Subgroups — #33096428

Order 18688.

946 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×#17988640	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	#22610650	C₂	1	4	yes	maximal, complemented
4	#22610111	C₂	1	4	yes	maximal
5	#22610083	C₂	1	2	yes	maximal, complemented
6	#22610048	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×#10596831	C₂²	1	2	yes	—
9	#17988640	—	2	4	no	—
10	C₂×C₄⋊#10596831	—	2	4	no	—
11	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
12	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	C₂²	1	8	yes	—
16	C₂×C₂²⋊#10596831	C₂²	1	2	yes	—
17	C₂×C₄×#10596831	C₂²	1	2	yes	—
18	#17988342	C₂²	1	8	yes	—
19	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
20	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
21	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₇₃	C₂²	1	2	yes	—
26	#376×C₇₃	C₄	1	2	yes	—
27	C₄³×C₇₃	C₂²	1	2	yes	—
28	C₂²×#10596831	—	2	4	no	—
29	C₂²×#10596831	—	4	4	no	—
30	C₂²×#10596831	C₂³	1	2	yes	—
31	C₂²×#10596831	C₂³	1	4	yes	—
32	C₂²×#10596831	C₂³	1	2	yes	—
33	C₄⋊#10596831	—	2	4	no	—
34	C₄⋊#10596831	—	2	4	no	—
35	C₄⋊#10596831	—	2	2	no	—
36	C₄⋊#10596831	—	2	2	no	—
37	C₄⋊#10596831	—	4	8	no	—
38	C₂²⋊#10596831	—	2	4	no	—
39	C₄×#10596831	—	2	4	no	—
40	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	#117×C₇₃	—	2	4	no	—
44	#117×C₇₃	C₈	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₃	D₄	1	4	yes	—
55	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	#95×C₇₃	C₂×C₄	1	4	yes	—
60	#95×C₇₃	C₂×C₄	1	4	yes	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	—	2	4	no	—
64	C₂×#10596831	—	2	4	no	—
65	C₂×#10596831	—	2	8	no	—
66	C₂×#10596831	—	2	4	no	—
67	C₂×#10596831	—	4	8	no	—
68	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	2	2	no	—
70	C₂²×C₄×C₇₃	—	2	2	no	—
71	C₂²×C₄×C₇₃	—	4	2	no	—
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
73	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
75	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₇₃	—	2	4	no	—
87	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	complemented
88	C₂²⋊C₄×C₇₃	—	2	4	no	—
89	C₂²⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
90	C₄²×C₇₃	—	2	4	no	—
91	C₄²×C₇₃	—	2	4	no	—
92	C₄²×C₇₃	—	2	4	no	—
93	C₄²×C₇₃	—	4	4	no	—
94	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
95	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
96	C₄²×C₇₃	C₂×C₈	1	8	yes	complemented
97	C₄²×C₇₃	C₂×D₄	1	2	yes	—
98	C₄²×C₇₃	C₂×D₄	1	2	yes	—
99	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
100	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
101	C₂³×C₇₃	—	2	2	no	—
102	C₂³×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
105	C₂³×C₇₃	C₂×M₁₆	1	2	yes	—
106	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	4	no	—
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	4	4	no	—
119	C₂×C₄×C₇₃	C₄⋊C₈	1	16	yes	complemented
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
133	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
134	#10596831	—	4	2	no	—
135	#10596831	—	4	4	no	—
136	#10596831	—	4	2	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	4	no	—
140	C₂²×C₇₃	—	2	4	no	—
141	C₂²×C₇₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₇₃	#520	1	2	yes	—
144	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₇₃	#568	1	2	yes	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	2	8	no	—
151	C₄×C₇₃	—	4	4	no	—
152	C₄×C₇₃	C₂×C₄⋊C₈	1	4	yes	—
153	C₄×C₇₃	C₂×C₄⋊C₈	1	4	yes	—
154	C₄×C₇₃	#535	1	1	yes	characteristic
155	C₄×C₇₃	#536	1	1	yes	characteristic
156	C₄×C₇₃	#544	1	2	yes	—
157	#21972	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	4	no	—
159	C₂×C₇₃	#2317	1	1	yes	characteristic
160	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₇₃	#2905	1	1	yes	characteristic
162	C₂×C₇₃	#2988	1	2	yes	—
163	C₂×C₇₃	#2989	1	2	yes	—
164	C₂×#431	—	73	1	no	—
165	C₂²×C₄⋊C₈	—	73	1	no	—
166	C₄×#117	D₇₃	1	1	yes	characteristic
167	#1939	—	73	4	no	—
168	#1850	—	73	4	no	—
169	#1842	—	73	2	no	—
170	#1831	—	73	2	no	—
171	C₇₃	#21972	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₈	—	73	2	no	—
173	C₂×#117	Dic₇₃	1	1	yes	characteristic
174	C₂×#117	Dic₇₃	1	1	yes	characteristic
175	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
176	#431	—	146	4	no	—
177	C₂×C₄⋊C₈	—	73	8	no	—
178	C₂×C₄⋊C₈	—	73	1	no	—
179	C₂×C₄⋊C₈	—	73	2	no	—
180	C₂×C₄⋊C₈	—	73	1	no	—
181	C₂×C₄⋊C₈	—	73	2	no	—
182	C₂×C₄⋊C₈	—	146	4	no	—
183	C₂×C₂²⋊C₈	—	73	2	no	—
184	C₂×C₄×C₈	—	73	2	no	—
185	C₄×C₄⋊C₄	Dic₇₃	1	2	yes	—
186	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
187	C₄×C₂²⋊C₄	Dic₇₃	1	2	yes	—
188	C₄×C₂²⋊C₄	C₂×D₇₃	1	2	yes	—
189	#376	Dic₇₃	1	2	yes	—
190	C₄³	C₂×D₇₃	1	2	yes	—
191	#336	—	73	8	no	—
192	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
195	C₂²×C₈	—	292	4	no	—
196	C₂²×C₈	—	73	4	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	146	4	no	—
200	#117	—	2	4	no	—
201	#117	#10596831	1	8	yes	complemented
202	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
204	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
205	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
206	C₂×C₄²	Dic₁₄₆	1	4	yes	—
207	C₂×C₄²	D₂₉₂	1	4	yes	—
208	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
209	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
210	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
213	C₂×C₄²	C₂²×D₇₃	1	4	yes	—
214	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
216	C₄⋊C₈	—	292	8	no	—
217	C₄⋊C₈	—	146	4	no	—
218	C₄⋊C₈	—	146	4	no	—
219	C₄⋊C₈	—	146	2	no	—
220	C₄⋊C₈	—	146	2	no	—
221	C₂²⋊C₈	—	146	4	no	—
222	C₄×C₈	—	146	4	no	—
223	#95	C₂×Dic₇₃	1	4	yes	—
224	#95	C₂×Dic₇₃	1	4	yes	—
225	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
230	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
231	C₂²×C₄	C₂×#10596831	1	4	yes	—
232	C₂²×C₄	#11924012	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
234	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
238	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
242	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
243	C₂×C₈	—	292	8	no	—
244	C₂×C₈	—	146	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	—	146	4	no	—
247	C₂×C₈	—	146	4	no	—
248	C₂×C₈	—	146	8	no	—
249	C₂×C₈	—	146	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂×#10596831	1	8	yes	complemented
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	C₂×#10596831	1	8	yes	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₄⋊Dic₇₃	1	4	yes	—
259	C₄²	C₄⋊Dic₇₃	1	4	yes	—
260	C₄²	C₂×#10596831	1	8	yes	complemented
261	C₄²	C₂×D₂₉₂	1	2	yes	—
262	C₄²	C₂×D₂₉₂	1	2	yes	—
263	C₄²	C₂×Dic₁₄₆	1	2	yes	—
264	C₄²	C₂×Dic₁₄₆	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
268	C₂³	C₂²×#10596831	1	2	yes	—
269	C₂³	C₂×#11924012	1	2	yes	—
270	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	C₄⋊#10596831	1	16	yes	complemented
284	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
289	C₂×C₄	C₂²×#10596831	1	8	yes	—
290	C₂×C₄	C₂²×#10596831	1	2	yes	—
291	C₂×C₄	C₂²×#10596831	1	2	yes	—
292	C₂×C₄	C₂×#11924012	1	2	yes	—
293	C₂×C₄	C₂×#11924012	1	2	yes	—
294	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
296	C₂×C₄	#14556637	1	2	yes	—
297	C₂×C₄	#14556638	1	2	yes	—
298	C₈	—	292	4	no	—
299	C₈	—	292	2	no	—
300	C₈	—	292	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	C₂×C₄⋊#10596831	1	4	yes	—
306	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
307	C₂²	#17989070	1	2	yes	—
308	C₂²	C₂³×#10596831	1	1	yes	characteristic
309	C₂²	C₂²×#11924012	1	1	yes	characteristic
310	C₂²	#17989354	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	8	no	—
315	C₄	—	4	4	no	—
316	C₄	C₂×C₄⋊#10596831	1	4	yes	—
317	C₄	C₂×C₄⋊#10596831	1	4	yes	—
318	C₄	#17989156	1	1	yes	characteristic
319	C₄	#17989163	1	1	yes	characteristic
320	C₄	#17989215	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#22612554	1	1	yes	characteristic
323	C₂	C₂²×C₄⋊#10596831	1	1	yes	characteristic
324	C₂	#22618495	1	1	yes	characteristic
325	C₂	#22619042	1	2	yes	—
326	C₂	#22619048	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×#117×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×#117	D₇₃
hypocenter	C₇₃	#21972
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #22610650	C₂	semidirect
5. #22610083	C₂	semidirect
6. #22610048	C₂	semidirect
44. #117×C₇₃	C₈	semidirect
87. C₄⋊C₄×C₇₃	C₂×C₈	semidirect
96. C₄²×C₇₃	C₂×C₈	semidirect
119. C₂×C₄×C₇₃	C₄⋊C₈	semidirect
171. C₇₃	#21972	semidirect
201. #117	#10596831	semidirect
251. C₄⋊C₄	C₂×#10596831	semidirect
260. C₄²	C₂×#10596831	semidirect
283. C₂×C₄	C₄⋊#10596831	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988640	2
C₂²×C₄⋊#10596831	2
#22610650	2
#22610111	2
#22610083	2
#22610048	2
C₄×#117×C₇₃	2
#21972	73