Subgroups — #33104460

Order 18688.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#22618816	C₂	1	2	yes	maximal
3	#22618523	C₂	1	2	yes	maximal, complemented
4	#22610711	C₂	1	2	yes	maximal, complemented
5	#22610158	C₂	1	4	yes	maximal, complemented
6	#22610088	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609987	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	#17988700	—	2	2	no	—
14	#17988695	—	2	2	no	—
15	C₄×#11924017	—	2	4	no	—
16	C₂×#14556527	C₂²	1	1	yes	characteristic
17	#17988630	C₂²	1	4	yes	—
18	#17988628	—	2	4	no	—
19	#17988622	—	2	2	no	—
20	C₂×#14556523	C₂²	1	2	yes	—
21	C₂×#14556523	C₂²	1	2	yes	—
22	#106×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	#14556524	—	2	4	no	—
52	#14556523	—	2	2	no	—
53	#14556523	—	2	2	no	—
54	#14556523	—	2	2	no	—
55	#14556523	—	2	2	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	2	yes	—
67	#106×C₇₃	D₄	1	2	yes	—
68	#97×C₇₃	—	2	2	no	—
69	C₄×C₈×C₇₃	—	2	2	no	—
70	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
71	#11924028	—	2	2	no	—
72	#11924028	—	2	2	no	—
73	#11924028	C₂²×C₄	1	4	yes	—
74	#11924028	C₂×Q₈	1	4	yes	—
75	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
76	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
77	C₂×C₂²⋊D₇₃	—	2	2	no	—
78	C₂²×Dic₇₃	—	2	4	no	—
79	C₂×C₄×D₇₃	—	2	2	no	—
80	C₂×C₄×D₇₃	—	4	4	no	—
81	Dic₂₉₂	C₄⋊C₄	1	4	yes	complemented
82	#11924017	C₄⋊C₄	1	8	yes	complemented
83	D₅₈₄	C₄⋊C₄	1	4	yes	complemented
84	#11924012	—	2	2	no	—
85	#11924012	C₂²×C₄	1	4	yes	—
86	#11924011	—	2	8	no	—
87	C₂×#10596831	—	2	2	no	—
88	C₈×D₇₃	—	2	8	no	—
89	C₄⋊Dic₇₃	—	2	4	no	—
90	C₄⋊Dic₇₃	—	2	4	no	—
91	C₄⋊Dic₇₃	—	8	4	no	—
92	#11924007	—	4	4	no	—
93	#11924005	—	4	4	no	—
94	C₄×Dic₇₃	—	4	4	no	—
95	C₂²×C₄×C₇₃	—	2	1	no	—
96	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₇₃	—	2	2	no	—
98	M₁₆×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
107	C₂×C₈×C₇₃	#41	1	2	yes	—
108	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₇₃	—	2	2	no	—
110	C₄²×C₇₃	—	2	2	no	—
111	C₄²×C₇₃	—	2	1	no	—
112	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
113	C₂³×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×D₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²⋊D₇₃	—	2	4	no	—
125	C₂²⋊D₇₃	—	2	4	no	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	—	4	4	no	—
128	C₂×Dic₇₃	—	8	4	no	—
129	D₂₉₂	—	2	2	no	—
130	D₂₉₂	C₂×C₄⋊C₄	1	4	yes	—
131	C₄×D₇₃	—	2	4	no	—
132	C₄×D₇₃	—	2	4	no	—
133	C₄×D₇₃	—	16	4	no	—
134	Dic₁₄₆	—	2	2	no	—
135	Dic₁₄₆	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₇₃	—	4	2	no	—
137	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₇₃	C₄×D₄	1	2	yes	—
140	C₈×C₇₃	C₄×D₄	1	2	yes	—
141	#10596831	—	2	4	no	—
142	C₂²×C₇₃	—	2	2	no	—
143	C₂²×C₇₃	#520	1	1	yes	characteristic
144	C₂²×C₇₃	D₄²	1	1	yes	characteristic
145	C₂²×C₇₃	#554	1	1	yes	characteristic
146	C₂×D₇₃	—	2	4	no	—
147	C₂×D₇₃	—	4	4	no	—
148	C₄×C₇₃	—	2	4	no	—
149	C₄×C₇₃	—	4	4	no	—
150	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₃	#547	1	1	yes	characteristic
153	C₄×C₇₃	D₄×Q₈	1	1	yes	characteristic
154	Dic₇₃	—	2	4	no	—
155	Dic₇₃	—	8	4	no	—
156	#22689	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	—	2	1	no	—
158	C₂×C₇₃	—	2	1	no	—
159	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₇₃	#3291	1	2	yes	—
161	D₇₃	—	4	4	no	—
162	C₂×#443	—	73	1	no	—
163	#2952	—	73	2	no	—
164	#2912	—	73	2	no	—
165	#1951	—	73	2	no	—
166	#1859	—	73	4	no	—
167	#1844	—	73	1	no	—
168	#1815	—	73	2	no	—
169	#1772	D₇₃	1	1	yes	characteristic, complemented
170	C₇₃	#22689	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	—	73	1	no	—
172	C₂×#131	—	73	2	no	—
173	#520	—	73	4	no	—
174	#443	—	73	8	no	—
175	#442	—	146	2	no	—
176	#441	—	146	2	no	—
177	C₄×SD₁₆	—	146	4	no	—
178	C₂×#108	—	73	1	no	—
179	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
180	#428	—	73	4	no	—
181	#428	—	146	4	no	—
182	#427	—	146	2	no	—
183	#427	C₂×D₇₃	1	1	yes	characteristic
184	C₂×#106	—	73	2	no	—
185	C₂×#106	—	73	2	no	—
186	C₂×#106	—	146	2	no	—
187	C₂×#104	—	73	2	no	—
188	#417	—	73	4	no	—
189	#405	C₂×D₇₃	1	2	yes	—
190	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
191	#384	—	73	2	no	—
192	#343	—	73	4	no	—
193	#339	C₂×D₇₃	1	2	yes	—
194	C₂×#41	—	73	2	no	—
195	#135	—	73	4	no	—
196	#135	—	73	4	no	—
197	C₂×Q₁₆	—	73	1	no	—
198	C₂×SD₁₆	—	73	2	no	—
199	C₂×D₈	—	73	1	no	—
200	#131	—	73	8	no	—
201	#131	—	146	4	no	—
202	C₂×M₁₆	—	73	2	no	—
203	C₂×M₁₆	—	146	2	no	—
204	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
205	C₂²×C₈	—	146	2	no	—
206	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×Q₈	—	146	4	no	—
209	C₄×D₄	—	292	4	no	—
210	C₄×D₄	—	146	4	no	—
211	#117	—	146	4	no	—
212	#117	C₂²×D₇₃	1	2	yes	—
213	C₂×C₄⋊C₄	—	292	2	no	—
214	C₂×C₄⋊C₄	—	73	4	no	—
215	C₂×C₄⋊C₄	—	146	4	no	—
216	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	—	73	4	no	—
219	#107	—	146	4	no	—
220	#107	D₂₉₂	1	2	yes	—
221	#106	—	146	2	no	—
222	#106	—	146	2	no	—
223	#106	—	146	2	no	—
224	#106	—	146	2	no	—
225	#106	—	146	4	no	—
226	#106	D₂₉₂	1	2	yes	—
227	#104	—	73	8	no	—
228	#103	—	146	4	no	—
229	#102	—	146	4	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	C₄×D₇₃	1	4	yes	complemented
233	#41	—	73	4	no	—
234	#41	—	73	4	no	—
235	#41	—	146	2	no	—
236	#41	—	146	2	no	—
237	C₂×Q₈	—	73	2	no	—
238	C₂×D₄	—	73	2	no	—
239	C₂×D₄	—	146	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	292	4	no	—
242	C₂²×C₄	—	146	4	no	—
243	C₂²×C₄	—	146	2	no	—
244	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
245	Q₁₆	—	73	4	no	—
246	SD₁₆	—	73	8	no	—
247	D₈	—	73	4	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	73	4	no	—
251	M₁₆	—	146	2	no	—
252	M₁₆	—	146	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	—	146	8	no	—
256	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
258	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₇₃	1	2	yes	—
263	C₂×C₈	#11924028	1	2	yes	—
264	C₄⋊C₄	—	292	4	no	—
265	C₄⋊C₄	—	584	4	no	—
266	C₄⋊C₄	—	146	4	no	—
267	C₄⋊C₄	—	146	4	no	—
268	C₄⋊C₄	C₂×D₂₉₂	1	4	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	292	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	292	4	no	—
274	C₄²	C₂×C₄×D₇₃	1	2	yes	—
275	C₂³	—	146	2	no	—
276	C₂³	#14556637	1	1	yes	characteristic
277	Q₈	—	73	4	no	—
278	Q₈	—	146	2	no	—
279	D₄	—	73	4	no	—
280	D₄	—	146	4	no	—
281	D₄	—	146	4	no	—
282	D₄	—	146	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	292	4	no	—
287	C₂×C₄	—	584	4	no	—
288	C₂×C₄	—	1168	4	no	—
289	C₂×C₄	—	146	2	no	—
290	C₂×C₄	—	146	4	no	—
291	C₂×C₄	—	146	4	no	—
292	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
297	C₂×C₄	#14556638	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	146	4	no	—
300	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
301	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
302	C₈	C₄×D₂₉₂	1	2	yes	—
303	C₈	C₄×D₂₉₂	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	292	4	no	—
306	C₂²	—	146	4	no	—
307	C₂²	#17989067	1	1	yes	characteristic
308	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
309	C₂²	#17989297	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	584	4	no	—
313	C₄	—	146	4	no	—
314	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
315	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
316	C₄	#17989242	1	1	yes	characteristic
317	C₄	Q₈×D₂₉₂	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	292	4	no	—
321	C₂	C₄⋊C₄×D₂₉₂	1	1	yes	characteristic
322	C₂	#22621725	1	2	yes	—

	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	#22689
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. #22618523	C₂	semidirect
4. #22610711	C₂	semidirect
5. #22610158	C₂	semidirect
6. #22610088	C₂	semidirect
7. #22609987	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
70. C₄×C₈×C₇₃	C₂×C₄	semidirect
81. Dic₂₉₂	C₄⋊C₄	semidirect
82. #11924017	C₄⋊C₄	semidirect
83. D₅₈₄	C₄⋊C₄	semidirect
169. #1772	D₇₃	semidirect
170. C₇₃	#22689	semidirect
232. 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
#22618816	2
#22618523	2
#22610711	2
#22610158	2
#22610088	2
#22609987	2
C₇₃×#1772	2
#22689	73