	no	—
89	C₄⋊C₄×C₇₃	—	2	24	no	—
90	C₄⋊C₄×C₇₃	—	2	24	no	—
91	C₄²×C₇₃	—	2	24	no	—
92	C₄²×C₇₃	—	2	24	no	—
93	C₄²×C₇₃	—	4	4	no	—
94	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
95	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
96	C₂³×C₇₃	C₂²×Q₈	1	3	yes	—
97	C₂³×C₇₃	C₂×#41	1	3	yes	—
98	C₂³×C₇₃	C₂×#41	1	3	yes	—
99	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
100	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
101	Q₈×C₇₃	—	2	32	no	—
102	C₂×C₄×C₇₃	—	2	6	no	—
103	C₂×C₄×C₇₃	—	2	12	no	—
104	C₂×C₄×C₇₃	—	2	6	no	—
105	C₂×C₄×C₇₃	—	2	6	no	—
106	C₂×C₄×C₇₃	—	2	6	no	—
107	C₂×C₄×C₇₃	—	2	6	no	—
108	C₂×C₄×C₇₃	—	2	6	no	—
109	C₂×C₄×C₇₃	—	2	12	no	—
110	C₂×C₄×C₇₃	—	2	6	no	—
111	C₂×C₄×C₇₃	—	2	6	no	—
112	C₂×C₄×C₇₃	—	2	6	no	—
113	C₂×C₄×C₇₃	—	2	6	no	—
114	C₂×C₄×C₇₃	—	2	6	no	—
115	C₂×C₄×C₇₃	—	2	6	no	—
116	C₂×C₄×C₇₃	—	2	6	no	—
117	C₂×C₄×C₇₃	—	2	6	no	—
118	C₂×C₄×C₇₃	—	2	12	no	—
119	C₂×C₄×C₇₃	—	4	1	no	—
120	C₂×C₄×C₇₃	—	4	1	no	—
121	C₂×C₄×C₇₃	—	4	6	no	—
122	C₂×C₄×C₇₃	—	4	3	no	—
123	C₂×C₄×C₇₃	—	4	3	no	—
124	C₂×C₄×C₇₃	—	4	6	no	—
125	C₂×C₄×C₇₃	—	4	3	no	—
126	C₂×C₄×C₇₃	—	4	6	no	—
127	C₂×C₄×C₇₃	—	4	3	no	—
128	C₂×C₄×C₇₃	C₄×D₄	1	12	yes	—
129	C₂×C₄×C₇₃	C₄×Q₈	1	12	yes	complemented
130	C₂×C₄×C₇₃	#120	1	4	yes	—
131	C₂×C₄×C₇₃	C₂²⋊Q₈	1	12	yes	—
132	C₂×C₄×C₇₃	C₄⋊Q₈	1	12	yes	—
133	C₂²×C₇₃	C₂×C₄×D₄	1	3	yes	—
134	C₂²×C₇₃	C₂×C₄×Q₈	1	3	yes	—
135	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	3	yes	—
138	C₂²×C₇₃	#525	1	3	yes	—
139	C₂²×C₇₃	#525	1	3	yes	—
140	C₂²×C₇₃	#529	1	3	yes	—
141	C₂²×C₇₃	C₂×C₄⋊Q₈	1	3	yes	—
142	C₂²×C₇₃	#532	1	3	yes	—
143	C₂²×C₇₃	#533	1	6	yes	—
144	C₂²×C₇₃	#533	1	3	yes	—
145	C₄×C₇₃	—	2	12	no	—
146	C₄×C₇₃	—	2	12	no	—
147	C₄×C₇₃	—	4	6	no	—
148	C₄×C₇₃	—	4	2	no	—
149	C₄×C₇₃	—	4	6	no	—
150	C₄×C₇₃	—	4	2	no	—
151	C₄×C₇₃	#395	1	8	yes	direct product factor
152	C₄×#395	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₇₃	C₄×#120	1	1	yes	characteristic
154	C₂×C₇₃	C₄×C₂²⋊Q₈	1	3	yes	—
155	C₂×C₇₃	C₄×C₄⋊Q₈	1	3	yes	—
156	C₂×C₇₃	C₂×#395	1	1	yes	characteristic
157	C₂×C₇₃	#2388	1	1	yes	characteristic
158	C₂×C₇₃	#2396	1	3	yes	—
159	C₂×C₇₃	#2399	1	3	yes	—
160	C₂²×C₄×Q₈	C₂×C₇₃	1	1	yes	characteristic
161	C₂×#395	C₂×C₇₃	1	1	yes	characteristic
162	C₂×C₄×C₄⋊C₄	C₂×C₇₃	1	3	yes	—
163	#1446	C₂×C₇₃	1	1	yes	characteristic
164	#1444	C₂×C₇₃	1	3	yes	—
165	#1441	C₂×C₇₃	1	3	yes	—
166	C₄×#95	C₂×C₇₃	1	3	yes	—
167	C₇₃	C₄×#395	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×Q₈	C₂²×C₇₃	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	2	12	no	—
170	C₂²×C₄⋊C₄	C₂²×C₇₃	1	3	yes	—
171	C₂²×C₄⋊C₄	C₂²×C₇₃	1	3	yes	—
172	C₂²×C₄⋊C₄	C₂²×C₇₃	1	3	yes	—
173	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₇₃	1	3	yes	—
175	C₂²×C₄²	C₂²×C₇₃	1	3	yes	—
176	#395	C₄×C₇₃	1	8	yes	direct product factor
177	C₄×C₄⋊C₄	—	2	12	no	—
178	C₂×#95	C₂²×C₇₃	1	6	yes	—
179	C₂×#95	C₂²×C₇₃	1	3	yes	—
180	C₂×#95	C₂²×C₇₃	1	3	yes	—
181	C₂×#95	C₂²×C₇₃	1	3	yes	—
182	C₂×#95	C₂²×C₇₃	1	3	yes	—
183	C₂²×Q₈	—	2	6	no	—
184	C₂²×Q₈	—	2	6	no	—
185	C₂²×Q₈	C₂×C₄×C₇₃	1	4	yes	—
186	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₇₃	1	3	yes	—
188	C₂³×C₄	C₂³×C₇₃	1	3	yes	—
189	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
190	C₂³×C₄	C₂³×C₇₃	1	3	yes	—
191	C₂³×C₄	C₂³×C₇₃	1	3	yes	—
192	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
193	C₄×Q₈	—	2	32	no	—
194	C₂×C₄⋊C₄	—	2	12	no	—
195	C₂×C₄⋊C₄	—	2	6	no	—
196	C₂×C₄⋊C₄	—	2	6	no	—
197	C₂×C₄⋊C₄	—	2	6	no	—
198	C₂×C₄⋊C₄	—	2	6	no	—
199	C₂×C₄⋊C₄	—	2	6	no	—
200	C₂×C₄⋊C₄	—	2	6	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	12	yes	—
202	C₂×C₄²	—	2	12	no	—
203	C₂×C₄²	—	2	6	no	—
204	C₂×C₄²	—	2	6	no	—
205	C₂×C₄²	—	2	6	no	—
206	C₂×C₄²	D₄×C₇₃	1	12	yes	—
207	C₂×C₄²	Q₈×C₇₃	1	12	yes	complemented
208	#95	—	2	6	no	—
209	#95	—	2	6	no	—
210	#95	—	2	6	no	—
211	#95	—	2	12	no	—
212	#95	C₂×C₄×C₇₃	1	12	yes	—
213	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
214	C₂×Q₈	—	2	8	no	—
215	C₂×Q₈	—	2	24	no	—
216	C₂×Q₈	—	2	24	no	—
217	C₂²×C₄	—	2	3	no	—
218	C₂²×C₄	—	2	6	no	—
219	C₂²×C₄	—	2	3	no	—
220	C₂²×C₄	—	2	3	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	—	2	3	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	—	2	6	no	—
230	C₂²×C₄	—	2	3	no	—
231	C₂²×C₄	—	2	3	no	—
232	C₂²×C₄	—	2	3	no	—
233	C₂²×C₄	—	2	3	no	—
234	C₂²×C₄	—	2	3	no	—
235	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₇₃	1	6	yes	—
237	C₂²×C₄	C₂²×C₄×C₇₃	1	6	yes	—
238	C₂²×C₄	C₂×D₄×C₇₃	1	6	yes	—
239	C₂²×C₄	C₂×D₄×C₇₃	1	6	yes	—
240	C₂²×C₄	C₂×D₄×C₇₃	1	6	yes	—
241	C₂²×C₄	C₂×Q₈×C₇₃	1	6	yes	—
242	C₂²×C₄	C₂×Q₈×C₇₃	1	6	yes	—
243	C₂²×C₄	C₂×Q₈×C₇₃	1	6	yes	—
244	C₂²×C₄	#41×C₇₃	1	2	yes	—
245	C₂²×C₄	#41×C₇₃	1	6	yes	—
246	C₂²×C₄	#41×C₇₃	1	6	yes	—
247	C₂²×C₄	#41×C₇₃	1	2	yes	—
248	C₄⋊C₄	—	2	24	no	—
249	C₄⋊C₄	—	2	24	no	—
250	C₄⋊C₄	—	2	24	no	—
251	C₄²	—	2	24	no	—
252	C₄²	—	2	24	no	—
253	C₄²	—	4	4	no	—
254	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
255	C₂³	C₂²×D₄×C₇₃	1	3	yes	—
256	C₂³	C₂²×Q₈×C₇₃	1	3	yes	—
257	C₂³	C₂×#41×C₇₃	1	3	yes	—
258	C₂³	C₂×#41×C₇₃	1	3	yes	—
259	C₂³	C₂×#41×C₇₃	1	1	yes	characteristic
260	C₂³	C₂×#41×C₇₃	1	1	yes	characteristic
261	Q₈	—	2	32	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	12	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	2	6	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	6	no	—
275	C₂×C₄	—	2	6	no	—
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	12	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	3	no	—
282	C₂×C₄	—	4	6	no	—
283	C₂×C₄	—	4	3	no	—
284	C₂×C₄	—	4	6	no	—
285	C₂×C₄	—	4	3	no	—
286	C₂×C₄	—	4	6	no	—
287	C₂×C₄	—	4	3	no	—
288	C₂×C₄	C₄×D₄×C₇₃	1	12	yes	—
289	C₂×C₄	C₄×Q₈×C₇₃	1	12	yes	complemented
290	C₂×C₄	#120×C₇₃	1	4	yes	—
291	C₂×C₄	C₂²⋊Q₈×C₇₃	1	12	yes	—
292	C₂×C₄	C₄⋊Q₈×C₇₃	1	12	yes	—
293	C₂²	C₂×C₄×D₄×C₇₃	1	3	yes	—
294	C₂²	C₂×C₄×Q₈×C₇₃	1	3	yes	—
295	C₂²	C₄×#41×C₇₃	1	1	yes	characteristic
296	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
297	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	3	yes	—
298	C₂²	#525×C₇₃	1	3	yes	—
299	C₂²	#525×C₇₃	1	3	yes	—
300	C₂²	#529×C₇₃	1	3	yes	—
301	C₂²	C₂×C₄⋊Q₈×C₇₃	1	3	yes	—
302	C₂²	#532×C₇₃	1	3	yes	—
303	C₂²	#533×C₇₃	1	3	yes	—
304	C₂²	#533×C₇₃	1	6	yes	—
305	C₄	—	2	12	no	—
306	C₄	—	2	12	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	6	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	6	no	—
311	C₄	#395×C₇₃	1	8	yes	direct product factor
312	C₂	C₄×#120×C₇₃	1	1	yes	characteristic
313	C₂	C₄×C₂²⋊Q₈×C₇₃	1	3	yes	—
314	C₂	C₄×C₄⋊Q₈×C₇₃	1	3	yes	—
315	C₂	C₂×#395×C₇₃	1	1	yes	characteristic
316	C₂	C₇₃×#2388	1	1	yes	characteristic
317	C₂	C₇₃×#2396	1	3	yes	—
318	C₂	C₇₃×#2399	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇₃	C₂³
commutator	C₂³	C₂³×C₄×C₇₃
Frattini, cosocle	C₂⁴	C₂⁴×C₇₃
socle, prime radical	C₂⁴×C₇₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
16. #395×C₇₃	C₄	direct (N × H)
47. C₂×C₄²×C₇₃	Q₈	semidirect
129. C₂×C₄×C₇₃	C₄×Q₈	semidirect
151. C₄×C₇₃	#395	direct (N × H)
152. C₄×#395	C₇₃	direct (N × H)
167. C₇₃	C₄×#395	direct (N × H)
176. #395	C₄×C₇₃	direct (N × H)
207. C₂×C₄²	Q₈×C₇₃	semidirect
289. C₂×C₄	C₄×Q₈×C₇₃	semidirect
311. C₄	#395×C₇₃	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×C₄×Q₈×C₇₃	2
C₂×#395×C₇₃	2
C₂×C₄×C₄⋊C₄×C₇₃	2
C₇₃×#1446	2
C₇₃×#1444	2
C₇₃×#1441	2
C₄×#95×C₇₃	2
C₄×#395	73

Subgroups — C4×#395×C73 #32923264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#395×C₇₃ #32923264