Subgroups — #33239171

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989003	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618641	C₂	1	1	yes	maximal, characteristic
3	#22611761	C₂	1	4	yes	maximal, complemented
4	C₁₆×C₂²⋊D₇₃	C₂	1	4	yes	maximal, complemented
5	#22611686	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611546	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556597	C₂²	1	1	yes	characteristic
11	#17989133	C₄	1	1	yes	characteristic
12	#17989003	—	2	4	no	—
13	C₂×#14556531	C₂²	1	1	yes	characteristic
14	C₂×#14556530	C₂²	1	2	yes	—
15	C₂²×#11924003	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
17	#17988663	C₄	1	4	yes	—
18	C₈×C₂²⋊D₇₃	C₂²	1	4	yes	—
19	#17988610	C₄	1	1	yes	characteristic
20	#17988555	C₄	1	1	yes	characteristic
21	#17988554	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
23	#17988405	C₂²	1	4	yes	—
24	C₂²⋊#11924003	C₂²	1	4	yes	—
25	#17988366	C₂²	1	4	yes	—
26	C₁₆×Dic₇₃	C₂²	1	2	yes	—
27	C₁₆×Dic₇₃	C₂²	1	2	yes	—
28	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
29	#504×C₇₃	—	2	4	no	—
30	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
31	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
33	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
35	#14556597	—	2	2	no	—
36	#14556597	—	2	2	no	—
37	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924011	C₂×C₄	1	2	yes	—
39	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	2	yes	—
41	#14556589	C₈	1	1	yes	characteristic
42	#14556587	C₈	1	1	yes	characteristic
43	#14556580	C₈	1	1	yes	characteristic
44	#14556578	C₈	1	1	yes	characteristic
45	#14556563	C₈	1	4	yes	—
46	C₄×C₂²⋊D₇₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
48	#14556531	—	2	2	no	—
49	#14556530	—	2	4	no	—
50	C₂×#11924003	—	2	2	no	—
51	C₂×#11924003	C₂³	1	2	yes	—
52	C₂×#11924003	C₂³	1	2	yes	—
53	C₁₆×D₇₃	—	2	4	no	—
54	#14556520	C₂×C₄	1	4	yes	—
55	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
56	#14556503	C₂×C₄	1	4	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₃	C₂³	1	2	yes	—
59	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
60	#131×C₇₃	—	2	2	no	—
61	#131×C₇₃	—	2	2	no	—
62	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
64	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	M₃₂×C₇₃	—	2	2	no	—
67	M₃₂×C₇₃	—	2	4	no	—
68	C₂×C₁₆×C₇₃	—	2	4	no	—
69	C₂×C₁₆×C₇₃	D₄	1	4	yes	complemented
70	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
73	#11924028	—	2	2	no	—
74	#11924028	—	2	2	no	—
75	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
76	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
77	C₂×C₂²⋊D₇₃	C₂×C₈	1	4	yes	—
78	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
79	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
81	#11924012	—	2	2	no	—
82	#11924011	—	2	4	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₈×D₇₃	—	2	2	no	—
87	C₈×D₇₃	—	2	2	no	—
88	#11924007	C₂×C₈	1	4	yes	—
89	C₂²⋊Dic₇₃	C₂×C₈	1	4	yes	—
90	#11924005	C₂×C₈	1	4	yes	—
91	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
94	#11924003	—	2	2	no	—
95	#11924003	—	2	2	no	—
96	#41×C₇₃	—	2	2	no	—
97	#41×C₇₃	—	2	2	no	—
98	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
100	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	M₁₆×C₇₃	—	2	2	no	—
104	M₁₆×C₇₃	—	2	4	no	—
105	C₂×C₈×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₇₃	#41	1	2	yes	—
111	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
112	C₁₆×C₇₃	—	2	2	no	—
113	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
114	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
115	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
118	Q₈×C₇₃	—	2	2	no	—
119	D₄×C₇₃	—	2	4	no	—
120	D₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	#131	1	2	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	2	4	no	—
129	C₂²⋊D₇₃	—	2	8	no	—
130	C₂×Dic₇₃	—	2	4	no	—
131	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
132	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
133	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
134	D₂₉₂	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	—	2	2	no	—
137	Dic₁₄₆	—	2	2	no	—
138	C₈×C₇₃	—	2	2	no	—
139	C₈×C₇₃	C₄×D₄	1	2	yes	—
140	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
142	#10596831	—	2	2	no	—
143	#10596831	—	2	2	no	—
144	C₂²×C₇₃	—	2	2	no	—
145	C₂²×C₇₃	—	2	4	no	—
146	C₂²×C₇₃	—	2	2	no	—
147	C₂²×C₇₃	C₈×D₄	1	2	yes	—
148	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₇₃	—	2	4	no	—
150	C₂×D₇₃	—	2	2	no	—
151	C₄×C₇₃	—	2	2	no	—
152	C₄×C₇₃	C₈×D₄	1	2	yes	—
153	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
155	Dic₇₃	—	2	2	no	—
156	Dic₇₃	—	2	2	no	—
157	Dic₇₃	#504	1	4	yes	complemented
158	#33434	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	—	4	2	no	—
161	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
162	C₂×C₇₃	C₂×#504	1	2	yes	—
163	D₇₃	—	4	2	no	—
164	C₂×#504	—	73	1	no	—
165	C₂×#504	D₇₃	1	1	yes	characteristic, complemented
166	#2931	—	73	1	no	—
167	#2169	—	73	4	no	—
168	D₄×C₁₆	—	73	4	no	—
169	#2150	—	73	1	no	—
170	#2113	—	73	1	no	—
171	#2113	—	73	1	no	—
172	C₂×C₄×C₁₆	—	73	1	no	—
173	C₇₃	#33434	1	1	yes	characteristic, complemented, Sylow
174	C₂×#131	—	73	1	no	—
175	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
176	#532	—	73	1	no	—
177	#504	—	2	4	no	—
178	#504	—	146	4	no	—
179	C₂×M₃₂	—	73	1	no	—
180	C₂×M₃₂	—	73	2	no	—
181	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
182	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
183	C₂²×C₁₆	—	73	1	no	—
184	C₂²×C₁₆	—	73	2	no	—
185	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
186	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
187	#436	—	73	4	no	—
188	C₈×D₄	—	73	4	no	—
189	#423	—	73	1	no	—
190	#408	—	73	1	no	—
191	#408	—	73	1	no	—
192	C₂×C₄×C₈	—	73	1	no	—
193	C₄⋊C₁₆	—	73	4	no	—
194	C₂²⋊C₁₆	—	73	4	no	—
195	C₂²⋊C₁₆	—	73	4	no	—
196	C₄×C₁₆	—	73	2	no	—
197	C₄×C₁₆	—	73	2	no	—
198	C₂×#41	—	73	1	no	—
199	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	146	2	no	—
203	#131	—	146	2	no	—
204	C₂×M₁₆	—	73	1	no	—
205	C₂×M₁₆	—	73	2	no	—
206	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
207	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
211	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	73	1	no	—
213	#127	—	73	1	no	—
214	#124	—	73	1	no	—
215	#124	—	73	1	no	—
216	#121	—	73	4	no	—
217	C₄×D₄	—	73	4	no	—
218	C₂×C₄²	—	73	1	no	—
219	M₃₂	—	2	2	no	—
220	M₃₂	—	2	4	no	—
221	M₃₂	—	146	2	no	—
222	M₃₂	—	146	4	no	—
223	C₂×C₁₆	—	2	4	no	—
224	C₂×C₁₆	—	73	2	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	146	2	no	—
227	C₂×C₁₆	—	146	4	no	—
228	C₂×C₁₆	C₂²⋊D₇₃	1	4	yes	complemented
229	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
230	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
232	C₄⋊C₈	—	73	4	no	—
233	C₂²⋊C₈	—	73	4	no	—
234	C₂²⋊C₈	—	73	4	no	—
235	C₄×C₈	—	73	2	no	—
236	C₄×C₈	—	73	2	no	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	146	2	no	—
240	#41	—	146	2	no	—
241	C₂×Q₈	—	73	1	no	—
242	C₂×Q₈	C₈×D₇₃	1	1	yes	characteristic
243	C₂×D₄	—	73	4	no	—
244	C₂×D₄	—	73	2	no	—
245	C₂×D₄	—	73	1	no	—
246	C₂×D₄	C₈×D₇₃	1	2	yes	—
247	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	146	2	no	—
255	M₁₆	—	146	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	—	146	2	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
266	C₂×C₈	#11924028	1	2	yes	—
267	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
268	C₄⋊C₄	—	73	4	no	—
269	C₂²⋊C₄	—	73	4	no	—
270	C₂²⋊C₄	—	73	4	no	—
271	C₄²	—	73	1	no	—
272	C₄²	—	73	1	no	—
273	C₄²	—	73	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	146	2	no	—
276	C₁₆	—	146	2	no	—
277	C₁₆	C₂×C₂²⋊D₇₃	1	2	yes	—
278	C₁₆	C₂×C₂²⋊D₇₃	1	2	yes	—
279	C₂³	—	73	2	no	—
280	C₂³	C₂×C₈×D₇₃	1	2	yes	—
281	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	146	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	—	146	8	no	—
287	D₄	—	146	4	no	—
288	D₄	—	146	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	73	2	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	146	4	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
298	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
300	C₂×C₄	#14556597	1	2	yes	—
301	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	146	2	no	—
304	C₈	—	146	2	no	—
305	C₈	C₄×C₂²⋊D₇₃	1	2	yes	—
306	C₈	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
307	C₈	C₂×#11924028	1	1	yes	characteristic
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	146	4	no	—
312	C₂²	—	146	2	no	—
313	C₂²	C₈×C₂²⋊D₇₃	1	2	yes	—
314	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	73	4	no	—
317	C₄	—	146	2	no	—
318	C₄	—	146	2	no	—
319	C₄	C₈×C₂²⋊D₇₃	1	2	yes	—
320	C₄	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
321	C₄	C₂×#14556597	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	292	2	no	—
325	C₂	C₂×C₈×C₂²⋊D₇₃	1	1	yes	characteristic
326	C₂	#504×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#504×C₇₃	C₂
prime radical	#17989133	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#504	D₇₃
hypocenter	C₇₃	#33434
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#17989003	C₂	semidirect
3. #22611761	C₂	semidirect
4. C₁₆×C₂²⋊D₇₃	C₂	semidirect
5. #22611686	C₂	semidirect
6. #22611546	C₂	semidirect
7. #22611545	C₂	semidirect
8. C₂×C₁₆×Dic₇₃	C₂	semidirect
9. C₂×#504×C₇₃	C₂	semidirect
69. C₂×C₁₆×C₇₃	D₄	semidirect
157. Dic₇₃	#504	semidirect
165. C₂×#504	D₇₃	semidirect
173. C₇₃	#33434	semidirect
228. C₂×C₁₆	C₂²⋊D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989003	2
#22618641	2
#22611761	2
C₁₆×C₂²⋊D₇₃	2
#22611686	2
#22611546	2
#22611545	2
C₂×C₁₆×Dic₇₃	2
C₂×#504×C₇₃	2
#33434	73