Subgroups — #33068494

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619927	C₂	1	2	yes	maximal, complemented
2	C₂×#17988666	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609417	C₂	1	2	yes	maximal, complemented
4	#22609407	C₂	1	2	yes	maximal, complemented
5	#22609393	C₂	1	4	yes	maximal, complemented
6	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608580	C₂	1	2	yes	maximal, complemented
8	C₇₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924011	—	2	1	no	—
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#17988898	—	2	2	no	—
12	#17988864	C₂²	1	4	yes	—
13	#17988779	C₂²	1	4	yes	—
14	#17988720	—	2	2	no	—
15	C₄×#11924013	—	2	4	no	—
16	#17988666	—	2	1	no	—
17	#17988666	—	2	1	no	—
18	#17988666	C₂²	1	2	yes	—
19	#17988666	C₂²	1	2	yes	—
20	#17988640	C₂²	1	2	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
22	C₂×#14556503	—	2	1	no	—
23	#17988340	C₂²	1	2	yes	complemented
24	#17988338	C₂²	1	2	yes	complemented
25	#17988333	C₂²	1	4	yes	complemented
26	#17988331	C₂²	1	4	yes	complemented
27	C₂×#135×C₇₃	—	2	1	no	—
28	C₂×#135×C₇₃	—	2	1	no	—
29	#532×C₇₃	C₂²	1	2	yes	—
30	#495×C₇₃	—	2	1	no	—
31	#495×C₇₃	—	2	1	no	—
32	#494×C₇₃	C₂²	1	1	yes	characteristic, complemented
33	#493×C₇₃	C₂²	1	1	yes	characteristic, complemented
34	#492×C₇₃	C₂²	1	2	yes	complemented
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₇₃	—	2	1	no	—
38	C₂×#11924013	—	2	4	no	—
39	C₂×#11924011	—	2	2	no	—
40	C₂×#11924011	—	2	2	no	—
41	C₂×#11924011	—	4	4	no	—
42	C₂²×#10596831	—	2	1	no	—
43	C₄×D₂₉₂	—	2	1	no	—
44	C₄×D₂₉₂	—	2	1	no	—
45	C₄×D₂₉₂	D₄	1	2	yes	—
46	C₄×D₂₉₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924005	—	2	1	no	—
49	#14556524	—	2	4	no	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	C₂³	1	2	yes	—
52	C₄⋊#10596831	C₂³	1	4	yes	—
53	#14556512	—	2	4	no	—
54	#14556511	—	2	4	no	—
55	C₂²⋊#10596831	—	2	2	no	—
56	#14556503	—	2	2	no	—
57	#14556503	—	2	2	no	—
58	C₄×#10596831	—	2	2	no	—
59	C₂×#41×C₇₃	—	2	2	no	—
60	#135×C₇₃	—	4	4	no	—
61	#135×C₇₃	—	4	4	no	—
62	C₂×Q₁₆×C₇₃	—	2	1	no	—
63	C₂×Q₁₆×C₇₃	—	2	1	no	—
64	C₂×SD₁₆×C₇₃	—	2	2	no	—
65	C₂×SD₁₆×C₇₃	—	2	2	no	—
66	C₂×D₈×C₇₃	—	2	1	no	—
67	C₂×D₈×C₇₃	—	2	1	no	—
68	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
71	#127×C₇₃	C₂³	1	2	yes	—
72	#124×C₇₃	—	2	2	no	—
73	#121×C₇₃	—	2	2	no	—
74	#121×C₇₃	—	2	2	no	—
75	C₄×D₄×C₇₃	—	2	2	no	—
76	C₄×D₄×C₇₃	—	2	2	no	—
77	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
79	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
80	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₂³×D₇₃	—	2	1	no	—
83	C₂×D₂₉₂	—	2	1	no	—
84	C₂×D₂₉₂	—	4	2	no	—
85	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
86	C₂²×Dic₇₃	—	2	1	no	—
87	C₂×C₄×D₇₃	—	2	2	no	—
88	C₂×C₄×D₇₃	—	2	2	no	—
89	C₂×C₄×D₇₃	—	4	2	no	—
90	C₂×C₄×D₇₃	—	4	2	no	—
91	#11924013	—	4	8	no	—
92	#11924011	—	4	4	no	—
93	#11924011	—	4	4	no	—
94	C₂×#10596831	—	2	4	no	—
95	C₂×#10596831	—	2	2	no	—
96	C₂×#10596831	—	4	2	no	—
97	C₄⋊Dic₇₃	—	2	1	no	—
98	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
99	#11924005	—	2	2	no	—
100	#11924005	—	2	2	no	—
101	#41×C₇₃	—	4	4	no	—
102	#41×C₇₃	—	4	4	no	—
103	C₂×Q₈×C₇₃	—	2	2	no	—
104	C₂×D₄×C₇₃	—	2	2	no	—
105	C₂×D₄×C₇₃	—	2	2	no	—
106	C₂×D₄×C₇₃	—	2	2	no	—
107	C₂×D₄×C₇₃	—	4	2	no	—
108	C₂²×C₄×C₇₃	—	4	2	no	—
109	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
112	Q₁₆×C₇₃	—	4	2	no	—
113	Q₁₆×C₇₃	—	4	2	no	—
114	SD₁₆×C₇₃	—	4	4	no	—
115	SD₁₆×C₇₃	—	4	4	no	—
116	D₈×C₇₃	—	4	2	no	—
117	D₈×C₇₃	—	4	2	no	—
118	C₂×C₈×C₇₃	—	2	2	no	—
119	C₂×C₈×C₇₃	D₈	1	4	yes	complemented
120	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	—	2	2	no	—
126	C₂²⋊C₄×C₇₃	—	4	2	no	—
127	C₂²⋊C₄×C₇₃	—	4	2	no	—
128	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₇₃	#41	1	1	yes	characteristic
131	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
132	C₂³×C₇₃	—	4	2	no	—
133	C₂³×C₇₃	#121	1	1	yes	characteristic
134	C₂²×D₇₃	—	2	2	no	—
135	C₂²×D₇₃	—	4	2	no	—
136	C₂²×D₇₃	—	4	2	no	—
137	C₂²×D₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	4	2	no	—
139	Q₈×C₇₃	—	4	2	no	—
140	D₄×C₇₃	—	4	2	no	—
141	D₄×C₇₃	—	4	2	no	—
142	D₄×C₇₃	—	4	4	no	—
143	D₄×C₇₃	—	8	2	no	—
144	D₄×C₇₃	—	8	4	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	—	4	1	no	—
147	C₂×C₄×C₇₃	—	4	2	no	—
148	C₂×C₄×C₇₃	—	8	4	no	—
149	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₇₃	—	2	2	no	—
158	C₂×Dic₇₃	—	4	2	no	—
159	D₂₉₂	—	2	4	no	—
160	D₂₉₂	—	4	2	no	—
161	C₄×D₇₃	—	4	2	no	—
162	C₄×D₇₃	—	4	2	no	—
163	C₄×D₇₃	—	4	4	no	—
164	C₈×C₇₃	—	2	1	no	—
165	C₈×C₇₃	—	2	1	no	—
166	C₈×C₇₃	C₂×D₈	1	2	yes	—
167	C₈×C₇₃	C₂×D₈	1	2	yes	—
168	#10596831	—	4	2	no	—
169	#10596831	—	4	4	no	—
170	C₂²×C₇₃	—	2	1	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	8	2	no	—
173	C₂²×C₇₃	—	8	2	no	—
174	C₂²×C₇₃	—	8	2	no	—
175	C₂²×C₇₃	#459	1	2	yes	—
176	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	4	2	no	—
180	C₂×D₇₃	—	8	4	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	2	1	no	—
184	C₄×C₇₃	—	8	2	no	—
185	C₄×C₇₃	#459	1	2	yes	—
186	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₇₃	#575	1	2	yes	—
189	Dic₇₃	—	4	2	no	—
190	#19133	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	—	2	2	no	—
192	C₂×C₇₃	—	16	2	no	—
193	C₂×C₇₃	C₂×#459	1	1	yes	characteristic
194	C₂×C₇₃	#3054	1	1	yes	characteristic
195	C₂×C₇₃	#3067	1	1	yes	characteristic
196	D₇₃	—	8	2	no	—
197	#3147	D₇₃	1	1	yes	characteristic, complemented
198	#3100	—	73	2	no	—
199	C₂×#436	—	73	1	no	—
200	#1689	—	73	2	no	—
201	#1687	—	73	2	no	—
202	#1685	—	73	4	no	—
203	#1564	—	73	1	no	—
204	#1540	—	73	2	no	—
205	C₇₃	#19133	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	—	2	1	no	—
207	C₂×#135	—	2	1	no	—
208	C₂²×M₁₆	—	146	1	no	—
209	#532	C₂×D₇₃	1	2	yes	—
210	C₂×C₄×D₄	—	73	1	no	—
211	#495	—	2	1	no	—
212	#495	—	2	1	no	—
213	#494	C₂×D₇₃	1	1	yes	characteristic, complemented
214	#493	C₂×D₇₃	1	1	yes	characteristic, complemented
215	#492	C₂×D₇₃	1	2	yes	complemented
216	#480	—	146	2	no	—
217	#474	—	73	4	no	—
218	#459	—	73	4	no	—
219	#447	—	146	2	no	—
220	C₄×D₈	—	146	4	no	—
221	#436	—	73	2	no	—
222	#436	—	73	2	no	—
223	#436	—	146	1	no	—
224	#436	—	146	1	no	—
225	#431	—	73	2	no	—
226	C₂×C₄⋊C₈	—	73	1	no	—
227	C₂×C₂²⋊C₈	—	146	1	no	—
228	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
229	#335	—	73	2	no	—
230	#334	—	73	2	no	—
231	#332	—	73	4	no	—
232	#331	—	73	4	no	—
233	C₂×#41	—	2	2	no	—
234	C₂²×D₄	—	73	1	no	—
235	C₂³×C₄	—	146	1	no	—
236	#135	—	4	4	no	—
237	#135	—	4	4	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	2	2	no	—
242	C₂×D₈	—	2	1	no	—
243	C₂×D₈	—	2	1	no	—
244	C₂×D₈	—	146	4	no	—
245	C₂×M₁₆	—	292	4	no	—
246	C₂×M₁₆	—	146	2	no	—
247	C₂×M₁₆	—	146	2	no	—
248	C₂²×C₈	—	146	1	no	—
249	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
251	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
252	#127	C₂²×D₇₃	1	2	yes	—
253	#124	—	2	2	no	—
254	#121	—	2	2	no	—
255	#121	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	—	73	2	no	—
259	C₄×D₄	—	73	2	no	—
260	C₄×D₄	—	146	1	no	—
261	C₄×D₄	—	146	1	no	—
262	C₂×C₄⋊C₄	—	73	1	no	—
263	C₂×C₂²⋊C₄	—	146	1	no	—
264	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
265	#107	—	146	4	no	—
266	C₄⋊C₈	—	73	2	no	—
267	C₄⋊C₈	—	73	4	no	—
268	C₄⋊C₈	—	146	1	no	—
269	#102	—	146	4	no	—
270	#102	—	146	4	no	—
271	C₂²⋊C₈	—	146	2	no	—
272	C₂²⋊C₈	—	146	2	no	—
273	C₂²⋊C₈	—	146	2	no	—
274	C₄×C₈	—	146	2	no	—
275	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
276	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
279	C₂⁴	—	146	1	no	—
280	#41	—	4	4	no	—
281	#41	—	4	4	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	292	2	no	—
288	C₂×D₄	—	73	2	no	—
289	C₂×D₄	—	146	1	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	292	2	no	—
292	C₂²×C₄	—	292	2	no	—
293	C₂²×C₄	—	146	1	no	—
294	C₂²×C₄	—	146	2	no	—
295	C₂²×C₄	—	146	2	no	—
296	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
298	C₂²×C₄	#11924027	1	1	yes	characteristic
299	Q₁₆	—	4	2	no	—
300	Q₁₆	—	4	2	no	—
301	SD₁₆	—	4	4	no	—
302	SD₁₆	—	4	4	no	—
303	D₈	—	4	2	no	—
304	D₈	—	4	2	no	—
305	D₈	—	292	8	no	—
306	M₁₆	—	292	4	no	—
307	M₁₆	—	292	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	292	2	no	—
310	C₂×C₈	—	146	4	no	—
311	C₂×C₈	—	146	2	no	—
312	C₂×C₈	#11924013	1	4	yes	complemented
313	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	73	2	no	—
320	C₄⋊C₄	—	146	1	no	—
321	C₂²⋊C₄	—	4	2	no	—
322	C₂²⋊C₄	—	4	2	no	—
323	C₂²⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	146	2	no	—
325	C₄²	D₄×D₇₃	1	1	yes	characteristic
326	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
327	C₄²	#11924027	1	1	yes	characteristic
328	C₄²	C₂³×D₇₃	1	1	yes	characteristic
329	C₂³	—	4	2	no	—
330	C₂³	—	292	2	no	—
331	C₂³	—	292	2	no	—
332	C₂³	—	292	2	no	—
333	C₂³	—	146	2	no	—
334	C₂³	#14556562	1	1	yes	characteristic
335	Q₈	—	4	2	no	—
336	Q₈	—	4	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	8	2	no	—
341	D₄	—	8	4	no	—
342	D₄	—	292	2	no	—
343	D₄	—	146	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	4	no	—
348	C₂×C₄	—	292	2	no	—
349	C₂×C₄	—	292	4	no	—
350	C₂×C₄	—	292	2	no	—
351	C₂×C₄	—	292	2	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	#14556562	1	1	yes	characteristic
354	C₂×C₄	#14556562	1	1	yes	characteristic
355	C₂×C₄	#14556562	1	1	yes	characteristic
356	C₂×C₄	C₂×#11924013	1	2	yes	—
357	C₂×C₄	#14556617	1	2	yes	—
358	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
359	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
360	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	292	4	no	—
364	C₈	—	292	2	no	—
365	C₈	C₂×#11924013	1	2	yes	—
366	C₈	C₂×#11924013	1	2	yes	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	8	2	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	292	2	no	—
373	C₂²	—	292	2	no	—
374	C₂²	—	292	2	no	—
375	C₂²	—	584	4	no	—
376	C₂²	#17988779	1	2	yes	—
377	C₂²	C₂×#14556562	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	8	2	no	—
382	C₄	—	292	2	no	—
383	C₄	#17988779	1	2	yes	—
384	C₄	C₂²×#11924013	1	1	yes	characteristic
385	C₄	C₂×#14556617	1	1	yes	characteristic
386	C₄	#17989384	1	2	yes	—
387	C₂	—	2	2	no	—
388	C₂	—	16	2	no	—
389	C₂	—	584	2	no	—
390	C₂	C₂×#17988779	1	1	yes	characteristic
391	C₂	#22619536	1	1	yes	characteristic
392	C₂	#22619647	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556562
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3147	C₂
socle	C₂²×C₇₃	C₂×#121
hypercenter	#3147	D₇₃
hypocenter	C₇₃	#19133
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22619927	C₂	semidirect
2. C₂×#17988666	C₂	semidirect
3. #22609417	C₂	semidirect
4. #22609407	C₂	semidirect
5. #22609393	C₂	semidirect
6. #22608679	C₂	semidirect
7. #22608580	C₂	semidirect
8. C₇₃×#3147	C₂	semidirect
23. #17988340	C₂²	semidirect
24. #17988338	C₂²	semidirect
25. #17988333	C₂²	semidirect
26. #17988331	C₂²	semidirect
32. #494×C₇₃	C₂²	semidirect
33. #493×C₇₃	C₂²	semidirect
34. #492×C₇₃	C₂²	semidirect
119. C₂×C₈×C₇₃	D₈	semidirect
197. #3147	D₇₃	semidirect
205. C₇₃	#19133	semidirect
213. #494	C₂×D₇₃	semidirect
214. #493	C₂×D₇₃	semidirect
215. #492	C₂×D₇₃	semidirect
312. C₂×C₈	#11924013	semidirect

Maximal subgroups

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

subgroup	index
#22619927	2
C₂×#17988666	2
#22609417	2
#22609407	2
#22609393	2
#22608679	2
#22608580	2
C₇₃×#3147	2
#19133	73