Subgroups — C₄×#108×D₇₃ #33084078

Order 18688.

1086 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₇₃	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₇₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#108×Dic₇₃	C₂	1	2	yes	maximal, complemented
5	C₄×#14556526	C₂	1	2	yes	maximal, complemented
6	C₄×#14556527	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609832	C₂	1	2	yes	maximal, complemented
8	#341×D₇₃	C₂	1	2	yes	maximal
9	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₇₃	C₂²	1	4	yes	—
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
14	C₂×#14556527	C₂²	1	2	yes	—
15	C₂×#14556526	C₂²	1	4	yes	—
16	#108×D₇₃	C₄	1	16	yes	direct product factor
17	C₄×#11924012	C₂²	1	2	yes	—
18	#17988539	—	2	2	no	—
19	M₁₆×Dic₇₃	C₂²	1	4	yes	—
20	C₄×#11924011	—	2	2	no	—
21	#97×D₇₃	—	2	2	no	—
22	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₇₃	—	2	2	no	—
26	C₄×C₈×D₇₃	C₄	1	4	yes	—
27	C₄×C₈×D₇₃	C₄	1	4	yes	—
28	#17988352	C₂²	1	2	yes	—
29	#17988351	C₂²	1	4	yes	—
30	C₂×#108×C₇₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	#341×C₇₃	C₂²	1	2	yes	—
34	C₂²×C₄×D₇₃	Q₈	1	2	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
36	C₂×#11924012	C₂³	1	4	yes	—
37	C₂×#11924011	—	2	4	no	—
38	M₁₆×D₇₃	—	2	8	no	—
39	M₁₆×D₇₃	C₂×C₄	1	16	yes	—
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂²×#10596831	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₇₃	—	2	4	no	—
43	C₂×C₈×D₇₃	—	2	4	no	—
44	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
50	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
51	C₄²×D₇₃	C₂×C₄	1	2	yes	—
52	C₄²×D₇₃	C₂×C₄	1	2	yes	—
53	C₄²×D₇₃	D₄	1	1	yes	characteristic
54	C₄²×D₇₃	D₄	1	1	yes	characteristic
55	C₄²×D₇₃	D₄	1	1	yes	characteristic
56	C₄²×D₇₃	D₄	1	1	yes	characteristic
57	#14556527	C₂×C₄	1	8	yes	complemented
58	#14556526	C₂×C₄	1	16	yes	complemented
59	#14556502	—	2	2	no	—
60	#14556501	—	2	4	no	—
61	C₄×#10596831	—	2	2	no	—
62	C₄×#10596831	C₂×C₄	1	4	yes	—
63	C₈×Dic₇₃	—	2	4	no	—
64	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
65	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
70	#108×C₇₃	C₂×C₄	1	8	yes	complemented
71	#97×C₇₃	—	2	2	no	—
72	C₄×C₈×C₇₃	—	2	2	no	—
73	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
74	C₂³×D₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×Dic₇₃	C₂×Q₈	1	2	yes	—
76	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₄×D₇₃	—	2	8	no	—
78	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
85	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×D₇₃	#41	1	4	yes	—
88	#11924012	—	2	4	no	—
89	#11924012	C₂²×C₄	1	8	yes	—
90	#11924011	—	2	16	no	—
91	C₂×#10596831	—	2	4	no	—
92	C₂×#10596831	—	2	2	no	—
93	C₂×#10596831	C₂²×C₄	1	2	yes	—
94	C₂×#10596831	C₂²×C₄	1	2	yes	—
95	C₂×#10596831	C₂²×C₄	1	4	yes	—
96	C₈×D₇₃	—	2	4	no	—
97	C₈×D₇₃	—	2	4	no	—
98	C₈×D₇₃	—	2	16	no	—
99	C₈×D₇₃	C₄²	1	8	yes	—
100	C₈×D₇₃	C₄²	1	8	yes	—
101	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
109	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₇₃	—	2	4	no	—
111	M₁₆×C₇₃	C₂²×C₄	1	8	yes	—
112	C₂×C₈×C₇₃	—	2	4	no	—
113	C₂×C₈×C₇₃	—	2	2	no	—
114	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
117	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
118	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂²×D₇₃	—	2	4	no	—
122	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₇₃	C₄×Q₈	1	4	yes	—
125	C₂×C₄×C₇₃	—	2	4	no	—
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
132	C₂×Dic₇₃	—	2	4	no	—
133	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₇₃	C₄×Q₈	1	4	yes	—
138	C₄×D₇₃	—	2	4	no	—
139	C₄×D₇₃	—	2	4	no	—
140	C₄×D₇₃	—	2	4	no	—
141	C₄×D₇₃	#108	1	8	yes	direct product factor
142	C₄×D₇₃	C₂×C₄²	1	2	yes	—
143	C₄×D₇₃	C₂×C₄²	1	2	yes	—
144	C₄×D₇₃	#117	1	2	yes	—
145	C₄×D₇₃	#117	1	2	yes	—
146	C₄×D₇₃	C₄×D₄	1	4	yes	—
147	C₄×D₇₃	C₄×D₄	1	4	yes	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	—	2	8	no	—
150	C₈×C₇₃	C₂×C₄²	1	4	yes	—
151	#10596831	—	2	2	no	—
152	#10596831	—	2	8	no	—
153	#10596831	C₂×C₄²	1	4	yes	—
154	C₂²×C₇₃	—	2	2	no	—
155	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₇₃	C₂×C₄×Q₈	1	2	yes	—
157	C₂×D₇₃	—	2	2	no	—
158	C₂×D₇₃	—	2	8	no	—
159	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
160	C₂×D₇₃	C₄×C₄⋊C₄	1	2	yes	—
161	C₂×D₇₃	C₂×#108	1	4	yes	—
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	C₂×#108	1	4	yes	complemented
164	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₇₃	C₂×C₄×D₄	1	2	yes	—
167	Dic₇₃	—	2	2	no	—
168	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
169	Dic₇₃	C₄×C₄⋊C₄	1	2	yes	—
170	Dic₇₃	C₂×#108	1	4	yes	complemented
171	C₂×C₄×#108	—	73	1	no	maximal, Sylow
172	C₂×C₇₃	—	2	2	no	—
173	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₇₃	C₂²×#108	1	2	yes	—
175	D₇₃	—	2	2	no	—
176	D₇₃	C₄×#108	1	4	yes	direct product factor
177	C₂²×#108	—	73	2	no	—
178	C₂×C₄×M₁₆	—	73	2	no	—
179	C₂²×C₄×C₈	—	73	1	no	—
180	C₄×#108	—	73	2	no	—
181	C₄×#108	—	73	2	no	—
182	C₄×#108	—	73	1	no	—
183	C₄×#108	—	73	2	no	—
184	C₄×#108	D₇₃	1	1	yes	characteristic, direct product factor
185	C₂×#341	—	73	2	no	—
186	C₇₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
187	C₂²×M₁₆	—	73	4	no	—
188	C₂³×C₈	—	73	1	no	—
189	C₂³×C₈	—	73	1	no	—
190	C₂²×C₄²	—	73	1	no	—
191	C₂×#108	—	73	4	no	—
192	C₂×#108	—	73	2	no	—
193	C₂×#108	—	73	16	no	—
194	C₂×#108	C₂×D₇₃	1	2	yes	—
195	C₄×M₁₆	—	73	4	no	—
196	C₄×M₁₆	—	73	2	no	—
197	C₄×M₁₆	—	146	2	no	—
198	C₄×M₁₆	—	146	2	no	—
199	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
200	C₂×#97	—	146	2	no	—
201	C₂×C₄×C₈	—	73	1	no	—
202	C₂×C₄×C₈	—	73	1	no	—
203	C₂×C₄×C₈	—	73	1	no	—
204	C₂×C₄×C₈	—	73	4	no	—
205	C₂×C₄×C₈	—	73	4	no	—
206	C₂×C₄×C₈	—	146	2	no	—
207	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
208	#341	—	73	4	no	—
209	#341	—	73	2	no	—
210	#341	C₂×D₇₃	1	2	yes	—
211	C₂³×C₄	—	73	1	no	—
212	C₂³×C₄	—	73	2	no	—
213	C₂×M₁₆	—	73	4	no	—
214	C₂×M₁₆	—	73	16	no	—
215	C₂×M₁₆	—	146	4	no	—
216	C₂×M₁₆	—	146	8	no	—
217	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	2	no	—
220	C₂²×C₈	—	73	2	no	—
221	C₂²×C₈	—	73	8	no	—
222	C₂²×C₈	—	73	1	no	—
223	C₂²×C₈	—	73	2	no	—
224	C₂²×C₈	—	73	2	no	—
225	C₂²×C₈	—	146	4	no	—
226	C₂²×C₈	—	146	4	no	—
227	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
229	C₂×C₄²	—	73	2	no	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	2	no	—
232	C₂×C₄²	—	73	2	no	—
233	C₂×C₄²	—	73	1	no	—
234	C₂×C₄²	—	73	1	no	—
235	C₂×C₄²	—	73	1	no	—
236	C₂×C₄²	—	73	1	no	—
237	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
238	#108	—	73	16	no	—
239	#108	—	73	8	no	—
240	#108	C₄×D₇₃	1	8	yes	direct product factor
241	#97	—	2	2	no	—
242	#97	—	146	4	no	—
243	#97	—	146	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	73	4	no	—
246	C₄×C₈	—	73	4	no	—
247	C₄×C₈	—	73	4	no	—
248	C₄×C₈	—	146	4	no	—
249	C₄×C₈	—	146	2	no	—
250	C₄×C₈	C₄×D₇₃	1	4	yes	—
251	C₂⁴	—	73	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	2	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	4	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	4	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	2	no	—
263	C₂²×C₄	—	73	2	no	—
264	C₂²×C₄	—	146	8	no	—
265	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
266	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
267	M₁₆	—	2	4	no	—
268	M₁₆	—	73	8	no	—
269	M₁₆	—	146	16	no	—
270	M₁₆	—	146	4	no	—
271	M₁₆	C₂×C₄×D₇₃	1	8	yes	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	73	2	no	—
275	C₂×C₈	—	73	4	no	—
276	C₂×C₈	—	73	8	no	—
277	C₂×C₈	—	73	8	no	—
278	C₂×C₈	—	73	2	no	—
279	C₂×C₈	—	146	4	no	—
280	C₂×C₈	—	146	4	no	—
281	C₂×C₈	—	146	16	no	—
282	C₂×C₈	—	146	4	no	—
283	C₂×C₈	—	146	2	no	—
284	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
285	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
287	C₄²	—	73	2	no	—
288	C₄²	—	73	2	no	—
289	C₄²	—	73	2	no	—
290	C₄²	—	73	2	no	—
291	C₄²	—	73	2	no	—
292	C₄²	—	73	1	no	—
293	C₄²	—	73	1	no	—
294	C₄²	C₂×C₄×D₇₃	1	2	yes	—
295	C₄²	D₄×D₇₃	1	1	yes	characteristic
296	C₄²	D₄×D₇₃	1	1	yes	characteristic
297	C₂³	—	73	1	no	—
298	C₂³	—	73	1	no	—
299	C₂³	—	73	4	no	—
300	C₂³	—	146	4	no	—
301	C₂³	C₂×Q₈×D₇₃	1	1	yes	characteristic
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	73	4	no	—
307	C₂×C₄	—	73	8	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	2	no	—
310	C₂×C₄	—	73	2	no	—
311	C₂×C₄	—	73	2	no	—
312	C₂×C₄	—	73	4	no	—
313	C₂×C₄	—	73	4	no	—
314	C₂×C₄	—	73	2	no	—
315	C₂×C₄	—	146	4	no	—
316	C₂×C₄	—	146	4	no	—
317	C₂×C₄	—	146	4	no	—
318	C₂×C₄	—	146	4	no	—
319	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
320	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
323	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₇₃	1	2	yes	—
325	C₈	—	2	2	no	—
326	C₈	—	2	8	no	—
327	C₈	—	73	4	no	—
328	C₈	—	146	2	no	—
329	C₈	—	146	8	no	—
330	C₈	C₄²×D₇₃	1	4	yes	—
331	C₂²	—	2	2	no	—
332	C₂²	—	73	2	no	—
333	C₂²	—	73	4	no	—
334	C₂²	—	73	2	no	—
335	C₂²	—	146	8	no	—
336	C₂²	—	146	2	no	—
337	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
338	C₂²	C₄×Q₈×D₇₃	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	73	2	no	—
341	C₄	—	73	2	no	—
342	C₄	—	73	4	no	—
343	C₄	—	146	2	no	—
344	C₄	#108×D₇₃	1	4	yes	direct product factor
345	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
346	C₄	#117×D₇₃	1	1	yes	characteristic
347	C₄	C₄×D₄×D₇₃	1	2	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	73	4	no	—
350	C₂	—	146	2	no	—
351	C₂	C₄×C₄⋊C₄×D₇₃	1	1	yes	characteristic
352	C₂	C₂×#108×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	C₂×C₄×#108
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #108×Dic₇₃	C₂	semidirect
5. C₄×#14556526	C₂	semidirect
6. C₄×#14556527	C₂	semidirect
7. #22609832	C₂	semidirect
9. C₄×#108×C₇₃	C₂	semidirect
16. #108×D₇₃	C₄	direct (N × H)
57. #14556527	C₂×C₄	semidirect
58. #14556526	C₂×C₄	semidirect
70. #108×C₇₃	C₂×C₄	semidirect
141. C₄×D₇₃	#108	direct (N × H)
163. C₄×C₇₃	C₂×#108	semidirect
170. Dic₇₃	C₂×#108	semidirect
176. D₇₃	C₄×#108	direct (N × H)
184. C₄×#108	D₇₃	direct (N × H)
186. C₇₃	C₂×C₄×#108	semidirect
240. #108	C₄×D₇₃	direct (N × H)
344. C₄	#108×D₇₃	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×#108×D₇₃	2
C₄×M₁₆×D₇₃	2
C₂×C₄×C₈×D₇₃	2
#108×Dic₇₃	2
C₄×#14556526	2
C₄×#14556527	2
#22609832	2
#341×D₇₃	2
C₄×#108×C₇₃	2
C₂×C₄×#108	73

Subgroups — C4×#108×D73 #33084078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#108×D₇₃ #33084078