Subgroups — #33086448

Order 18688.

994 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988635	C₂	1	4	yes	maximal
2	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
4	#22610105	C₂	1	4	yes	maximal, complemented
5	#22610102	C₂	1	4	yes	maximal, complemented
6	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	C₂²	1	2	yes	—
8	C₂²×#11924012	C₂²	1	2	yes	—
9	#17988635	—	2	16	no	—
10	C₂×#14556526	C₂²	1	16	yes	—
11	#17988611	—	2	2	no	—
12	#17988611	C₂²	1	2	yes	—
13	#17988611	C₂²	1	2	yes	—
14	C₂×C₄⋊#10596831	—	2	2	no	—
15	C₄×#11924012	C₄	1	2	yes	—
16	C₄×#11924012	C₄	1	2	yes	—
17	C₄×#11924012	C₄	1	2	yes	—
18	C₄×#11924012	C₄	1	2	yes	—
19	C₂×#14556502	C₂²	1	2	yes	—
20	C₂×C₄×#10596831	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	#423×C₇₃	—	2	2	no	—
24	#423×C₇₃	C₂²	1	2	yes	—
25	#423×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
27	C₂×#11924012	—	2	2	no	—
28	C₂×#11924012	—	2	8	no	—
29	C₂×#11924012	C₂×C₄	1	4	yes	—
30	C₂×#11924012	C₂×C₄	1	4	yes	—
31	C₂×#11924012	D₄	1	8	yes	—
32	C₂×#11924012	Q₈	1	8	yes	—
33	C₂×#11924012	C₂³	1	4	yes	—
34	C₂²×#10596831	—	2	2	no	—
35	C₂²×#10596831	C₂³	1	4	yes	—
36	#14556526	—	2	32	no	—
37	C₄⋊#10596831	—	2	4	no	—
38	C₄⋊#10596831	—	2	4	no	—
39	#14556502	C₂×C₄	1	4	yes	—
40	#14556502	C₂×C₄	1	4	yes	—
41	C₄×#10596831	C₂×C₄	1	4	yes	—
42	C₄×#10596831	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
44	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₇₃	—	2	2	no	—
46	C₂×M₁₆×C₇₃	—	2	8	no	—
47	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	D₄	1	2	yes	—
54	C₂×C₄²×C₇₃	D₄	1	2	yes	—
55	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₃	—	2	4	no	—
59	C₄⋊C₈×C₇₃	—	2	4	no	—
60	#11924012	—	2	8	no	—
61	#11924012	—	4	4	no	—
62	#11924012	C₄⋊C₄	1	16	yes	—
63	#11924012	C₄⋊C₄	1	16	yes	—
64	C₂×#10596831	—	2	4	no	—
65	C₂×#10596831	—	4	4	no	—
66	C₂×#10596831	C₂²×C₄	1	8	yes	—
67	C₂×#10596831	C₂×D₄	1	8	yes	—
68	C₂×#10596831	C₂×Q₈	1	8	yes	—
69	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	—	4	4	no	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₇₃	—	2	8	no	—
86	M₁₆×C₇₃	—	4	4	no	—
87	C₂×C₈×C₇₃	—	2	4	no	—
88	C₂×C₈×C₇₃	—	4	4	no	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×D₄	1	2	yes	—
98	C₄²×C₇₃	C₂×D₄	1	2	yes	—
99	C₂³×C₇₃	—	4	2	no	—
100	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
103	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	—	2	4	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	4	2	no	—
108	C₂×C₄×C₇₃	—	4	2	no	—
109	C₂×C₄×C₇₃	—	4	2	no	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇₃	#127	1	2	yes	—
119	C₂×C₄×C₇₃	#127	1	2	yes	—
120	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
127	C₈×C₇₃	—	4	4	no	—
128	#10596831	—	4	4	no	—
129	#10596831	C₂×C₄⋊C₄	1	16	yes	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	4	4	no	—
134	C₂²×C₇₃	#384	1	2	yes	—
135	C₂²×C₇₃	#384	1	2	yes	—
136	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	#384	1	2	yes	—
142	C₄×C₇₃	#384	1	2	yes	—
143	C₄×C₇₃	#430	1	4	yes	complemented
144	C₄×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
145	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
146	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	#20806	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	—	2	2	no	—
151	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#430	1	2	yes	—
153	C₂×#430	—	73	4	no	—
154	C₂×#423	—	73	1	no	—
155	C₂×#423	D₇₃	1	1	yes	characteristic
156	C₂×C₄×M₁₆	—	73	1	no	—
157	#1849	—	73	4	no	—
158	#1848	—	73	4	no	—
159	C₇₃	#20806	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	73	2	no	—
161	C₂²×M₁₆	—	73	2	no	—
162	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
163	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
164	#430	—	146	16	no	—
165	C₂×#108	—	73	16	no	—
166	#423	—	2	2	no	—
167	#423	—	73	2	no	—
168	#423	—	73	2	no	—
169	#423	—	146	2	no	—
170	#423	C₂×D₇₃	1	2	yes	—
171	#423	C₂×D₇₃	1	2	yes	—
172	C₂×C₄⋊C₈	—	2	2	no	—
173	C₂×C₄⋊C₈	—	146	2	no	—
174	C₄×M₁₆	—	73	2	no	—
175	C₄×M₁₆	—	73	2	no	—
176	C₄×M₁₆	—	73	2	no	—
177	C₄×M₁₆	—	73	2	no	—
178	C₂×#97	—	73	2	no	—
179	C₂×C₄×C₈	—	73	2	no	—
180	C₂³×C₄	Dic₁₄₆	1	2	yes	—
181	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	8	no	—
184	C₂×M₁₆	—	73	4	no	—
185	C₂×M₁₆	—	73	4	no	—
186	C₂×M₁₆	—	73	8	no	—
187	C₂×M₁₆	—	73	8	no	—
188	C₂×M₁₆	—	73	4	no	—
189	C₂×M₁₆	—	146	8	no	—
190	C₂×M₁₆	—	146	2	no	—
191	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	73	4	no	—
194	C₂²×C₈	—	146	2	no	—
195	C₂×C₄²	Dic₁₄₆	1	2	yes	—
196	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
204	#108	—	146	32	no	—
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	146	4	no	—
208	C₄⋊C₈	—	146	4	no	—
209	#97	—	73	4	no	—
210	#97	—	73	4	no	—
211	C₄×C₈	—	73	4	no	—
212	C₄×C₈	—	73	4	no	—
213	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	#11924007	1	2	yes	—
216	C₂²×C₄	#11924007	1	2	yes	—
217	C₂²×C₄	#11924007	1	2	yes	—
218	C₂²×C₄	#11924007	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₇₃	1	4	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₇₃	1	4	yes	—
225	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
227	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	M₁₆	—	292	4	no	—
232	M₁₆	—	73	16	no	—
233	M₁₆	—	73	16	no	—
234	M₁₆	—	146	8	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	292	4	no	—
238	C₂×C₈	—	73	8	no	—
239	C₂×C₈	—	73	8	no	—
240	C₂×C₈	—	73	8	no	—
241	C₂×C₈	—	146	4	no	—
242	C₄²	#11924007	1	2	yes	—
243	C₄²	#11924007	1	2	yes	—
244	C₄²	#11924007	1	2	yes	—
245	C₄²	#11924007	1	2	yes	—
246	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
251	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×#11924007	1	1	yes	characteristic
254	C₂³	C₂×#11924007	1	1	yes	characteristic
255	C₂³	#14556590	1	4	yes	—
256	C₂³	C₂²×Dic₁₄₆	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
264	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
265	C₂×C₄	C₂×#11924007	1	2	yes	—
266	C₂×C₄	C₂×#11924007	1	2	yes	—
267	C₂×C₄	C₂×#11924007	1	2	yes	—
268	C₂×C₄	C₂×#11924007	1	2	yes	—
269	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
270	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
271	C₂×C₄	#14556587	1	2	yes	—
272	C₂×C₄	#14556587	1	2	yes	—
273	C₂×C₄	#14556589	1	2	yes	—
274	C₂×C₄	#14556589	1	2	yes	—
275	C₂×C₄	#14556590	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
278	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
280	C₈	—	4	4	no	—
281	C₈	—	292	4	no	—
282	C₈	—	73	16	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	#17988456	1	2	yes	—
288	C₂²	#17988456	1	2	yes	—
289	C₂²	C₂²×#11924007	1	1	yes	characteristic
290	C₂²	C₂×#14556590	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	#17988456	1	2	yes	—
295	C₄	#17988456	1	2	yes	—
296	C₄	#17988635	1	4	yes	complemented
297	C₄	C₂×C₄⋊C₄×D₇₃	1	2	yes	—
298	C₄	C₂×#14556587	1	1	yes	characteristic
299	C₄	C₂×#14556589	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	C₂×#17988456	1	1	yes	characteristic
304	C₂	C₂×#17988635	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#20806
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #22610105	C₂	semidirect
5. #22610102	C₂	semidirect
143. C₄×C₇₃	#430	semidirect
159. C₇₃	#20806	semidirect
296. C₄	#17988635	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988635	2
C₂×#17988611	2
C₂×C₄×#11924012	2
#22610105	2
#22610102	2
C₂×#423×C₇₃	2
#20806	73