Subgroups — #33238985

Order 18688.

684 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623033	C₂	1	2	yes	maximal
2	C₄×#14556597	C₂	1	1	yes	maximal, characteristic
3	#22611696	C₂	1	2	yes	maximal, complemented
4	#22611695	C₂	1	4	yes	maximal, complemented
5	C₂×#17988408	C₂	1	1	yes	maximal, characteristic, complemented
6	#364×D₇₃	C₂	1	4	yes	maximal, complemented
7	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556597	C₄	1	1	yes	characteristic
9	C₂×#14556597	C₂²	1	1	yes	characteristic
10	C₄×#11924028	C₄	1	1	yes	characteristic
11	#17989005	—	2	4	no	—
12	#17989003	—	2	4	no	—
13	C₂×#14556531	C₂²	1	2	yes	—
14	M₃₂×D₇₃	C₂²	1	8	yes	—
15	#17988542	C₄	1	1	yes	characteristic
16	#17988542	C₂²	1	1	yes	characteristic
17	#17988541	C₄	1	2	yes	—
18	#17988541	C₂²	1	2	yes	—
19	C₄×#11924012	C₄	1	1	yes	characteristic
20	C₄×#11924011	C₄	1	2	yes	—
21	C₄×C₈×D₇₃	C₂²	1	2	yes	—
22	#17988408	C₂²	1	2	yes	—
23	#17988408	C₂²	1	2	yes	—
24	#17988407	C₂²	1	8	yes	—
25	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	#364×C₇₃	C₂²	1	2	yes	—
28	#364×C₇₃	C₂²	1	2	yes	—
29	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
30	#14556597	C₈	1	4	yes	—
31	#14556597	C₈	1	4	yes	—
32	#14556597	D₄	1	2	yes	—
33	#14556597	D₄	1	2	yes	—
34	#14556597	Q₈	1	2	yes	—
35	#14556597	Q₈	1	2	yes	—
36	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924011	C₂×C₄	1	2	yes	—
39	C₂×#11924011	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂³	1	2	yes	—
42	C₄×Dic₁₄₆	—	2	1	no	—
43	C₄×D₂₉₂	—	2	1	no	—
44	C₄×C₂²⋊D₇₃	—	2	2	no	—
45	#14556550	C₂×C₄	1	1	yes	characteristic
46	#14556548	C₂×C₄	1	2	yes	—
47	C₄²×D₇₃	C₂×C₄	1	2	yes	—
48	#14556531	—	2	2	no	—
49	#14556531	C₂³	1	4	yes	—
50	#14556530	—	2	8	no	—
51	C₂×#11924003	—	2	2	no	—
52	C₁₆×D₇₃	—	2	8	no	—
53	#14556502	C₂×C₄	1	2	yes	—
54	#14556501	C₂×C₄	1	2	yes	—
55	#14556501	C₂×C₄	1	2	yes	—
56	C₄×#10596831	C₂×C₄	1	2	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₇₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₇₃	—	2	2	no	—
63	M₃₂×C₇₃	C₂³	1	4	yes	—
64	C₂×C₁₆×C₇₃	—	2	2	no	—
65	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	#11924028	C₄⋊C₄	1	2	yes	—
70	#11924028	C₄⋊C₄	1	2	yes	—
71	#11924028	C₂×C₈	1	2	yes	—
72	#11924028	C₂×C₈	1	2	yes	—
73	C₂×Dic₁₄₆	M₁₆	1	1	yes	characteristic
74	C₂×D₂₉₂	M₁₆	1	1	yes	characteristic
75	C₂×C₂²⋊D₇₃	M₁₆	1	2	yes	—
76	C₂×C₄×D₇₃	—	2	2	no	—
77	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
78	#11924012	C₄⋊C₄	1	2	yes	—
79	#11924012	C₄⋊C₄	1	2	yes	—
80	#11924012	C₂×C₈	1	2	yes	—
81	#11924012	C₂×C₈	1	2	yes	—
82	#11924011	C₄⋊C₄	1	4	yes	—
83	#11924011	C₄⋊C₄	1	4	yes	—
84	#11924011	C₂×C₈	1	8	yes	—
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₂×#10596831	C₂²×C₄	1	2	yes	—
87	C₈×D₇₃	C₂×C₈	1	8	yes	—
88	C₈×D₇₃	C₂×D₄	1	4	yes	—
89	C₈×D₇₃	C₂×Q₈	1	4	yes	—
90	C₄⋊Dic₇₃	—	2	1	no	—
91	#11924007	—	2	2	no	—
92	C₂²⋊Dic₇₃	—	2	1	no	—
93	#11924005	—	2	2	no	—
94	C₄×Dic₇₃	—	2	2	no	—
95	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
96	#11924003	—	2	4	no	—
97	C₂²×C₄×C₇₃	—	2	1	no	—
98	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
109	C₄²×C₇₃	—	2	1	no	—
110	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₁₆×C₇₃	—	2	4	no	—
113	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂²×D₇₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₂²⋊D₇₃	C₄⋊C₈	1	4	yes	—
125	C₂²⋊D₇₃	C₄⋊C₈	1	4	yes	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
128	D₂₉₂	C₄⋊C₈	1	2	yes	—
129	D₂₉₂	C₄⋊C₈	1	2	yes	—
130	C₄×D₇₃	—	4	4	no	—
131	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₄×D₇₃	C₂²×C₈	1	4	yes	—
133	Dic₁₄₆	C₄⋊C₈	1	2	yes	—
134	Dic₁₄₆	C₄⋊C₈	1	2	yes	—
135	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈×C₇₃	C₂²×C₈	1	2	yes	—
138	C₈×C₇₃	C₂²×C₈	1	2	yes	—
139	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
141	#10596831	C₂×C₄⋊C₄	1	4	yes	—
142	#10596831	C₂²×C₈	1	4	yes	—
143	C₂²×C₇₃	—	4	1	no	—
144	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×D₇₃	—	4	2	no	—
148	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	4	1	no	—
151	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
155	Dic₇₃	—	4	2	no	—
156	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
157	#33401	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	1	no	—
159	C₂×C₇₃	—	2	1	no	—
160	C₂×C₇₃	—	2	1	no	—
161	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
162	D₇₃	—	2	4	no	—
163	#3407	—	73	2	no	—
164	C₄×#131	—	73	1	no	—
165	#2153	—	73	2	no	—
166	#2153	—	73	4	no	—
167	C₂×#364	—	73	1	no	—
168	C₂×#364	—	73	4	no	—
169	C₂×#364	D₇₃	1	1	yes	characteristic, complemented
170	C₇₃	#33401	1	1	yes	characteristic, complemented, Sylow
171	C₂×#131	—	73	1	no	—
172	C₂×#131	—	73	1	no	—
173	C₄×#41	—	73	1	no	—
174	#504	—	146	4	no	—
175	#504	—	146	4	no	—
176	C₂×M₃₂	—	73	2	no	—
177	C₂×M₃₂	—	73	8	no	—
178	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
179	#405	—	73	1	no	—
180	#405	—	73	1	no	—
181	#405	—	73	2	no	—
182	#405	—	73	2	no	—
183	C₄×M₁₆	—	73	1	no	—
184	C₄×M₁₆	—	73	2	no	—
185	C₂×C₄×C₈	—	73	2	no	—
186	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
187	#364	—	73	8	no	—
188	#364	—	73	2	no	—
189	#364	—	73	2	no	—
190	#364	C₂×D₇₃	1	2	yes	—
191	#364	C₂×D₇₃	1	2	yes	—
192	C₂×#41	—	73	1	no	—
193	#131	—	73	4	no	—
194	#131	—	73	4	no	—
195	#131	—	73	2	no	—
196	#131	—	73	2	no	—
197	#131	—	73	2	no	—
198	#131	—	73	2	no	—
199	C₂×M₁₆	—	73	1	no	—
200	C₂×M₁₆	—	73	1	no	—
201	C₂×M₁₆	—	73	2	no	—
202	C₂×M₁₆	—	73	2	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
207	C₄×Q₈	—	146	1	no	—
208	C₄×D₄	—	146	1	no	—
209	C₄×D₄	—	146	2	no	—
210	#117	—	73	1	no	—
211	#117	—	73	2	no	—
212	C₂×C₄²	—	73	2	no	—
213	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
214	M₃₂	—	2	2	no	—
215	M₃₂	—	73	4	no	—
216	M₃₂	—	146	8	no	—
217	M₃₂	—	146	2	no	—
218	M₃₂	C₂²×D₇₃	1	4	yes	—
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	—	146	8	no	—
221	C₂×C₁₆	—	146	2	no	—
222	#97	—	73	2	no	—
223	#97	—	73	2	no	—
224	#97	—	73	2	no	—
225	C₄×C₈	—	73	2	no	—
226	C₄×C₈	—	73	2	no	—
227	C₄×C₈	—	73	2	no	—
228	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
229	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
232	#41	—	73	2	no	—
233	#41	—	73	2	no	—
234	#41	—	73	2	no	—
235	#41	—	73	2	no	—
236	C₂×Q₈	—	73	1	no	—
237	C₂×D₄	—	73	2	no	—
238	C₂×D₄	—	73	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	146	2	no	—
242	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
243	M₁₆	—	73	2	no	—
244	M₁₆	—	73	2	no	—
245	M₁₆	—	73	2	no	—
246	M₁₆	—	73	2	no	—
247	M₁₆	—	73	4	no	—
248	M₁₆	—	73	4	no	—
249	M₁₆	—	73	8	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	8	no	—
253	C₂×C₈	—	73	4	no	—
254	C₂×C₈	—	73	4	no	—
255	C₂×C₈	C₈×D₇₃	1	2	yes	—
256	C₂×C₈	C₈×D₇₃	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
260	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
261	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	1	no	—
267	C₂²⋊C₄	—	146	2	no	—
268	C₂²⋊C₄	—	146	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	73	2	no	—
271	C₄²	—	146	2	no	—
272	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₁₆	—	146	4	no	—
276	C₂³	—	73	2	no	—
277	C₂³	M₁₆×D₇₃	1	1	yes	characteristic
278	Q₈	—	73	2	no	—
279	Q₈	—	73	2	no	—
280	D₄	—	73	4	no	—
281	D₄	—	73	4	no	—
282	D₄	—	73	2	no	—
283	D₄	—	73	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	292	4	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	4	no	—
290	C₂×C₄	—	73	4	no	—
291	C₂×C₄	—	146	2	no	—
292	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
293	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
298	C₈	—	73	4	no	—
299	C₈	—	73	4	no	—
300	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
301	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
302	C₈	C₂×C₈×D₇₃	1	2	yes	—
303	C₈	C₂×C₈×D₇₃	1	2	yes	—
304	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
305	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	292	2	no	—
308	C₂²	—	73	4	no	—
309	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
310	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
311	C₂²	C₂×M₁₆×D₇₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	292	2	no	—
315	C₄	—	73	4	no	—
316	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
317	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
318	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
319	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	146	4	no	—
324	C₂	C₂×C₄⋊C₈×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂×C₇₃	C₂²×C₄⋊C₈
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#33401
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #22611696	C₂	semidirect
4. #22611695	C₂	semidirect
5. C₂×#17988408	C₂	semidirect
6. #364×D₇₃	C₂	semidirect
7. C₂×#364×C₇₃	C₂	semidirect
169. C₂×#364	D₇₃	semidirect
170. C₇₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#22623033	2
C₄×#14556597	2
#22611696	2
#22611695	2
C₂×#17988408	2
#364×D₇₃	2
C₂×#364×C₇₃	2
#33401	73