Subgroups — #32995693

Order 18688.

402 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618638	C₂	1	1	yes	maximal, characteristic
2	#22611685	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607887	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607885	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556597	—	2	1	no	—
9	#17989134	C₄	1	1	yes	characteristic
10	C₂×#14556530	—	2	1	no	—
11	C₂×C₁₆×D₇₃	—	2	1	no	—
12	#17988665	C₄	1	1	yes	characteristic
13	#17988664	C₄	1	1	yes	characteristic
14	#17988662	—	2	1	no	—
15	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
16	C₈×C₂²⋊D₇₃	—	2	1	no	—
17	#17988609	C₄	1	1	yes	characteristic
18	#17988552	—	2	1	no	—
19	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊#11924003	C₂²	1	1	yes	characteristic
21	#17988405	—	2	1	no	—
22	C₄×#11924003	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₇₃	—	2	1	no	—
24	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#11924028	—	2	1	no	—
28	#14556597	—	4	2	no	—
29	#14556597	—	4	2	no	—
30	C₂×#11924012	—	2	1	no	—
31	C₂×#11924011	—	2	1	no	—
32	C₂×#11924011	—	2	1	no	—
33	C₂×C₈×D₇₃	—	2	1	no	—
34	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
36	#14556590	C₈	1	1	yes	characteristic
37	#14556587	C₈	1	1	yes	characteristic
38	#14556577	—	2	1	no	—
39	#14556566	C₈	1	1	yes	characteristic
40	#14556564	—	2	1	no	—
41	#14556562	C₈	1	1	yes	characteristic
42	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
43	C₄×C₂²⋊D₇₃	—	2	1	no	—
44	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
45	#14556530	—	4	2	no	—
46	C₂×#11924003	—	2	1	no	—
47	C₂×#11924003	—	2	1	no	—
48	C₁₆×D₇₃	—	2	4	no	—
49	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
50	#14556520	—	2	1	no	—
51	C₂²⋊#10596831	—	2	1	no	—
52	#14556503	—	2	1	no	—
53	#14556503	—	2	1	no	—
54	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₇₃	D₄	1	1	yes	characteristic
56	C₈×Dic₇₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₃	—	2	1	no	—
58	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₃	—	2	1	no	—
60	C₂×C₁₆×C₇₃	—	2	1	no	—
61	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₇₃	—	2	1	no	—
63	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	#11924028	—	4	2	no	—
65	#11924028	—	4	2	no	—
66	C₂×Dic₁₄₆	—	2	1	no	—
67	C₂×D₂₉₂	—	2	1	no	—
68	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
69	C₂×C₂²⋊D₇₃	—	2	1	no	—
70	C₂×C₂²⋊D₇₃	—	2	1	no	—
71	C₂×C₂²⋊D₇₃	—	2	1	no	—
72	C₂×C₄×D₇₃	—	2	1	no	—
73	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
76	#11924012	—	4	2	no	—
77	#11924011	—	4	2	no	—
78	#11924011	—	4	2	no	—
79	C₂×#10596831	—	2	1	no	—
80	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
82	C₈×D₇₃	—	2	2	no	—
83	C₈×D₇₃	—	4	1	no	—
84	C₈×D₇₃	—	4	1	no	—
85	C₈×D₇₃	D₈	1	2	yes	—
86	C₈×D₇₃	SD₁₆	1	2	yes	—
87	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
88	#11924007	—	2	1	no	—
89	C₂²⋊Dic₇₃	—	2	1	no	—
90	#11924005	—	2	1	no	—
91	#11924005	—	2	1	no	—
92	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
95	#11924003	—	2	2	no	—
96	#11924003	—	4	1	no	—
97	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	—	2	1	no	—
99	C₂×C₈×C₇₃	—	2	1	no	—
100	C₂×C₈×C₇₃	—	4	1	no	—
101	C₂×C₈×C₇₃	—	4	1	no	—
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₇₃	—	2	1	no	—
106	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₁₆×C₇₃	—	2	2	no	—
108	C₁₆×C₇₃	—	4	1	no	—
109	C₂³×C₇₃	—	2	1	no	—
110	C₂²×D₇₃	—	2	1	no	—
111	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
112	D₄×C₇₃	—	2	1	no	—
113	D₄×C₇₃	—	2	1	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	4	1	no	—
116	C₂×C₄×C₇₃	—	4	1	no	—
117	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂²⋊D₇₃	—	4	2	no	—
121	C₂²⋊D₇₃	—	4	2	no	—
122	C₂²⋊D₇₃	—	4	2	no	—
123	C₂×Dic₇₃	—	2	1	no	—
124	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
127	D₂₉₂	—	2	1	no	—
128	D₂₉₂	—	2	1	no	—
129	D₂₉₂	—	4	2	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	2	2	no	—
133	C₄×D₇₃	—	4	1	no	—
134	C₄×D₇₃	—	4	1	no	—
135	C₄×D₇₃	#102	1	2	yes	—
136	C₄×D₇₃	#104	1	2	yes	—
137	Dic₁₄₆	—	4	2	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	4	1	no	—
140	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
141	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
142	#10596831	—	2	1	no	—
143	#10596831	—	4	1	no	—
144	#10596831	#102	1	2	yes	—
145	C₂²×C₇₃	—	4	1	no	—
146	C₂²×C₇₃	—	4	1	no	—
147	C₂²×C₇₃	—	4	1	no	—
148	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	4	1	no	—
152	C₂×D₇₃	—	4	1	no	—
153	C₂×D₇₃	—	4	1	no	—
154	C₂×D₇₃	D₄⋊C₈	1	2	yes	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	4	1	no	—
157	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₇₃	C₂×#104	1	1	yes	characteristic
159	C₄×C₇₃	#504	1	1	yes	characteristic
160	C₄×C₇₃	#504	1	1	yes	characteristic
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	—	4	1	no	—
163	Dic₇₃	D₄⋊C₈	1	2	yes	—
164	Dic₇₃	#350	1	2	yes	—
165	#11584	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	8	1	no	—
167	C₂×C₇₃	C₂×D₄⋊C₈	1	1	yes	characteristic
168	C₂×C₇₃	#2113	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
170	D₇₃	—	2	2	no	—
171	D₇₃	—	8	1	no	—
172	#2931	—	73	1	no	—
173	#2150	—	73	1	no	—
174	C₂×C₄×C₁₆	—	73	1	no	—
175	D₄⋊C₁₆	—	73	1	no	—
176	D₄⋊C₁₆	—	73	1	no	—
177	D₄⋊C₁₆	—	73	1	no	—
178	D₄⋊C₁₆	D₇₃	1	1	yes	characteristic, complemented
179	C₇₃	#11584	1	1	yes	characteristic, complemented, Sylow
180	C₂×#131	—	146	1	no	—
181	#532	—	73	1	no	—
182	C₂×M₃₂	—	146	1	no	—
183	C₂²×C₁₆	—	146	1	no	—
184	#436	—	73	1	no	—
185	#436	—	73	1	no	—
186	#436	—	146	1	no	—
187	C₈×D₄	—	73	1	no	—
188	C₈×D₄	—	146	1	no	—
189	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
190	#423	—	73	1	no	—
191	#408	—	146	1	no	—
192	C₂×C₄×C₈	—	73	1	no	—
193	C₄⋊C₁₆	—	73	1	no	—
194	C₄⋊C₁₆	—	146	1	no	—
195	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
196	C₄×C₁₆	—	73	1	no	—
197	C₄×C₁₆	—	146	1	no	—
198	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
199	C₂×#41	—	146	1	no	—
200	#131	—	292	2	no	—
201	#131	—	292	2	no	—
202	C₂×M₁₆	—	146	1	no	—
203	C₂×M₁₆	—	146	1	no	—
204	C₂×M₁₆	—	146	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	146	1	no	—
209	C₄⋊Q₈	—	73	1	no	—
210	#127	—	73	1	no	—
211	#124	—	146	1	no	—
212	#121	—	73	1	no	—
213	#121	—	73	1	no	—
214	#121	—	146	1	no	—
215	C₄×D₄	—	73	1	no	—
216	C₄×D₄	—	146	1	no	—
217	C₄×D₄	C₄×D₇₃	1	1	yes	characteristic
218	C₂×C₄²	—	73	1	no	—
219	M₃₂	—	292	2	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	146	4	no	—
223	C₂×C₁₆	—	146	1	no	—
224	C₂×C₁₆	—	146	1	no	—
225	C₄⋊C₈	—	73	1	no	—
226	C₄⋊C₈	—	146	1	no	—
227	C₄⋊C₈	C₄×D₇₃	1	1	yes	characteristic
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	146	1	no	—
230	C₂²⋊C₈	—	146	1	no	—
231	C₂²⋊C₈	—	146	1	no	—
232	C₄×C₈	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	1	no	—
235	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
236	#41	—	292	2	no	—
237	#41	—	292	2	no	—
238	C₂×Q₈	—	146	1	no	—
239	C₂×D₄	—	73	1	no	—
240	C₂×D₄	—	146	1	no	—
241	C₂×D₄	—	146	1	no	—
242	C₂×D₄	—	146	1	no	—
243	C₂×D₄	—	146	1	no	—
244	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	146	1	no	—
250	M₁₆	—	292	2	no	—
251	M₁₆	—	292	2	no	—
252	M₁₆	—	292	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	292	1	no	—
257	C₂×C₈	—	292	1	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	—	146	1	no	—
264	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
265	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
266	C₄⋊C₄	—	73	1	no	—
267	C₄⋊C₄	—	146	1	no	—
268	C₄⋊C₄	C₈×D₇₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	146	1	no	—
271	C₂²⋊C₄	—	146	1	no	—
272	C₂²⋊C₄	—	146	1	no	—
273	C₄²	—	73	1	no	—
274	C₄²	—	73	1	no	—
275	C₄²	—	73	1	no	—
276	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	4	1	no	—
279	C₁₆	—	292	1	no	—
280	C₁₆	—	146	2	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	73	1	no	—
283	C₂³	—	146	1	no	—
284	Q₈	—	292	2	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	292	2	no	—
288	D₄	—	292	2	no	—
289	D₄	—	292	2	no	—
290	D₄	—	292	2	no	—
291	D₄	—	146	1	no	—
292	D₄	—	146	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	292	1	no	—
297	C₂×C₄	—	292	1	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	2	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	146	2	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	1	no	—
306	C₂×C₄	—	146	1	no	—
307	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	292	1	no	—
313	C₈	—	73	2	no	—
314	C₈	—	146	1	no	—
315	C₈	D₈×D₇₃	1	1	yes	characteristic
316	C₈	SD₁₆×D₇₃	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	292	1	no	—
321	C₂²	—	292	1	no	—
322	C₂²	—	292	1	no	—
323	C₂²	—	73	2	no	—
324	C₂²	—	146	1	no	—
325	C₂²	—	146	1	no	—
326	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	292	1	no	—
330	C₄	—	73	2	no	—
331	C₄	—	73	2	no	—
332	C₄	—	146	1	no	—
333	C₄	#102×D₇₃	1	1	yes	characteristic
334	C₄	#104×D₇₃	1	1	yes	characteristic
335	C₄	#17989003	1	1	yes	characteristic
336	C₄	#17989005	1	1	yes	characteristic
337	C₂	—	8	1	no	—
338	C₂	—	584	1	no	—
339	C₂	—	146	2	no	—
340	C₂	D₄⋊C₈×D₇₃	1	1	yes	characteristic
341	C₂	#22611543	1	1	yes	characteristic
342	C₂	#350×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×D₄⋊C₁₆	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	#17989134	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	D₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#11584
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22611685	C₂	semidirect
3. C₄×C₁₆×D₇₃	C₂	semidirect
4. #22607887	C₂	semidirect
5. D₄⋊#11924003	C₂	semidirect
6. #22607885	C₂	semidirect
7. C₇₃×D₄⋊C₁₆	C₂	semidirect
178. D₄⋊C₁₆	D₇₃	semidirect
179. C₇₃	#11584	semidirect

Maximal subgroups

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

subgroup	index
#22618638	2
#22611685	2
C₄×C₁₆×D₇₃	2
#22607887	2
D₄⋊#11924003	2
#22607885	2
C₇₃×D₄⋊C₁₆	2
#11584	73