Subgroups — #33084059

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988629	C₂	1	2	yes	maximal
2	#22612287	C₂	1	2	yes	maximal
3	#22609828	C₂	1	4	yes	maximal, complemented
4	C₄×#14556525	C₂	1	2	yes	maximal, complemented
5	C₄×#14556523	C₂	1	2	yes	maximal, complemented
6	#22609648	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988350	C₂	1	1	yes	maximal, characteristic
8	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556550	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#17988629	C₄	1	16	yes	complemented
12	C₂×#14556525	C₂²	1	4	yes	—
13	C₂×#14556523	C₂²	1	4	yes	—
14	C₄⋊C₄×Dic₇₃	—	2	2	no	—
15	C₄⋊C₄×Dic₇₃	—	2	2	no	—
16	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
18	C₂²⋊C₄×Dic₇₃	—	2	4	no	—
19	#17988350	C₂²	1	1	yes	characteristic
20	#17988350	C₂²	1	2	yes	—
21	#17988350	C₂²	1	1	yes	characteristic
22	#17988350	C₂²	1	2	yes	—
23	#17988350	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
27	#405×C₇₃	C₄	1	8	yes	complemented
28	C₂×#97×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
30	C₂³×Dic₇₃	—	2	2	no	—
31	#14556550	—	2	4	no	—
32	#14556550	C₂×C₄	1	8	yes	—
33	C₂×C₄⋊Dic₇₃	—	2	2	no	—
34	C₂×C₄⋊Dic₇₃	—	2	4	no	—
35	C₂×C₄⋊Dic₇₃	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
40	C₂×C₄×Dic₇₃	—	2	2	no	—
41	C₂×C₄×Dic₇₃	—	2	4	no	—
42	C₂×C₄×Dic₇₃	—	2	4	no	—
43	#14556525	C₂×C₄	1	16	yes	complemented
44	#14556523	C₂×C₄	1	16	yes	complemented
45	#14556498	—	2	4	no	—
46	#14556498	—	2	4	no	—
47	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₇₃	—	2	4	no	—
51	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
55	#117×C₇₃	C₂×C₄	1	4	yes	—
56	#117×C₇₃	Q₈	1	4	yes	—
57	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
61	#97×C₇₃	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
63	C₂²×Dic₇₃	—	2	2	no	—
64	C₂²×Dic₇₃	—	2	2	no	—
65	C₂²×Dic₇₃	—	2	4	no	—
66	C₂²×Dic₇₃	—	2	8	no	—
67	C₂²×Dic₇₃	—	8	2	no	—
68	C₂²×Dic₇₃	—	8	2	no	—
69	C₄⋊Dic₇₃	—	2	8	no	—
70	C₄⋊Dic₇₃	—	2	4	no	—
71	C₄⋊Dic₇₃	—	2	2	no	—
72	C₄⋊Dic₇₃	—	2	2	no	—
73	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
74	C₄⋊Dic₇₃	C₂²×C₄	1	8	yes	—
75	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
76	C₂²⋊Dic₇₃	—	2	8	no	—
77	C₄×Dic₇₃	—	2	8	no	—
78	C₄×Dic₇₃	—	8	4	no	—
79	C₄×Dic₇₃	—	8	4	no	—
80	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₇₃	#41	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₇₃	C₄²	1	16	yes	complemented
90	C₂×C₈×C₇₃	—	2	8	no	—
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	C₂²×C₄	1	8	yes	—
94	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
97	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
98	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
99	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
100	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
102	C₂³×C₇₃	—	2	1	no	—
103	C₂³×C₇₃	—	2	2	no	—
104	C₂³×C₇₃	—	2	1	no	—
105	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₇₃	C₄×D₄	1	4	yes	—
107	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	4	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇₃	#117	1	4	yes	—
116	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₇₃	C₄×Q₈	1	8	yes	—
118	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
122	C₂×Dic₇₃	—	2	4	no	—
123	C₂×Dic₇₃	—	2	4	no	—
124	C₂×Dic₇₃	—	2	8	no	—
125	C₂×Dic₇₃	—	4	2	no	—
126	C₂×Dic₇₃	—	4	4	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	4	4	no	—
129	C₂×Dic₇₃	—	4	2	no	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	8	8	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	—	2	2	no	—
134	C₈×C₇₃	C₂×C₄²	1	8	yes	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	—	2	4	no	—
138	C₂²×C₇₃	—	2	4	no	—
139	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
140	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×C₄×D₄	1	2	yes	—
142	C₂²×C₇₃	#576	1	1	yes	characteristic
143	C₂²×C₇₃	#577	1	2	yes	—
144	C₂²×C₇₃	#578	1	1	yes	characteristic
145	C₄×C₇₃	—	2	4	no	—
146	C₄×C₇₃	—	2	4	no	—
147	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
148	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
151	Dic₇₃	—	4	4	no	—
152	Dic₇₃	—	4	8	no	—
153	Dic₇₃	—	4	4	no	—
154	#20453	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	4	no	—
156	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₇₃	#2912	1	1	yes	characteristic
158	C₂×C₇₃	#2913	1	1	yes	characteristic
159	C₂×C₇₃	#2946	1	2	yes	—
160	C₂×C₇₃	#2947	1	2	yes	—
161	C₂×#428	—	73	2	no	—
162	C₂×#405	D₇₃	1	1	yes	characteristic
163	#2274	—	73	2	no	—
164	#1776	—	73	4	no	—
165	C₄×#107	—	73	2	no	—
166	C₄×#106	—	73	2	no	—
167	#1735	—	73	1	no	—
168	C₂×#340	—	73	1	no	—
169	C₇₃	#20453	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
172	C₂×#117	—	73	2	no	—
173	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	—	73	2	no	—
175	#428	—	73	16	no	—
176	C₂×#107	—	73	4	no	—
177	C₂×#106	—	73	4	no	—
178	#405	Dic₇₃	1	8	yes	complemented
179	C₂×#97	C₂×D₇₃	1	2	yes	—
180	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
181	C₄×C₄⋊C₄	—	73	4	no	—
182	C₄×C₄⋊C₄	—	73	4	no	—
183	C₄×C₄⋊C₄	—	146	2	no	—
184	C₄×C₄⋊C₄	—	146	2	no	—
185	C₄×C₂²⋊C₄	—	146	4	no	—
186	#340	—	73	2	no	—
187	#340	—	73	2	no	—
188	#340	—	73	1	no	—
189	#340	—	73	1	no	—
190	#340	—	73	2	no	—
191	C₂³×C₄	—	146	2	no	—
192	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
193	C₂×M₁₆	C₄×D₇₃	1	8	yes	—
194	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
197	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
200	#117	—	73	8	no	—
201	#117	—	146	4	no	—
202	#117	Dic₁₄₆	1	4	yes	—
203	#117	C₂×Dic₇₃	1	4	yes	—
204	C₂×C₄⋊C₄	—	73	8	no	—
205	C₂×C₄⋊C₄	—	73	4	no	—
206	C₂×C₄⋊C₄	—	73	2	no	—
207	C₂×C₄⋊C₄	—	73	2	no	—
208	C₂×C₄⋊C₄	—	146	2	no	—
209	C₂×C₄⋊C₄	—	146	4	no	—
210	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	146	2	no	—
213	C₂×C₂²⋊C₄	D₂₉₂	1	2	yes	—
214	C₂×C₄²	—	146	4	no	—
215	C₂×C₄²	—	146	4	no	—
216	C₂×C₄²	—	146	2	no	—
217	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
218	#107	—	73	16	no	—
219	#106	—	73	16	no	—
220	#97	C₂×Dic₇₃	1	8	yes	complemented
221	C₄×C₈	C₂×Dic₇₃	1	8	yes	complemented
222	#95	—	146	4	no	—
223	#95	—	146	4	no	—
224	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
225	C₂²×C₄	—	584	2	no	—
226	C₂²×C₄	—	584	2	no	—
227	C₂²×C₄	—	146	2	no	—
228	C₂²×C₄	—	146	2	no	—
229	C₂²×C₄	—	146	4	no	—
230	C₂²×C₄	—	146	8	no	—
231	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
232	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
233	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
234	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
235	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
236	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
237	C₂²×C₄	#11924028	1	4	yes	—
238	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
239	M₁₆	C₄×Dic₇₃	1	16	yes	complemented
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×C₄×D₇₃	1	8	yes	—
244	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
245	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
246	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
247	C₄⋊C₄	—	73	4	no	—
248	C₄⋊C₄	—	73	8	no	—
249	C₄⋊C₄	—	73	4	no	—
250	C₄⋊C₄	—	146	8	no	—
251	C₄⋊C₄	—	146	4	no	—
252	C₄⋊C₄	—	146	2	no	—
253	C₄⋊C₄	—	146	2	no	—
254	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
255	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
256	C₂²⋊C₄	—	146	8	no	—
257	C₂²⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
258	C₄²	—	584	4	no	—
259	C₄²	—	584	4	no	—
260	C₄²	—	146	8	no	—
261	C₄²	C₂²×Dic₇₃	1	4	yes	—
262	C₄²	C₂×Dic₁₄₆	1	4	yes	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	1	no	—
266	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
267	C₂³	C₄×D₂₉₂	1	4	yes	—
268	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	292	2	no	—
273	C₂×C₄	—	292	4	no	—
274	C₂×C₄	—	292	2	no	—
275	C₂×C₄	—	292	4	no	—
276	C₂×C₄	—	292	2	no	—
277	C₂×C₄	—	292	2	no	—
278	C₂×C₄	—	584	8	no	—
279	C₂×C₄	—	146	4	no	—
280	C₂×C₄	—	146	4	no	—
281	C₂×C₄	—	146	8	no	—
282	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
283	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
286	C₂×C₄	#14556550	1	4	yes	—
287	C₂×C₄	C₄×D₂₉₂	1	4	yes	—
288	C₂×C₄	C₄×Dic₁₄₆	1	8	yes	—
289	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
290	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
292	C₂×C₄	C₂×#11924028	1	2	yes	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	C₄×C₄⋊Dic₇₃	1	4	yes	—
301	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
302	C₂²	C₂×C₄×D₂₉₂	1	2	yes	—
303	C₂²	#17989392	1	1	yes	characteristic
304	C₂²	#17989399	1	2	yes	—
305	C₂²	#17989404	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	4	no	—
308	C₄	—	292	8	no	—
309	C₄	—	292	4	no	—
310	C₄	—	292	4	no	—
311	C₄	C₄×C₄⋊Dic₇₃	1	4	yes	—
312	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
313	C₄	C₂×#14556550	1	1	yes	characteristic
314	C₄	C₂×C₄×Dic₁₄₆	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
317	C₂	#22618526	1	1	yes	characteristic
318	C₂	#22618531	1	1	yes	characteristic
319	C₂	#22618765	1	2	yes	—
320	C₂	#22618776	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#405×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#20453
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #22609828	C₂	semidirect
4. C₄×#14556525	C₂	semidirect
5. C₄×#14556523	C₂	semidirect
11. #17988629	C₄	semidirect
27. #405×C₇₃	C₄	semidirect
43. #14556525	C₂×C₄	semidirect
44. #14556523	C₂×C₄	semidirect
61. #97×C₇₃	C₂×C₄	semidirect
62. C₄×C₈×C₇₃	C₂×C₄	semidirect
89. M₁₆×C₇₃	C₄²	semidirect
169. C₇₃	#20453	semidirect
178. #405	Dic₇₃	semidirect
220. #97	C₂×Dic₇₃	semidirect
221. C₄×C₈	C₂×Dic₇₃	semidirect
239. M₁₆	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988629	2
#22612287	2
#22609828	2
C₄×#14556525	2
C₄×#14556523	2
#22609648	2
C₂×#17988350	2
C₂×#405×C₇₃	2
#20453	73