Subgroups — #34029818

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	2	yes	maximal
2	#22626154	C₂	1	2	yes	maximal, complemented
3	#22626153	C₂	1	4	yes	maximal, complemented
4	#22626152	C₂	1	2	yes	maximal, complemented
5	#22626148	C₂	1	4	yes	maximal, complemented
6	#405×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556666	C₄	1	4	yes	complemented
8	#17989643	C₄	1	4	yes	complemented
9	#17989643	C₄	1	4	yes	complemented
10	#17989643	C₂²	1	4	yes	—
11	#17989643	C₂²	1	4	yes	—
12	C₂×#14556665	C₂²	1	4	yes	—
13	C₂×#14556664	C₄	1	4	yes	complemented
14	#17989480	C₂²	1	8	yes	complemented
15	#17989479	C₂²	1	8	yes	complemented
16	C₂×M₁₆×D₇₃	C₄	1	1	yes	characteristic
17	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	#117×D₇₃	C₄	1	1	yes	characteristic
19	#17988545	C₂²	1	1	yes	characteristic
20	#17988544	C₄	1	2	yes	—
21	#17988543	C₂²	1	1	yes	characteristic
22	#17988542	C₄	1	1	yes	characteristic
23	#17988541	C₂²	1	2	yes	—
24	#97×D₇₃	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₄	1	2	yes	—
26	#405×C₇₃	C₄	1	1	yes	characteristic
27	#14556666	C₂×C₄	1	8	yes	—
28	#14556666	C₂×C₄	1	8	yes	complemented
29	#14556665	C₂×C₄	1	8	yes	complemented
30	#14556665	C₂³	1	8	yes	—
31	#14556664	C₂×C₄	1	8	yes	—
32	#14556664	C₂×C₄	1	8	yes	complemented
33	C₂×#11924031	C₂×C₄	1	8	yes	complemented
34	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924011	C₂×C₄	1	2	yes	—
37	M₁₆×D₇₃	—	2	2	no	—
38	M₁₆×D₇₃	—	2	2	no	—
39	C₂²×#10596831	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
49	#14556550	C₂×C₄	1	1	yes	characteristic
50	#14556549	—	2	1	no	—
51	#14556548	C₂×C₄	1	2	yes	—
52	#14556547	—	2	1	no	—
53	C₄⋊C₄×D₇₃	—	2	1	no	—
54	C₂²⋊C₄×D₇₃	—	2	1	no	—
55	C₄²×D₇₃	C₂×C₄	1	2	yes	—
56	#14556502	C₂³	1	2	yes	—
57	#14556501	C₂×C₄	1	2	yes	—
58	#14556501	C₂³	1	2	yes	—
59	C₄×#10596831	C₂×C₄	1	2	yes	—
60	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
62	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
65	#97×C₇₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
67	#11924031	C₂²×C₄	1	16	yes	—
68	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
69	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
70	C₂×C₄×D₇₃	—	2	2	no	—
71	C₂×C₄×D₇₃	C₄²	1	2	yes	—
72	C₂×C₄×D₇₃	C₄²	1	2	yes	—
73	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
76	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
77	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
79	#11924012	—	2	2	no	—
80	#11924011	—	2	4	no	—
81	C₂×#10596831	C₂²×C₄	1	2	yes	—
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₂×#10596831	C₂⁴	1	1	yes	characteristic
86	C₈×D₇₃	—	2	2	no	—
87	C₈×D₇₃	—	2	2	no	—
88	C₈×D₇₃	C₄²	1	2	yes	—
89	C₈×D₇₃	C₄²	1	1	yes	characteristic
90	C₈×D₇₃	C₄²	1	2	yes	—
91	C₈×D₇₃	C₄²	1	1	yes	characteristic
92	C₈×D₇₃	C₄²	1	1	yes	characteristic
93	C₈×D₇₃	C₄²	1	1	yes	characteristic
94	C₈×D₇₃	C₂²×C₄	1	2	yes	—
95	C₈×D₇₃	C₂²×C₄	1	2	yes	—
96	C₈×D₇₃	C₂²×C₄	1	2	yes	—
97	C₈×D₇₃	C₂²×C₄	1	2	yes	—
98	C₄⋊Dic₇₃	—	2	1	no	—
99	#11924007	—	2	2	no	—
100	C₂²⋊Dic₇₃	—	2	1	no	—
101	#11924005	—	2	2	no	—
102	C₄×Dic₇₃	—	2	2	no	—
103	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	M₁₆×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	C₄²	1	2	yes	—
107	C₂×C₈×C₇₃	C₄²	1	2	yes	—
108	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₇₃	—	2	1	no	—
112	C₂²⋊C₄×C₇₃	—	2	1	no	—
113	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂²×D₇₃	—	4	2	no	—
116	C₂²×D₇₃	#97	1	2	yes	—
117	C₂²×D₇₃	#97	1	2	yes	—
118	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	#97	1	2	yes	—
127	C₂×Dic₇₃	#97	1	2	yes	—
128	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
130	C₄×D₇₃	—	4	4	no	—
131	C₄×D₇₃	#97	1	2	yes	—
132	C₄×D₇₃	#97	1	2	yes	—
133	C₄×D₇₃	#97	1	2	yes	—
134	C₄×D₇₃	#97	1	2	yes	—
135	C₄×D₇₃	C₂×C₄²	1	2	yes	—
136	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
137	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
138	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
139	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
140	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
141	C₈×C₇₃	—	2	2	no	—
142	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
143	C₈×C₇₃	C₂×C₄²	1	2	yes	—
144	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
145	#10596831	—	2	2	no	—
146	#10596831	C₂×C₄²	1	1	yes	characteristic
147	#10596831	C₂×C₄²	1	1	yes	characteristic
148	#10596831	C₂³×C₄	1	2	yes	—
149	C₂²×C₇₃	—	4	1	no	—
150	C₂²×C₇₃	C₂×#97	1	2	yes	—
151	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	4	no	—
154	C₂×D₇₃	—	4	2	no	—
155	C₂×D₇₃	—	4	1	no	—
156	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
157	C₂×D₇₃	C₂×#97	1	2	yes	—
158	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
159	C₄×C₇₃	—	4	2	no	—
160	C₄×C₇₃	C₂×#97	1	2	yes	—
161	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
162	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
163	Dic₇₃	—	4	2	no	—
164	Dic₇₃	C₂×#97	1	1	yes	characteristic
165	Dic₇₃	C₂×#97	1	1	yes	characteristic
166	Dic₇₃	C₂×#97	1	2	yes	—
167	C₂×#2110	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	1	no	—
169	C₂×C₇₃	—	2	2	no	—
170	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
171	D₇₃	—	2	1	no	—
172	D₇₃	—	2	2	no	—
173	D₇₃	#2110	1	2	yes	complemented
174	C₂²×M₃₂	—	73	2	no	—
175	C₂×#405	—	73	1	no	—
176	#2110	—	73	2	no	—
177	#2110	—	73	4	no	—
178	#2110	—	73	2	no	—
179	C₂×#347	—	73	4	no	—
180	C₇₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
181	C₂²×M₁₆	—	73	1	no	—
182	C₂³×C₈	—	73	1	no	—
183	C₂×#117	—	73	1	no	—
184	C₂×M₃₂	—	73	4	no	—
185	C₂×M₃₂	—	73	4	no	—
186	C₂×M₃₂	—	73	4	no	—
187	C₂×M₃₂	—	73	4	no	—
188	C₂×M₃₂	—	73	4	no	—
189	C₂×M₃₂	—	73	4	no	—
190	C₂²×C₁₆	—	73	4	no	—
191	#405	—	73	1	no	—
192	#405	—	73	1	no	—
193	#405	—	73	1	no	—
194	#405	—	73	2	no	—
195	#405	—	73	2	no	—
196	#405	#64408	1	1	yes	characteristic
197	C₂×#97	—	73	2	no	—
198	C₂×C₄×C₈	—	73	2	no	—
199	#347	—	73	8	no	—
200	#347	—	73	8	no	—
201	C₂³×C₄	—	73	1	no	—
202	C₂×M₁₆	—	73	1	no	—
203	C₂×M₁₆	—	73	2	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂×M₁₆	—	146	2	no	—
206	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	—	73	1	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₂²×C₈	—	73	1	no	—
216	C₂²×C₈	—	73	2	no	—
217	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
218	#117	—	73	1	no	—
219	#117	—	73	2	no	—
220	#117	—	146	1	no	—
221	#117	—	146	1	no	—
222	#117	C₂×#64408	1	1	yes	characteristic
223	C₂×C₄⋊C₄	—	146	1	no	—
224	C₂×C₂²⋊C₄	—	146	1	no	—
225	C₂×C₄²	—	73	2	no	—
226	M₃₂	—	73	8	no	—
227	M₃₂	—	73	8	no	—
228	M₃₂	—	73	8	no	—
229	M₃₂	—	73	8	no	—
230	C₂×C₁₆	—	73	8	no	—
231	C₂×C₁₆	—	73	8	no	—
232	C₂×C₁₆	—	73	8	no	—
233	#97	—	73	2	no	—
234	#97	—	73	2	no	—
235	#97	—	73	2	no	—
236	#97	C₂×#64408	1	2	yes	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	C₂×#64408	1	2	yes	—
241	C₂⁴	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	2	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	2	no	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
253	M₁₆	—	2	2	no	—
254	M₁₆	—	146	4	no	—
255	M₁₆	—	146	2	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	73	1	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	73	2	no	—
271	C₂×C₈	—	146	2	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	C₄×#64408	1	2	yes	—
274	C₂×C₈	C₄×#64408	1	2	yes	—
275	C₂×C₈	C₂²×#64408	1	2	yes	—
276	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
277	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	146	2	no	—
280	C₄⋊C₄	—	146	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	146	2	no	—
283	C₂²⋊C₄	—	146	1	no	—
284	C₄²	—	73	2	no	—
285	C₄²	—	146	2	no	—
286	C₄²	C₂²×#64408	1	2	yes	—
287	C₁₆	—	73	16	no	—
288	C₂³	—	292	2	no	—
289	C₂³	—	73	1	no	—
290	C₂³	—	73	2	no	—
291	C₂³	—	73	1	no	—
292	C₂³	—	73	2	no	—
293	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	292	4	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	2	no	—
299	C₂×C₄	—	73	2	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	73	2	no	—
307	C₂×C₄	—	73	2	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	2	no	—
310	C₂×C₄	—	146	2	no	—
311	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
312	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
313	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
314	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	73	2	no	—
317	C₈	—	73	1	no	—
318	C₈	—	73	1	no	—
319	C₈	—	146	2	no	—
320	C₈	C₂×C₄×#64408	1	2	yes	—
321	C₈	C₂×C₄×#64408	1	1	yes	characteristic
322	C₈	C₂×C₄×#64408	1	1	yes	characteristic
323	C₂²	—	4	1	no	—
324	C₂²	—	292	1	no	—
325	C₂²	—	292	2	no	—
326	C₂²	—	73	1	no	—
327	C₂²	—	73	1	no	—
328	C₂²	—	73	2	no	—
329	C₂²	—	146	2	no	—
330	C₂²	—	146	4	no	—
331	C₂²	#17989546	1	2	yes	—
332	C₂²	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
333	C₄	—	4	2	no	—
334	C₄	—	292	2	no	—
335	C₄	—	73	1	no	—
336	C₄	—	73	1	no	—
337	C₄	—	73	2	no	—
338	C₄	#17989546	1	2	yes	—
339	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
340	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	2	no	—
343	C₂	—	73	2	no	—
344	C₂	—	146	1	no	—
345	C₂	—	146	2	no	—
346	C₂	C₂×#17989546	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#64408
Fitting	#405×C₇₃	C₄
socle	C₂×C₇₃	C₂²×#97
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×#10596831	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	C₂×#2110
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22626154	C₂	semidirect
3. #22626153	C₂	semidirect
4. #22626152	C₂	semidirect
5. #22626148	C₂	semidirect
7. C₂×#14556666	C₄	semidirect
8. #17989643	C₄	semidirect
9. #17989643	C₄	semidirect
13. C₂×#14556664	C₄	semidirect
14. #17989480	C₂²	semidirect
15. #17989479	C₂²	semidirect
28. #14556666	C₂×C₄	semidirect
29. #14556665	C₂×C₄	semidirect
32. #14556664	C₂×C₄	semidirect
33. C₂×#11924031	C₂×C₄	semidirect
173. D₇₃	#2110	semidirect
180. C₇₃	C₂×#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989643	2
#22626154	2
#22626153	2
#22626152	2
#22626148	2
#405×D₇₃	2
C₂×#2110	73