Subgroups — #34030057

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 328 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	#22626265	C₂	1	2	yes	maximal, complemented
3	#22626264	C₂	1	4	yes	maximal, complemented
4	#22626263	C₂	1	2	yes	maximal, complemented
5	#22626261	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556666	C₂²	1	4	yes	—
8	#17989643	—	2	8	no	—
9	C₂×#14556665	C₂²	1	4	yes	—
10	C₂×#14556664	C₂²	1	4	yes	—
11	#17989508	C₂²	1	8	yes	complemented
12	C₄⋊#11924031	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₃	C₄	1	2	yes	—
20	C₄×C₈×D₇₃	C₄	1	2	yes	—
21	C₄×C₈×D₇₃	C₂²	1	2	yes	—
22	C₄×C₈×D₇₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
24	#14556666	—	2	8	no	—
25	#14556665	—	2	8	no	—
26	#14556664	—	2	8	no	—
27	C₂×#11924031	C₂³	1	8	yes	—
28	C₂²×C₄×D₇₃	C₈	1	2	yes	—
29	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂²×#10596831	C₂³	1	1	yes	characteristic
32	C₂×C₈×D₇₃	—	2	4	no	—
33	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₇₃	D₄	1	4	yes	—
38	C₂×C₈×D₇₃	Q₈	1	4	yes	—
39	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
42	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₇₃	C₈	1	1	yes	characteristic
44	C₄²×D₇₃	C₈	1	1	yes	characteristic
45	C₄²×D₇₃	C₈	1	1	yes	characteristic
46	C₄²×D₇₃	C₈	1	1	yes	characteristic
47	C₄²×D₇₃	C₂×C₄	1	2	yes	—
48	C₄²×D₇₃	C₂×C₄	1	2	yes	—
49	C₄×#10596831	C₂×C₄	1	2	yes	—
50	C₄×#10596831	C₂³	1	2	yes	—
51	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₇₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
60	#11924031	—	2	8	no	—
61	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
63	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₇₃	—	2	8	no	—
65	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₇₃	M₁₆	1	4	yes	—
73	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	C₂²×C₄	1	2	yes	—
77	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
78	C₂×#10596831	C₂×D₄	1	2	yes	—
79	C₂×#10596831	C₂×Q₈	1	2	yes	—
80	C₂×#10596831	C₂⁴	1	1	yes	characteristic
81	C₈×D₇₃	—	2	4	no	—
82	C₈×D₇₃	—	2	4	no	—
83	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
84	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
85	C₈×D₇₃	C₂×D₄	1	2	yes	—
86	C₈×D₇₃	C₂×D₄	1	2	yes	—
87	C₈×D₇₃	C₂×Q₈	1	2	yes	—
88	C₈×D₇₃	C₂×Q₈	1	2	yes	—
89	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
90	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
91	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
92	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
93	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
97	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
104	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
105	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
106	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
107	C₂²×D₇₃	—	2	4	no	—
108	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
109	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
110	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	—	2	4	no	—
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₇₃	—	2	4	no	—
119	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
120	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
121	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
124	C₄×D₇₃	—	2	4	no	—
125	C₄×D₇₃	—	2	4	no	—
126	C₄×D₇₃	—	2	4	no	—
127	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
128	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
129	C₄×D₇₃	M₃₂	1	8	yes	complemented
130	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
133	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
134	C₈×C₇₃	—	2	2	no	—
135	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
137	#10596831	—	2	2	no	—
138	#10596831	C₂×C₄⋊C₄	1	2	yes	—
139	#10596831	C₂²×D₄	1	1	yes	characteristic
140	#10596831	C₂²×Q₈	1	1	yes	characteristic
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
143	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₇₃	—	2	2	no	—
145	C₂×D₇₃	—	2	8	no	—
146	C₂×D₇₃	C₂×C₄⋊C₈	1	2	yes	—
147	C₂×D₇₃	C₂×C₄⋊C₈	1	2	yes	—
148	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
151	C₄×C₇₃	C₂×M₃₂	1	4	yes	complemented
152	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
154	Dic₇₃	—	2	2	no	—
155	Dic₇₃	C₂×C₄⋊C₈	1	2	yes	—
156	Dic₇₃	C₂×C₄⋊C₈	1	2	yes	—
157	Dic₇₃	C₂×M₃₂	1	4	yes	complemented
158	C₂×#2150	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₇₃	C₂²×M₃₂	1	2	yes	—
162	D₇₃	—	2	2	no	—
163	D₇₃	#2150	1	4	yes	complemented
164	C₂²×M₃₂	—	73	2	no	—
165	C₂²×C₄×C₈	—	73	1	no	—
166	#2150	—	73	2	no	—
167	#2150	—	73	4	no	—
168	#2150	—	73	2	no	—
169	C₂×C₄⋊C₁₆	—	73	4	no	—
170	C₇₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₈	—	73	1	no	—
172	C₂³×C₈	—	73	1	no	—
173	C₂²×C₄²	—	73	1	no	—
174	C₂×M₃₂	—	73	4	no	—
175	C₂×M₃₂	—	73	4	no	—
176	C₂×M₃₂	—	146	8	no	—
177	C₂²×C₁₆	—	73	4	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₂×C₄×C₈	—	73	1	no	—
180	C₂×C₄×C₈	—	73	1	no	—
181	C₂×C₄×C₈	—	73	2	no	—
182	C₂×C₄×C₈	—	73	2	no	—
183	C₂×C₄×C₈	—	73	2	no	—
184	C₂×C₄×C₈	—	73	2	no	—
185	C₂×C₄×C₈	#64408	1	1	yes	characteristic
186	C₄⋊C₁₆	—	73	8	no	—
187	C₄⋊C₁₆	—	73	8	no	—
188	C₂³×C₄	—	73	1	no	—
189	C₂³×C₄	—	73	2	no	—
190	C₂²×C₈	—	73	1	no	—
191	C₂²×C₈	—	73	4	no	—
192	C₂²×C₈	—	73	4	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	73	1	no	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	73	1	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	2	no	—
200	C₂²×C₈	—	146	4	no	—
201	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
202	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
203	C₂×C₄²	—	73	2	no	—
204	C₂×C₄²	—	73	1	no	—
205	C₂×C₄²	—	73	1	no	—
206	C₂×C₄²	—	73	1	no	—
207	C₂×C₄²	—	73	1	no	—
208	C₂×C₄²	—	73	1	no	—
209	C₂×C₄²	—	73	2	no	—
210	C₂×C₄²	—	73	2	no	—
211	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
212	M₃₂	—	146	8	no	—
213	M₃₂	—	146	8	no	—
214	C₂×C₁₆	—	73	8	no	—
215	C₂×C₁₆	—	146	8	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	2	no	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	—	73	2	no	—
220	C₄×C₈	—	73	2	no	—
221	C₄×C₈	—	73	2	no	—
222	C₄×C₈	C₂×#64408	1	2	yes	—
223	C₄×C₈	C₂×#64408	1	2	yes	—
224	C₂⁴	—	73	1	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	2	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	73	1	no	—
229	C₂²×C₄	—	73	4	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	4	no	—
233	C₂²×C₄	—	73	2	no	—
234	C₂²×C₄	—	73	2	no	—
235	C₂²×C₄	—	73	2	no	—
236	C₂²×C₄	—	73	2	no	—
237	C₂²×C₄	—	146	8	no	—
238	C₂²×C₄	#11924036	1	2	yes	—
239	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	73	1	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	4	no	—
246	C₂×C₈	—	73	4	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	73	2	no	—
249	C₂×C₈	—	73	2	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	146	4	no	—
253	C₂×C₈	—	146	4	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	C₄⋊#64408	1	4	yes	—
256	C₂×C₈	C₂²×#64408	1	2	yes	—
257	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
258	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
259	C₄²	—	73	2	no	—
260	C₄²	—	73	2	no	—
261	C₄²	—	73	1	no	—
262	C₄²	—	73	2	no	—
263	C₄²	—	73	2	no	—
264	C₄²	—	73	1	no	—
265	C₄²	—	73	2	no	—
266	C₄²	#11924036	1	1	yes	characteristic
267	C₄²	#11924036	1	1	yes	characteristic
268	C₄²	C₂²×#64408	1	2	yes	—
269	C₁₆	—	146	8	no	—
270	C₂³	—	73	1	no	—
271	C₂³	—	73	1	no	—
272	C₂³	—	73	4	no	—
273	C₂³	—	146	4	no	—
274	C₂³	C₂×#11924036	1	1	yes	characteristic
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	73	1	no	—
277	C₂×C₄	—	73	1	no	—
278	C₂×C₄	—	73	2	no	—
279	C₂×C₄	—	73	4	no	—
280	C₂×C₄	—	73	8	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	73	2	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	73	2	no	—
285	C₂×C₄	—	73	4	no	—
286	C₂×C₄	—	73	4	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	146	4	no	—
289	C₂×C₄	—	146	4	no	—
290	C₂×C₄	—	146	4	no	—
291	C₂×C₄	—	146	4	no	—
292	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
293	C₂×C₄	C₂×#11924036	1	2	yes	—
294	C₂×C₄	C₂×#11924036	1	2	yes	—
295	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
296	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
297	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	73	1	no	—
300	C₈	—	73	2	no	—
301	C₈	—	73	1	no	—
302	C₈	—	146	2	no	—
303	C₈	C₂×C₄⋊#64408	1	2	yes	—
304	C₈	C₂×C₄⋊#64408	1	2	yes	—
305	C₂²	—	2	2	no	—
306	C₂²	—	73	2	no	—
307	C₂²	—	73	4	no	—
308	C₂²	—	73	2	no	—
309	C₂²	—	146	8	no	—
310	C₂²	—	146	2	no	—
311	C₂²	#17989588	1	2	yes	—
312	C₂²	C₂²×#11924036	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	73	2	no	—
315	C₄	—	73	2	no	—
316	C₄	—	73	4	no	—
317	C₄	—	146	2	no	—
318	C₄	#17989588	1	2	yes	—
319	C₄	#17989643	1	4	yes	complemented
320	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
321	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	73	4	no	—
324	C₂	—	146	2	no	—
325	C₂	C₂×#17989588	1	1	yes	characteristic
326	C₂	C₂×#17989643	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#2150
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22626265	C₂	semidirect
3. #22626264	C₂	semidirect
4. #22626263	C₂	semidirect
5. #22626261	C₂	semidirect
11. #17989508	C₂²	semidirect
12. C₄⋊#11924031	C₂²	semidirect
129. C₄×D₇₃	M₃₂	semidirect
151. C₄×C₇₃	C₂×M₃₂	semidirect
157. Dic₇₃	C₂×M₃₂	semidirect
163. D₇₃	#2150	semidirect
170. C₇₃	C₂×#2150	semidirect
319. C₄	#17989643	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
#22626265	2
#22626264	2
#22626263	2
#22626261	2
C₂×C₄×C₈×D₇₃	2
C₂×#2150	73