Subgroups — #33051578

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11924028	C₂	1	1	yes	maximal, characteristic
2	#22618887	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556599	C₂	1	1	yes	maximal, characteristic
4	#22608215	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608213	C₂	1	2	yes	maximal, complemented
6	#22608210	C₂	1	2	yes	maximal, complemented
7	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#11924011	C₂	1	2	yes	maximal, complemented
9	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
10	#322×D₇₃	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556599	C₄	1	2	yes	—
13	C₄×#11924028	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₄₆	C₄	1	2	yes	—
15	#17988711	C₄	1	2	yes	—
16	C₈×D₂₉₂	C₄	1	2	yes	—
17	#17988652	C₄	1	2	yes	—
18	C₈×C₂²⋊D₇₃	C₄	1	4	yes	—
19	#17988650	C₄	1	4	yes	—
20	#17988640	C₂²	1	1	yes	characteristic
21	#17988638	C₂²	1	2	yes	—
22	#17988638	C₂²	1	2	yes	—
23	#17988545	C₄	1	2	yes	—
24	#17988544	C₄	1	4	yes	—
25	C₄×#11924011	C₂²	1	2	yes	—
26	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₇₃	C₂²	1	2	yes	—
29	C₄×C₈×D₇₃	C₂²	1	2	yes	—
30	C₄×C₈×D₇₃	C₂²	1	2	yes	—
31	#17988309	C₂²	1	1	yes	characteristic
32	#17988309	C₂²	1	2	yes	—
33	#17988309	C₂²	1	1	yes	characteristic
34	#17988309	C₂²	1	1	yes	characteristic
35	#17988309	C₂²	1	1	yes	characteristic
36	#17988308	C₂²	1	2	yes	—
37	C₈×#10596831	C₂²	1	2	yes	—
38	C₈×#10596831	C₂²	1	2	yes	—
39	#431×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
42	#322×C₇₃	C₂²	1	2	yes	—
43	C₈²×C₇₃	C₂²	1	2	yes	—
44	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
45	#14556599	C₈	1	16	yes	complemented
46	C₂×#11924011	C₂×C₄	1	4	yes	—
47	C₂²×#10596831	C₂×C₄	1	2	yes	—
48	C₂²×#10596831	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
50	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
51	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
54	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
56	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
58	#14556550	C₂×C₄	1	1	yes	characteristic
59	#14556548	C₂×C₄	1	2	yes	—
60	C₄²×D₇₃	C₂³	1	2	yes	—
61	C₄⋊#10596831	C₂×C₄	1	2	yes	—
62	#14556520	C₂×C₄	1	4	yes	—
63	#14556520	C₂×C₄	1	4	yes	—
64	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
65	#14556503	C₂×C₄	1	4	yes	—
66	#14556503	C₂×C₄	1	4	yes	—
67	#14556501	C₂×C₄	1	4	yes	—
68	C₄×#10596831	C₂×C₄	1	2	yes	—
69	C₄×#10596831	C₂×C₄	1	2	yes	—
70	C₄×#10596831	C₂³	1	1	yes	characteristic
71	C₄×#10596831	C₂³	1	2	yes	—
72	C₄×#10596831	C₂³	1	1	yes	characteristic
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
76	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
77	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
79	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
83	#97×C₇₃	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	2	yes	—
88	C₄×C₈×C₇₃	C₂³	1	2	yes	—
89	#11924028	C₂×C₈	1	8	yes	—
90	C₂×Dic₁₄₆	C₄²	1	1	yes	characteristic
91	C₂×D₂₉₂	C₄²	1	1	yes	characteristic
92	C₂×C₂²⋊D₇₃	C₄²	1	2	yes	—
93	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
96	#11924011	C₂×C₈	1	16	yes	complemented
97	C₂×#10596831	—	4	2	no	—
98	C₂×#10596831	C₂×C₈	1	8	yes	complemented
99	C₂×#10596831	C₂²×C₄	1	2	yes	—
100	C₂×#10596831	C₂²×C₄	1	4	yes	—
101	C₂×#10596831	C₂²×C₄	1	2	yes	—
102	C₂×#10596831	C₂²×C₄	1	2	yes	—
103	C₂×#10596831	C₂²×C₄	1	2	yes	—
104	C₈×D₇₃	—	4	4	no	—
105	C₈×D₇₃	—	4	4	no	—
106	C₈×D₇₃	C₂×C₈	1	16	yes	complemented
107	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
108	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
109	#11924007	C₄²	1	2	yes	—
110	#11924007	C₄²	1	2	yes	—
111	C₂²⋊Dic₇₃	C₄²	1	2	yes	—
112	#11924005	C₄²	1	2	yes	—
113	#11924005	C₄²	1	2	yes	—
114	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
115	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
116	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	M₁₆×C₇₃	C₂×C₈	1	8	yes	complemented
121	C₂×C₈×C₇₃	—	4	2	no	—
122	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
125	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
126	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
131	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₇₃	—	2	2	no	—
133	C₂×C₄×C₇₃	—	2	2	no	—
134	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
138	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
141	C₂²⋊D₇₃	C₄×C₈	1	8	yes	—
142	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
143	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
144	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
145	D₂₉₂	C₄×C₈	1	4	yes	—
146	C₄×D₇₃	—	2	4	no	—
147	C₄×D₇₃	—	2	4	no	—
148	C₄×D₇₃	C₂²×C₈	1	8	yes	—
149	Dic₁₄₆	C₄×C₈	1	4	yes	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	—	2	2	no	—
152	C₈×C₇₃	—	2	4	no	—
153	C₈×C₇₃	C₂²×C₈	1	8	yes	—
154	#10596831	—	2	2	no	—
155	#10596831	—	2	4	no	—
156	#10596831	—	2	2	no	—
157	#10596831	C₂²×C₈	1	4	yes	—
158	#10596831	C₂²×C₈	1	4	yes	—
159	C₂²×C₇₃	—	2	1	no	—
160	C₂²×C₇₃	—	2	1	no	—
161	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
162	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	2	2	no	—
165	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	—	2	1	no	—
168	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
169	C₄×C₇₃	C₂³×C₈	1	2	yes	—
170	C₄×C₇₃	#568	1	2	yes	—
171	C₄×C₇₃	#568	1	2	yes	—
172	Dic₇₃	—	2	2	no	—
173	Dic₇₃	—	2	2	no	—
174	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
175	#17316	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	2	2	no	—
177	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
178	C₂×C₇₃	#2875	1	1	yes	characteristic
179	C₂×C₇₃	#2875	1	1	yes	characteristic
180	D₇₃	—	2	4	no	—
181	C₈×#41	—	73	1	no	—
182	C₈×#41	—	73	1	no	—
183	C₄×#131	—	73	1	no	—
184	#1450	—	73	1	no	—
185	#1450	—	73	2	no	—
186	C₈×M₁₆	—	73	1	no	—
187	C₈×M₁₆	—	73	2	no	—
188	C₈×M₁₆	—	73	2	no	—
189	C₈×M₁₆	D₇₃	1	1	yes	characteristic, complemented
190	C₂×#322	—	73	1	no	—
191	C₂×#322	—	73	2	no	—
192	C₇₃	#17316	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	73	2	no	—
194	C₄×#41	—	73	1	no	—
195	C₈×Q₈	—	73	2	no	—
196	C₈×Q₈	—	73	2	no	—
197	C₈×D₄	—	73	4	no	—
198	C₈×D₄	—	73	2	no	—
199	C₈×D₄	—	73	4	no	—
200	C₈×D₄	—	73	2	no	—
201	#431	—	73	1	no	—
202	#431	—	73	2	no	—
203	#431	—	73	2	no	—
204	#431	C₂×D₇₃	1	1	yes	characteristic
205	#405	—	73	2	no	—
206	#405	—	73	4	no	—
207	C₄×M₁₆	—	73	2	no	—
208	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
209	C₂×C₄×C₈	—	73	1	no	—
210	C₂×C₄×C₈	—	73	1	no	—
211	C₂×C₄×C₈	—	73	2	no	—
212	C₂×C₄×C₈	—	73	2	no	—
213	C₂×C₄×C₈	—	73	2	no	—
214	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
215	#322	—	73	1	no	—
216	#322	—	73	2	no	—
217	#322	—	73	1	no	—
218	#322	—	73	1	no	—
219	#322	—	73	2	no	—
220	#322	—	73	1	no	—
221	#322	C₂×D₇₃	1	2	yes	—
222	C₈²	—	73	2	no	—
223	C₈²	—	73	2	no	—
224	C₈²	C₂×D₇₃	1	2	yes	—
225	C₂×#41	—	73	1	no	—
226	#131	—	73	16	no	—
227	C₂×M₁₆	—	73	4	no	—
228	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
229	C₂²×C₈	—	73	2	no	—
230	C₂²×C₈	—	73	4	no	—
231	C₂²×C₈	—	73	4	no	—
232	C₂²×C₈	—	73	2	no	—
233	C₂²×C₈	—	73	4	no	—
234	C₂²×C₈	C₄×D₇₃	1	2	yes	—
235	C₄×Q₈	—	73	1	no	—
236	C₄×Q₈	—	73	1	no	—
237	C₄×D₄	—	73	2	no	—
238	C₄×D₄	—	73	2	no	—
239	C₄×D₄	—	73	1	no	—
240	C₄×D₄	—	73	1	no	—
241	#117	—	73	1	no	—
242	#117	—	73	2	no	—
243	C₂×C₄²	—	73	2	no	—
244	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
245	C₄⋊C₈	—	73	4	no	—
246	C₄⋊C₈	—	73	4	no	—
247	C₄⋊C₈	—	73	2	no	—
248	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
249	C₂²⋊C₈	—	73	2	no	—
250	C₂²⋊C₈	—	73	4	no	—
251	C₂²⋊C₈	—	73	4	no	—
252	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
253	#97	—	73	4	no	—
254	#97	C₄×D₇₃	1	2	yes	—
255	C₄×C₈	—	73	4	no	—
256	C₄×C₈	—	73	4	no	—
257	C₄×C₈	—	73	4	no	—
258	C₄×C₈	—	73	2	no	—
259	C₄×C₈	—	73	2	no	—
260	C₄×C₈	—	73	1	no	—
261	C₄×C₈	—	73	2	no	—
262	C₄×C₈	—	73	1	no	—
263	C₄×C₈	—	73	1	no	—
264	C₄×C₈	—	73	1	no	—
265	C₄×C₈	C₄×D₇₃	1	2	yes	—
266	C₄×C₈	C₂²×D₇₃	1	2	yes	—
267	C₄×C₈	C₂²×D₇₃	1	2	yes	—
268	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
270	#41	—	73	8	no	—
271	C₂×Q₈	—	73	1	no	—
272	C₂×D₄	—	73	2	no	—
273	C₂×D₄	—	73	1	no	—
274	C₂²×C₄	—	73	2	no	—
275	C₂²×C₄	—	73	2	no	—
276	C₂²×C₄	—	73	2	no	—
277	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
280	M₁₆	—	73	16	no	—
281	M₁₆	C₈×D₇₃	1	8	yes	complemented
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	292	4	no	—
284	C₂×C₈	—	292	4	no	—
285	C₂×C₈	—	292	2	no	—
286	C₂×C₈	—	73	8	no	—
287	C₂×C₈	—	73	2	no	—
288	C₂×C₈	—	73	2	no	—
289	C₂×C₈	—	73	16	no	—
290	C₂×C₈	—	73	2	no	—
291	C₂×C₈	—	73	2	no	—
292	C₂×C₈	—	73	4	no	—
293	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
295	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
296	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
297	C₄⋊C₄	—	73	2	no	—
298	C₄⋊C₄	—	73	2	no	—
299	C₄⋊C₄	—	73	1	no	—
300	C₄⋊C₄	—	73	1	no	—
301	C₂²⋊C₄	—	73	2	no	—
302	C₂²⋊C₄	—	73	2	no	—
303	C₂²⋊C₄	—	73	2	no	—
304	C₄²	—	73	2	no	—
305	C₄²	—	73	2	no	—
306	C₄²	—	73	2	no	—
307	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
310	C₄²	C₂³×D₇₃	1	1	yes	characteristic
311	C₂³	—	73	2	no	—
312	C₂³	C₄²×D₇₃	1	1	yes	characteristic
313	Q₈	—	73	4	no	—
314	D₄	—	73	8	no	—
315	D₄	—	73	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	8	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	2	no	—
322	C₂×C₄	—	146	4	no	—
323	C₂×C₄	—	146	4	no	—
324	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	73	4	no	—
335	C₈	—	73	4	no	—
336	C₈	—	146	2	no	—
337	C₈	—	146	2	no	—
338	C₈	—	146	4	no	—
339	C₈	C₂×C₈×D₇₃	1	8	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	73	4	no	—
343	C₂²	—	146	2	no	—
344	C₂²	—	146	2	no	—
345	C₂²	C₄×C₈×D₇₃	1	2	yes	—
346	C₂²	C₂×C₄²×D₇₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	73	4	no	—
350	C₄	—	146	2	no	—
351	C₄	—	146	2	no	—
352	C₄	C₄×C₈×D₇₃	1	2	yes	—
353	C₄	C₂²×C₈×D₇₃	1	2	yes	—
354	C₄	#17989352	1	2	yes	—
355	C₄	#17989353	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	146	4	no	—
358	C₂	C₂×C₄×C₈×D₇₃	1	1	yes	characteristic
359	C₂	#22618358	1	1	yes	characteristic
360	C₂	#22618360	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₃
commutator	C₂×C₇₃	C₂²×C₄×C₈
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄²
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#17316
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22608215	C₂	semidirect
5. #22608213	C₂	semidirect
6. #22608210	C₂	semidirect
7. M₁₆×#10596831	C₂	semidirect
8. C₈×#11924011	C₂	semidirect
9. C₂×#17988309	C₂	semidirect
10. #322×D₇₃	C₂	semidirect
11. C₈×M₁₆×C₇₃	C₂	semidirect
45. #14556599	C₈	semidirect
96. #11924011	C₂×C₈	semidirect
98. C₂×#10596831	C₂×C₈	semidirect
106. C₈×D₇₃	C₂×C₈	semidirect
120. M₁₆×C₇₃	C₂×C₈	semidirect
189. C₈×M₁₆	D₇₃	semidirect
192. C₇₃	#17316	semidirect
281. M₁₆	C₈×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924028	2
#22618887	2
C₄×#14556599	2
#22608215	2
#22608213	2
#22608210	2
M₁₆×#10596831	2
C₈×#11924011	2
C₂×#17988309	2
#322×D₇₃	2
C₈×M₁₆×C₇₃	2
#17316	73