Subgroups — #33104484

Order 18688.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988705	C₂	1	1	yes	maximal, characteristic
2	#22618808	C₂	1	2	yes	maximal
3	#22618528	C₂	1	2	yes	maximal, complemented
4	#22610716	C₂	1	2	yes	maximal, complemented
5	#22610158	C₂	1	4	yes	maximal, complemented
6	#22610091	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609992	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	C₂×#14556597	C₂²	1	2	yes	—
11	#17989067	C₂²	1	4	yes	—
12	#17988705	C₄	1	8	yes	complemented
13	#17988689	—	2	4	no	—
14	C₄×Dic₂₉₂	—	2	2	no	—
15	C₄×D₅₈₄	—	2	2	no	—
16	C₂×#14556527	C₂²	1	1	yes	characteristic
17	#17988630	C₂²	1	4	yes	—
18	#17988626	—	2	4	no	—
19	#17988624	—	2	2	no	—
20	C₂×#14556525	C₂²	1	2	yes	—
21	C₂×#14556525	C₂²	1	2	yes	—
22	#107×D₇₃	—	2	2	no	—
23	C₂×#14556519	C₂²	1	2	yes	—
24	#17988589	C₂²	1	4	yes	—
25	#17988457	C₂²	1	2	yes	—
26	#17988355	C₂²	1	4	yes	—
27	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
28	#427×C₇₃	C₂²	1	1	yes	characteristic
29	#405×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#339×C₇₃	C₂²	1	2	yes	—
32	C₂×#11924028	C₂³	1	2	yes	—
33	#14556613	C₂×C₄	1	4	yes	—
34	#14556613	Q₈	1	4	yes	—
35	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
36	C₂×#11924017	D₄	1	2	yes	—
37	C₂×D₅₈₄	D₄	1	1	yes	characteristic
38	#14556597	—	2	4	no	—
39	#14556597	C₂×C₄	1	8	yes	complemented
40	C₂×#11924012	C₂³	1	2	yes	—
41	C₂×#11924011	—	2	2	no	—
42	C₂×C₈×D₇₃	—	2	2	no	—
43	C₄×Dic₁₄₆	—	2	4	no	—
44	C₄×D₂₉₂	—	2	4	no	—
45	#14556551	—	4	4	no	—
46	#14556549	—	2	4	no	—
47	C₂×C₄⋊Dic₇₃	—	4	2	no	—
48	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
49	C₄⋊C₄×D₇₃	—	2	4	no	—
50	#14556527	C₂×C₄	1	4	yes	complemented
51	#14556525	—	2	2	no	—
52	#14556525	—	2	2	no	—
53	#14556525	—	2	2	no	—
54	#14556525	—	2	2	no	—
55	#14556524	—	2	4	no	—
56	#14556522	—	2	4	no	—
57	#14556519	C₂×C₄	1	8	yes	complemented
58	#14556516	—	2	4	no	—
59	#14556513	—	2	4	no	—
60	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
63	#117×C₇₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	#108×C₇₃	—	2	2	no	—
66	#107×C₇₃	D₄	1	1	yes	characteristic
67	#107×C₇₃	D₄	1	1	yes	characteristic
68	#106×C₇₃	D₄	1	2	yes	—
69	#97×C₇₃	—	2	2	no	—
70	C₄×C₈×C₇₃	—	2	2	no	—
71	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
72	#11924028	—	2	2	no	—
73	#11924028	—	2	2	no	—
74	#11924028	C₂²×C₄	1	4	yes	—
75	#11924028	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
77	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
78	C₂×C₂²⋊D₇₃	—	2	2	no	—
79	C₂²×Dic₇₃	—	2	4	no	—
80	C₂×C₄×D₇₃	—	2	2	no	—
81	C₂×C₄×D₇₃	—	4	4	no	—
82	Dic₂₉₂	C₄⋊C₄	1	4	yes	complemented
83	#11924017	C₄⋊C₄	1	8	yes	complemented
84	D₅₈₄	C₄⋊C₄	1	4	yes	complemented
85	#11924012	—	2	2	no	—
86	#11924012	C₂²×C₄	1	4	yes	—
87	#11924011	—	2	8	no	—
88	C₂×#10596831	—	2	2	no	—
89	C₈×D₇₃	—	2	8	no	—
90	C₄⋊Dic₇₃	—	2	4	no	—
91	C₄⋊Dic₇₃	—	2	4	no	—
92	C₄⋊Dic₇₃	—	8	4	no	—
93	#11924007	—	4	4	no	—
94	#11924005	—	4	4	no	—
95	C₄×Dic₇₃	—	4	4	no	—
96	C₂²×C₄×C₇₃	—	2	1	no	—
97	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₇₃	—	2	2	no	—
99	M₁₆×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	—	2	2	no	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₇₃	#41	1	2	yes	—
109	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₇₃	—	2	2	no	—
112	C₄²×C₇₃	—	2	2	no	—
113	C₄²×C₇₃	—	2	1	no	—
114	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×D₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	2	4	no	—
129	C₂²⋊D₇₃	—	2	4	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	4	4	no	—
132	C₂×Dic₇₃	—	8	4	no	—
133	D₂₉₂	—	2	2	no	—
134	D₂₉₂	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₇₃	—	2	4	no	—
136	C₄×D₇₃	—	2	4	no	—
137	C₄×D₇₃	—	16	4	no	—
138	Dic₁₄₆	—	2	2	no	—
139	Dic₁₄₆	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₇₃	—	4	2	no	—
141	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₇₃	C₄×D₄	1	2	yes	—
144	C₈×C₇₃	C₄×D₄	1	2	yes	—
145	#10596831	—	2	4	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	#520	1	1	yes	characteristic
149	C₂²×C₇₃	D₄²	1	1	yes	characteristic
150	C₂²×C₇₃	#554	1	1	yes	characteristic
151	C₂×D₇₃	—	2	4	no	—
152	C₂×D₇₃	—	4	2	no	—
153	C₂×D₇₃	—	4	2	no	—
154	C₄×C₇₃	—	2	4	no	—
155	C₄×C₇₃	—	4	2	no	—
156	C₄×C₇₃	—	4	2	no	—
157	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₇₃	#547	1	1	yes	characteristic
160	C₄×C₇₃	D₄×Q₈	1	1	yes	characteristic
161	Dic₇₃	—	2	4	no	—
162	Dic₇₃	—	8	4	no	—
163	#22692	—	73	1	no	maximal, Sylow
164	C₂×C₇₃	—	2	1	no	—
165	C₂×C₇₃	—	2	1	no	—
166	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₇₃	#3292	1	1	yes	characteristic
168	C₂×C₇₃	#3293	1	1	yes	characteristic
169	D₇₃	—	4	4	no	—
170	C₂×#443	—	73	1	no	—
171	#2951	—	73	2	no	—
172	#2913	—	73	2	no	—
173	#1952	—	73	2	no	—
174	#1859	—	73	4	no	—
175	#1845	—	73	1	no	—
176	#1816	—	73	2	no	—
177	#1772	D₇₃	1	1	yes	characteristic, complemented
178	C₇₃	#22692	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	73	1	no	—
180	C₂×#131	—	73	2	no	—
181	#520	—	73	4	no	—
182	#443	—	73	8	no	—
183	#440	—	146	4	no	—
184	C₄×Q₁₆	—	146	2	no	—
185	C₄×D₈	—	146	2	no	—
186	C₂×#108	—	73	1	no	—
187	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
188	#428	—	73	4	no	—
189	#428	—	146	4	no	—
190	#427	—	146	2	no	—
191	#427	C₂×D₇₃	1	1	yes	characteristic
192	C₂×#107	—	73	2	no	—
193	C₂×#107	—	73	2	no	—
194	C₂×#107	—	146	2	no	—
195	C₂×#104	—	73	2	no	—
196	#417	—	73	4	no	—
197	#405	C₂×D₇₃	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
199	#384	—	73	2	no	—
200	#343	—	73	4	no	—
201	#339	C₂×D₇₃	1	2	yes	—
202	C₂×#41	—	73	2	no	—
203	#135	—	73	4	no	—
204	#135	—	73	4	no	—
205	C₂×Q₁₆	—	73	1	no	—
206	C₂×SD₁₆	—	73	2	no	—
207	C₂×D₈	—	73	1	no	—
208	#131	—	73	8	no	—
209	#131	—	146	4	no	—
210	C₂×M₁₆	—	73	2	no	—
211	C₂×M₁₆	—	146	2	no	—
212	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
213	C₂²×C₈	—	146	2	no	—
214	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
216	C₄×Q₈	—	146	4	no	—
217	C₄×D₄	—	292	4	no	—
218	C₄×D₄	—	146	4	no	—
219	#117	—	146	4	no	—
220	#117	C₂²×D₇₃	1	2	yes	—
221	C₂×C₄⋊C₄	—	292	2	no	—
222	C₂×C₄⋊C₄	—	73	4	no	—
223	C₂×C₄⋊C₄	—	146	4	no	—
224	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	73	4	no	—
227	#107	—	146	2	no	—
228	#107	—	146	2	no	—
229	#107	—	146	2	no	—
230	#107	—	146	2	no	—
231	#107	—	146	4	no	—
232	#107	D₂₉₂	1	1	yes	characteristic
233	#107	D₂₉₂	1	1	yes	characteristic
234	#106	—	146	4	no	—
235	#106	D₂₉₂	1	2	yes	—
236	#104	—	73	8	no	—
237	#103	—	146	4	no	—
238	#102	—	146	4	no	—
239	#97	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	C₄×D₇₃	1	4	yes	complemented
242	#41	—	73	4	no	—
243	#41	—	73	4	no	—
244	#41	—	146	2	no	—
245	#41	—	146	2	no	—
246	C₂×Q₈	—	73	2	no	—
247	C₂×D₄	—	73	2	no	—
248	C₂×D₄	—	146	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	292	4	no	—
251	C₂²×C₄	—	146	4	no	—
252	C₂²×C₄	—	146	2	no	—
253	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
254	Q₁₆	—	73	4	no	—
255	SD₁₆	—	73	8	no	—
256	D₈	—	73	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	73	4	no	—
260	M₁₆	—	146	2	no	—
261	M₁₆	—	146	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	—	146	8	no	—
265	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
267	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
271	C₂×C₈	Q₈×D₇₃	1	2	yes	—
272	C₂×C₈	#11924028	1	2	yes	—
273	C₄⋊C₄	—	292	4	no	—
274	C₄⋊C₄	—	584	4	no	—
275	C₄⋊C₄	—	146	4	no	—
276	C₄⋊C₄	—	146	4	no	—
277	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
278	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	292	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	292	4	no	—
284	C₄²	C₂×C₄×D₇₃	1	2	yes	—
285	C₂³	—	146	2	no	—
286	C₂³	#14556637	1	1	yes	characteristic
287	Q₈	—	73	4	no	—
288	Q₈	—	146	2	no	—
289	D₄	—	73	4	no	—
290	D₄	—	146	4	no	—
291	D₄	—	146	4	no	—
292	D₄	—	146	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	292	4	no	—
299	C₂×C₄	—	584	4	no	—
300	C₂×C₄	—	1168	4	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	4	no	—
303	C₂×C₄	—	146	4	no	—
304	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
309	C₂×C₄	#14556638	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	146	4	no	—
312	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
313	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
314	C₈	C₄×D₂₉₂	1	2	yes	—
315	C₈	C₄×D₂₉₂	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	292	2	no	—
319	C₂²	—	292	2	no	—
320	C₂²	—	146	4	no	—
321	C₂²	#17989067	1	1	yes	characteristic
322	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
323	C₂²	#17989297	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	584	4	no	—
328	C₄	—	146	4	no	—
329	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
330	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
331	C₄	#17989242	1	1	yes	characteristic
332	C₄	Q₈×D₂₉₂	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	292	4	no	—
336	C₂	C₄⋊C₄×D₂₉₂	1	1	yes	characteristic
337	C₂	#22621731	1	1	yes	characteristic
338	C₂	#22621736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1772	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×#14556613	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1772	D₇₃
hypocenter	C₇₃	#22692
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #22618528	C₂	semidirect
4. #22610716	C₂	semidirect
5. #22610158	C₂	semidirect
6. #22610091	C₂	semidirect
7. #22609992	C₂	semidirect
8. C₇₃×#1772	C₂	semidirect
12. #17988705	C₄	semidirect
39. #14556597	C₂×C₄	semidirect
50. #14556527	C₂×C₄	semidirect
57. #14556519	C₂×C₄	semidirect
71. C₄×C₈×C₇₃	C₂×C₄	semidirect
82. Dic₂₉₂	C₄⋊C₄	semidirect
83. #11924017	C₄⋊C₄	semidirect
84. D₅₈₄	C₄⋊C₄	semidirect
177. #1772	D₇₃	semidirect
178. C₇₃	#22692	semidirect
241. C₄×C₈	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988705	2
#22618808	2
#22618528	2
#22610716	2
#22610158	2
#22610091	2
#22609992	2
C₇₃×#1772	2
#22692	73