Subgroups — #33057443

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618882	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22608827	C₂	1	2	yes	maximal, complemented
4	#22608780	C₂	1	4	yes	maximal, complemented
5	#22608737	C₂	1	2	yes	maximal, complemented
6	#22608359	C₂	1	2	yes	maximal
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic
8	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	#17988716	—	2	4	no	—
12	#17988714	C₂²	1	4	yes	—
13	#17988711	C₂²	1	4	yes	—
14	C₄×Dic₂₉₂	C₄	1	2	yes	—
15	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
16	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₉₂	C₂²	1	2	yes	—
19	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
20	#17988640	C₂²	1	1	yes	characteristic
21	C₂×#14556525	C₄	1	1	yes	characteristic
22	#17988610	—	2	2	no	—
23	C₂×#14556516	C₄	1	2	yes	—
24	C₄×#11924012	C₂²	1	1	yes	characteristic
25	M₁₆×Dic₇₃	—	2	2	no	—
26	#17988317	C₂²	1	4	yes	—
27	#17988317	C₂²	1	4	yes	—
28	#17988309	C₂²	1	2	yes	—
29	C₈×#10596831	C₂²	1	2	yes	—
30	#423×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	2	yes	—
33	#335×C₇₃	C₂²	1	2	yes	—
34	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₉₂	—	2	4	no	—
36	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924012	D₄	1	1	yes	characteristic
40	C₂×#11924012	D₄	1	1	yes	characteristic
41	C₄×Dic₁₄₆	—	2	2	no	—
42	C₄×Dic₁₄₆	—	2	2	no	—
43	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₄₆	C₂³	1	2	yes	—
46	C₄×Dic₁₄₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
48	C₂×#11924007	—	2	2	no	—
49	C₂×C₄×Dic₇₃	—	2	2	no	—
50	#14556525	C₂×C₄	1	2	yes	—
51	#14556525	C₂×C₄	1	1	yes	characteristic
52	#14556525	C₂×C₄	1	1	yes	characteristic
53	C₄⋊#10596831	—	2	1	no	—
54	#14556520	—	2	4	no	—
55	#14556520	—	2	4	no	—
56	#14556516	C₂×C₄	1	2	yes	—
57	#14556516	C₂×C₄	1	2	yes	—
58	#14556516	C₂×C₄	1	2	yes	—
59	#14556516	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596831	—	2	1	no	—
61	#14556502	D₄	1	2	yes	—
62	#14556501	—	2	4	no	—
63	C₄×#10596831	C₂³	1	2	yes	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₈×Dic₇₃	—	2	4	no	—
66	C₂×M₁₆×C₇₃	—	2	2	no	—
67	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	—	2	2	no	—
71	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₄₆	—	2	4	no	—
77	C₂×Dic₁₄₆	—	2	2	no	—
78	C₂×Dic₁₄₆	—	4	2	no	—
79	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₇₃	—	2	2	no	—
82	C₂²×Dic₇₃	—	2	2	no	—
83	Dic₂₉₂	—	2	4	no	—
84	Dic₂₉₂	—	2	2	no	—
85	Dic₂₉₂	M₁₆	1	4	yes	complemented
86	#11924012	—	2	2	no	—
87	#11924012	Q₁₆	1	4	yes	complemented
88	C₂×#10596831	—	2	2	no	—
89	C₂×#10596831	C₂×D₄	1	2	yes	—
90	C₂×#10596831	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₇₃	—	2	2	no	—
92	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
93	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
94	#11924007	—	2	2	no	—
95	#11924007	—	2	2	no	—
96	#11924007	—	2	2	no	—
97	#11924007	—	2	2	no	—
98	C₄×Dic₇₃	—	2	2	no	—
99	C₄×Dic₇₃	—	2	2	no	—
100	C₄×Dic₇₃	—	2	2	no	—
101	C₄×Dic₇₃	—	2	2	no	—
102	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
105	M₁₆×C₇₃	—	4	4	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	—	2	4	no	—
108	C₂×C₈×C₇₃	—	4	1	no	—
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₃	#41	1	1	yes	characteristic
116	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	4	1	no	—
121	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₇₃	#135	1	2	yes	—
126	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	C₂×Dic₇₃	—	2	2	no	—
133	C₂×Dic₇₃	—	4	4	no	—
134	C₂×Dic₇₃	—	8	2	no	—
135	Dic₁₄₆	—	2	2	no	—
136	Dic₁₄₆	—	2	2	no	—
137	Dic₁₄₆	—	2	2	no	—
138	Dic₁₄₆	—	4	4	no	—
139	Dic₁₄₆	C₂×M₁₆	1	4	yes	—
140	C₈×C₇₃	—	2	1	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	—	4	4	no	—
144	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
145	#10596831	—	2	2	no	—
146	#10596831	—	4	2	no	—
147	#10596831	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
156	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₇₃	#568	1	1	yes	characteristic
158	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
160	Dic₇₃	—	2	4	no	—
161	Dic₇₃	—	4	2	no	—
162	Dic₇₃	—	4	2	no	—
163	Dic₇₃	—	4	2	no	—
164	M₁₆×Q₁₆	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	2	2	no	—
166	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₇₃	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	—	73	2	no	—
170	C₂×C₄×Q₁₆	—	73	1	no	—
171	#1591	—	73	2	no	—
172	#1584	—	73	4	no	—
173	C₈×Q₁₆	—	73	2	no	—
174	#1567	D₇₃	1	1	yes	characteristic
175	#1487	—	73	2	no	—
176	C₈×M₁₆	—	73	1	no	—
177	C₇₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	73	1	no	—
179	C₂×C₄×Q₈	—	73	2	no	—
180	#446	—	73	4	no	—
181	#446	—	146	4	no	—
182	C₈×Q₈	—	73	4	no	—
183	C₄×Q₁₆	—	73	1	no	—
184	C₄×Q₁₆	—	73	2	no	—
185	C₄×Q₁₆	—	73	1	no	—
186	C₄×Q₁₆	—	73	1	no	—
187	C₄×Q₁₆	—	73	2	no	—
188	C₄×Q₁₆	—	73	1	no	—
189	#431	—	73	1	no	—
190	C₂×#107	—	73	1	no	—
191	#423	—	146	2	no	—
192	#423	C₂×D₇₃	1	2	yes	—
193	C₂×#103	—	73	2	no	—
194	C₄×M₁₆	—	73	1	no	—
195	C₄×M₁₆	—	146	2	no	—
196	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
197	#335	C₂×D₇₃	1	2	yes	—
198	#335	C₂×D₇₃	1	2	yes	—
199	Q₈⋊C₈	—	73	4	no	—
200	Q₈⋊C₈	—	73	4	no	—
201	#322	—	73	2	no	—
202	C₈²	—	73	2	no	—
203	C₂²×Q₈	—	73	2	no	—
204	C₂×Q₁₆	—	73	1	no	—
205	C₂×Q₁₆	—	73	2	no	—
206	C₂×Q₁₆	—	73	1	no	—
207	C₂×Q₁₆	—	146	4	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	73	1	no	—
210	C₂×M₁₆	—	73	1	no	—
211	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
213	C₄×Q₈	—	73	2	no	—
214	C₄×Q₈	—	73	2	no	—
215	C₄×Q₈	—	73	2	no	—
216	C₄×Q₈	—	73	2	no	—
217	C₄×Q₈	—	146	2	no	—
218	C₄×Q₈	—	146	2	no	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₄⋊C₄	—	146	2	no	—
221	C₂×C₄²	—	146	2	no	—
222	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
223	#107	—	73	2	no	—
224	#107	—	73	1	no	—
225	#107	—	73	1	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	146	4	no	—
228	C₄⋊C₈	—	146	4	no	—
229	C₄⋊C₈	—	146	1	no	—
230	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
231	#103	—	73	2	no	—
232	#103	—	73	2	no	—
233	#103	—	73	2	no	—
234	#103	—	73	2	no	—
235	C₂²⋊C₈	—	146	1	no	—
236	#97	—	73	2	no	—
237	#97	—	146	4	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	—	146	4	no	—
241	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	C₂×Q₈	—	292	2	no	—
246	C₂×Q₈	—	73	2	no	—
247	C₂×Q₈	—	73	2	no	—
248	C₂×Q₈	—	146	4	no	—
249	C₂×Q₈	—	146	2	no	—
250	C₂²×C₄	—	146	2	no	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
254	C₂²×C₄	#11924029	1	1	yes	characteristic
255	Q₁₆	—	73	4	no	—
256	Q₁₆	—	146	4	no	—
257	Q₁₆	—	146	2	no	—
258	M₁₆	—	4	4	no	—
259	M₁₆	—	73	4	no	—
260	M₁₆	—	146	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	146	2	no	—
267	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	—	73	2	no	—
273	C₄⋊C₄	—	146	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄²	—	146	2	no	—
279	C₄²	—	146	2	no	—
280	C₄²	—	146	2	no	—
281	C₄²	—	146	2	no	—
282	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
283	C₄²	D₄×D₇₃	1	1	yes	characteristic
284	C₄²	#11924029	1	1	yes	characteristic
285	C₄²	C₂³×D₇₃	1	1	yes	characteristic
286	C₂³	#14556553	1	1	yes	characteristic
287	Q₈	—	292	4	no	—
288	Q₈	—	73	4	no	—
289	Q₈	—	146	2	no	—
290	Q₈	—	146	2	no	—
291	Q₈	—	146	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	292	4	no	—
296	C₂×C₄	—	584	2	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	#14556553	1	1	yes	characteristic
302	C₂×C₄	#14556553	1	1	yes	characteristic
303	C₂×C₄	#14556553	1	1	yes	characteristic
304	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
305	C₂×C₄	#14556610	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	292	2	no	—
314	C₈	—	73	4	no	—
315	C₈	—	146	2	no	—
316	C₈	M₁₆×D₇₃	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	#17988681	1	2	yes	—
320	C₂²	C₂×#14556553	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	292	2	no	—
325	C₄	—	292	2	no	—
326	C₄	—	292	2	no	—
327	C₄	—	146	4	no	—
328	C₄	#17988681	1	2	yes	—
329	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
330	C₄	#17989353	1	1	yes	characteristic
331	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
332	C₄	C₂×#14556610	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	#22618666	1	1	yes	characteristic
335	C₂	C₂×#17988681	1	1	yes	characteristic
336	C₂	#22618847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1567	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1567	D₇₃
hypocenter	C₇₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22608827	C₂	semidirect
4. #22608780	C₂	semidirect
5. #22608737	C₂	semidirect
85. Dic₂₉₂	M₁₆	semidirect
87. #11924012	Q₁₆	semidirect
147. #10596831	C₂×Q₁₆	semidirect
177. C₇₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22618882	2
C₂×C₄×Dic₂₉₂	2
#22608827	2
#22608780	2
#22608737	2
#22608359	2
C₈×#11924012	2
C₇₃×#1567	2
M₁₆×Q₁₆	73