Subgroups — #32991788

Order 18688.

842 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988343	C₂	1	4	yes	maximal
4	C₇₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂×#14556550	C₂²	1	2	yes	—
7	C₄×#11924012	—	2	1	no	—
8	C₄×#11924012	—	2	1	no	—
9	C₄×#11924012	—	2	1	no	—
10	C₄×#11924012	—	2	1	no	—
11	C₂×#14556502	C₄	1	2	yes	complemented
12	C₂×C₄×#10596831	C₄	1	2	yes	complemented
13	#17988457	—	2	1	no	—
14	#17988453	—	2	1	no	—
15	#17988447	—	2	2	no	—
16	C₄×C₄⋊Dic₇₃	C₄	1	4	yes	complemented
17	#17988343	—	2	8	no	—
18	C₂×#117×C₇₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	#384×C₇₃	—	2	1	no	—
21	#382×C₇₃	—	2	1	no	—
22	#380×C₇₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₇₃	C₄	1	4	yes	complemented
24	C₂×#11924012	—	2	2	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	2	no	—
28	C₂×#11924012	D₄	1	4	yes	—
29	C₂×#11924012	Q₈	1	4	yes	—
30	C₂²×#10596831	C₂×C₄	1	4	yes	—
31	#14556550	—	2	4	no	—
32	#14556550	—	2	4	no	—
33	C₂×C₄⋊Dic₇₃	—	2	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
37	#14556502	—	2	2	no	—
38	#14556502	—	2	2	no	—
39	C₄×#10596831	—	2	2	no	—
40	C₄×#10596831	—	2	2	no	—
41	#14556498	—	2	4	no	—
42	C₂³×C₄×C₇₃	D₄	1	2	yes	—
43	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	#117×C₇₃	—	2	4	no	—
45	#117×C₇₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
59	#95×C₇₃	—	2	4	no	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	#11924012	—	2	4	no	—
62	#11924012	—	2	4	no	—
63	#11924012	—	2	4	no	—
64	#11924012	—	2	4	no	—
65	C₂×#10596831	—	2	4	no	—
66	C₂×#10596831	—	2	4	no	—
67	C₂×#10596831	C₄⋊C₄	1	8	yes	complemented
68	C₄⋊Dic₇₃	—	4	8	no	—
69	C₄⋊Dic₇₃	—	4	4	no	—
70	C₂²⋊Dic₇₃	—	4	2	no	—
71	C₂²⋊Dic₇₃	—	4	2	no	—
72	C₄×Dic₇₃	—	2	8	no	—
73	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	—	2	4	no	—
75	C₂²×C₄×C₇₃	—	4	4	no	—
76	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	#41	1	2	yes	—
91	C₂²×C₄×C₇₃	#41	1	2	yes	—
92	C₄⋊C₄×C₇₃	—	4	8	no	—
93	C₄⋊C₄×C₇₃	—	4	4	no	—
94	C₂²⋊C₄×C₇₃	—	4	2	no	—
95	C₂²⋊C₄×C₇₃	—	4	2	no	—
96	C₄²×C₇₃	—	2	8	no	—
97	C₄²×C₇₃	C₄²	1	2	yes	—
98	C₄²×C₇₃	C₄²	1	2	yes	—
99	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
104	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
106	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
107	C₂³×C₇₃	—	4	2	no	—
108	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₇₃	#124	1	1	yes	characteristic
112	C₂³×C₇₃	#124	1	1	yes	characteristic
113	C₂³×C₇₃	#127	1	1	yes	characteristic
114	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	4	2	no	—
121	C₂×C₄×C₇₃	—	4	8	no	—
122	C₂×C₄×C₇₃	—	4	4	no	—
123	C₂×C₄×C₇₃	#95	1	2	yes	—
124	C₂×C₄×C₇₃	#95	1	2	yes	—
125	C₂×C₄×C₇₃	#95	1	2	yes	—
126	C₂×C₄×C₇₃	#95	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	#117	1	2	yes	—
133	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₇₃	#121	1	2	yes	—
138	C₂×C₄×C₇₃	#121	1	2	yes	—
139	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₇₃	#125	1	2	yes	—
142	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
143	C₂×Dic₇₃	—	4	8	no	—
144	C₂×Dic₇₃	—	4	4	no	—
145	#10596831	—	2	8	no	—
146	C₂²×C₇₃	—	2	2	no	—
147	C₂²×C₇₃	—	2	2	no	—
148	C₂²×C₇₃	—	2	2	no	—
149	C₂²×C₇₃	—	4	1	no	—
150	C₂²×C₇₃	—	4	1	no	—
151	C₂²×C₇₃	—	4	1	no	—
152	C₂²×C₇₃	—	4	1	no	—
153	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
154	C₂²×C₇₃	#382	1	1	yes	characteristic
155	C₂²×C₇₃	#384	1	1	yes	characteristic
156	C₂²×C₇₃	#390	1	1	yes	characteristic
157	C₂²×C₇₃	#390	1	1	yes	characteristic
158	C₂²×C₇₃	#391	1	1	yes	characteristic
159	C₂²×C₇₃	#391	1	1	yes	characteristic
160	C₄×C₇₃	—	2	2	no	—
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	—	8	4	no	—
164	C₄×C₇₃	H(3,ℤ/4)	1	4	yes	complemented
165	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
166	C₄×C₇₃	#380	1	2	yes	—
167	C₄×C₇₃	#389	1	2	yes	—
168	C₄×C₇₃	#389	1	2	yes	—
169	Dic₇₃	—	8	4	no	—
170	#10837	—	73	1	no	maximal, Sylow
171	C₂×C₇₃	—	2	1	no	—
172	C₂×C₇₃	—	2	1	no	—
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	—	2	1	no	—
175	C₂×C₇₃	—	2	1	no	—
176	C₂×C₇₃	—	2	1	no	—
177	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
178	C₂×C₇₃	C₂×H(3,ℤ/4)	1	2	yes	—
179	C₂×C₄×M₁₆	—	73	1	no	—
180	#2285	—	73	1	no	—
181	#2285	D₇₃	1	1	yes	characteristic
182	C₂×H(3,ℤ/4)	—	73	4	no	—
183	C₇₃	#10837	1	1	yes	characteristic, complemented, Sylow
184	C₂²×M₁₆	—	73	2	no	—
185	C₂×#117	—	73	2	no	—
186	C₂×#117	C₂×D₇₃	1	2	yes	—
187	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
188	C₄×M₁₆	—	146	1	no	—
189	C₄×M₁₆	—	146	1	no	—
190	C₄×M₁₆	—	146	1	no	—
191	C₄×M₁₆	—	146	1	no	—
192	C₂×#97	—	73	2	no	—
193	C₂×C₄×C₈	—	73	2	no	—
194	#384	—	2	1	no	—
195	#384	—	146	1	no	—
196	#382	—	2	1	no	—
197	#382	—	146	1	no	—
198	#380	—	2	2	no	—
199	#380	—	146	2	no	—
200	C₄×C₄⋊C₄	—	73	4	no	—
201	C₄×C₄⋊C₄	Dic₇₃	1	4	yes	complemented
202	H(3,ℤ/4)	—	146	8	no	—
203	C₂³×C₄	C₂²⋊D₇₃	1	2	yes	—
204	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
205	C₂×M₁₆	—	73	4	no	—
206	C₂×M₁₆	—	73	4	no	—
207	C₂×M₁₆	—	146	2	no	—
208	C₂×M₁₆	—	146	2	no	—
209	C₂×M₁₆	—	146	2	no	—
210	C₂×M₁₆	—	146	2	no	—
211	C₂²×C₈	—	73	4	no	—
212	#117	—	2	4	no	—
213	#117	—	2	4	no	—
214	#117	—	146	4	no	—
215	#117	—	146	4	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	146	4	no	—
219	C₂×C₄⋊C₄	—	146	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	146	2	no	—
222	C₂×C₄²	—	73	4	no	—
223	C₂×C₄²	Dic₁₄₆	1	2	yes	—
224	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₇₃	1	2	yes	—
227	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
228	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
230	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
231	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
233	#97	—	146	2	no	—
234	#97	—	146	2	no	—
235	C₄×C₈	—	146	2	no	—
236	C₄×C₈	—	146	2	no	—
237	#95	—	2	4	no	—
238	#95	—	146	4	no	—
239	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	146	4	no	—
243	C₂²×C₄	#11924005	1	2	yes	—
244	C₂²×C₄	#11924005	1	2	yes	—
245	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
246	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
247	C₂²×C₄	#11924007	1	2	yes	—
248	C₂²×C₄	#11924007	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₇₃	1	2	yes	—
253	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
256	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
257	C₂²×C₄	#11924027	1	2	yes	—
258	C₂²×C₄	#11924029	1	2	yes	—
259	M₁₆	—	146	4	no	—
260	M₁₆	—	146	4	no	—
261	M₁₆	—	146	4	no	—
262	M₁₆	—	146	4	no	—
263	C₂×C₈	—	73	8	no	—
264	C₂×C₈	—	146	4	no	—
265	C₂×C₈	—	146	4	no	—
266	C₄⋊C₄	—	4	8	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	292	8	no	—
269	C₄⋊C₄	—	292	4	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	292	2	no	—
273	C₂²⋊C₄	—	292	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	—	146	8	no	—
276	C₄²	C₄×Dic₇₃	1	2	yes	—
277	C₄²	C₄×Dic₇₃	1	2	yes	—
278	C₄²	#11924005	1	1	yes	characteristic
279	C₄²	#11924005	1	1	yes	characteristic
280	C₄²	#11924005	1	1	yes	characteristic
281	C₄²	#11924005	1	1	yes	characteristic
282	C₄²	#11924007	1	2	yes	—
283	C₄²	#11924007	1	2	yes	—
284	C₄²	C₄⋊Dic₇₃	1	2	yes	—
285	C₄²	C₄⋊Dic₇₃	1	2	yes	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×#11924005	1	1	yes	characteristic
288	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
289	C₂³	C₂×#11924007	1	1	yes	characteristic
290	C₂³	#14556578	1	1	yes	characteristic
291	C₂³	#14556580	1	1	yes	characteristic
292	C₂³	#14556587	1	1	yes	characteristic
293	C₂³	#14556589	1	1	yes	characteristic
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	292	8	no	—
303	C₂×C₄	—	292	4	no	—
304	C₂×C₄	#14556498	1	2	yes	—
305	C₂×C₄	#14556498	1	2	yes	—
306	C₂×C₄	#14556498	1	2	yes	—
307	C₂×C₄	#14556498	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
311	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
313	C₂×C₄	#14556549	1	2	yes	—
314	C₂×C₄	#14556553	1	2	yes	—
315	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
316	C₂×C₄	#14556556	1	2	yes	—
317	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
318	C₂×C₄	#14556562	1	2	yes	—
319	C₂×C₄	#14556566	1	2	yes	—
320	C₂×C₄	#14556567	1	2	yes	—
321	C₂×C₄	#14556571	1	2	yes	—
322	C₂×C₄	#14556582	1	2	yes	—
323	C₂×C₄	#14556590	1	2	yes	—
324	C₈	—	146	8	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂×#14556498	1	1	yes	characteristic
333	C₂²	#17988452	1	1	yes	characteristic
334	C₂²	#17988456	1	1	yes	characteristic
335	C₂²	#17988488	1	1	yes	characteristic
336	C₂²	#17988490	1	1	yes	characteristic
337	C₂²	#17988492	1	1	yes	characteristic
338	C₂²	#17988494	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	8	4	no	—
343	C₄	—	584	4	no	—
344	C₄	#17988343	1	4	yes	complemented
345	C₄	C₄⋊C₄×Dic₇₃	1	2	yes	—
346	C₄	#17988444	1	2	yes	—
347	C₄	#17988483	1	2	yes	—
348	C₄	#17988486	1	2	yes	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	#22608160	1	1	yes	characteristic
356	C₂	C₂×#17988343	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#2285	C₂
socle	C₂²×C₇₃	C₂×#95
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#2285	D₇₃
hypocenter	C₇₃	#10837
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
11. C₂×#14556502	C₄	semidirect
12. C₂×C₄×#10596831	C₄	semidirect
16. C₄×C₄⋊Dic₇₃	C₄	semidirect
23. C₄×C₄⋊C₄×C₇₃	C₄	semidirect
67. C₂×#10596831	C₄⋊C₄	semidirect
164. C₄×C₇₃	H(3,ℤ/4)	semidirect
183. C₇₃	#10837	semidirect
201. C₄×C₄⋊C₄	Dic₇₃	semidirect
344. C₄	#17988343	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924012	2
#22612338	2
C₂×#17988343	2
C₇₃×#2285	2
#10837	73