Subgroups — #33086224

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
3	#22610071	C₂	1	4	yes	maximal, complemented
4	#22610068	C₂	1	4	yes	maximal, complemented
5	#22609657	C₂	1	4	yes	maximal
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	#17988611	—	2	2	no	—
10	#17988611	C₂²	1	2	yes	—
11	#17988611	C₂²	1	2	yes	—
12	C₂×C₄⋊#10596831	—	2	2	no	—
13	C₄×#11924012	C₄	1	2	yes	—
14	C₄×#11924012	C₄	1	2	yes	—
15	C₄×#11924012	C₄	1	2	yes	—
16	C₄×#11924012	C₄	1	2	yes	—
17	C₂×#14556502	C₂²	1	2	yes	—
18	C₂×C₄×#10596831	C₂²	1	2	yes	—
19	#17988351	C₂²	1	16	yes	—
20	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	#423×C₇₃	—	2	2	no	—
23	#423×C₇₃	C₂²	1	2	yes	—
24	#423×C₇₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	4	no	—
28	C₂×#11924012	—	2	4	no	—
29	C₂×#11924012	—	2	8	no	—
30	C₂×#11924012	C₂×C₄	1	4	yes	—
31	C₂×#11924012	C₂×C₄	1	4	yes	—
32	C₂×#11924012	C₂³	1	4	yes	—
33	C₂²×#10596831	—	2	2	no	—
34	C₂²×#10596831	C₂³	1	4	yes	—
35	C₄⋊#10596831	—	2	4	no	—
36	C₄⋊#10596831	—	2	4	no	—
37	#14556502	C₂×C₄	1	4	yes	—
38	#14556502	C₂×C₄	1	4	yes	—
39	C₄×#10596831	C₂×C₄	1	4	yes	—
40	C₄×#10596831	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
42	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₇₃	—	2	2	no	—
44	C₂×M₁₆×C₇₃	—	2	8	no	—
45	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
46	C₂²×C₈×C₇₃	—	2	2	no	—
47	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	D₄	1	2	yes	—
52	C₂×C₄²×C₇₃	D₄	1	2	yes	—
53	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₃	—	2	4	no	—
57	C₄⋊C₈×C₇₃	—	2	4	no	—
58	#11924012	—	2	8	no	—
59	#11924012	—	2	8	no	—
60	#11924012	—	2	8	no	—
61	#11924012	—	4	4	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	2	4	no	—
64	C₂×#10596831	—	2	4	no	—
65	C₂×#10596831	—	4	4	no	—
66	C₂×#10596831	C₂²×C₄	1	8	yes	—
67	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	—	4	4	no	—
69	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	#41	1	4	yes	—
81	C₂²×C₄×C₇₃	#41	1	4	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₇₃	—	2	8	no	—
84	M₁₆×C₇₃	—	4	4	no	—
85	C₂×C₈×C₇₃	—	2	4	no	—
86	C₂×C₈×C₇₃	—	4	4	no	—
87	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₃	C₂×D₄	1	2	yes	—
96	C₄²×C₇₃	C₂×D₄	1	2	yes	—
97	C₂³×C₇₃	—	4	2	no	—
98	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₇₃	#125	1	4	yes	—
101	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	4	2	no	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇₃	#117	1	4	yes	—
115	C₂×C₄×C₇₃	#117	1	4	yes	—
116	C₂×C₄×C₇₃	#124	1	2	yes	—
117	C₂×C₄×C₇₃	#124	1	2	yes	—
118	C₂×C₄×C₇₃	#124	1	2	yes	—
119	C₂×C₄×C₇₃	#124	1	2	yes	—
120	C₂×C₄×C₇₃	#125	1	4	yes	—
121	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
125	C₈×C₇₃	—	4	4	no	—
126	#10596831	—	2	8	no	—
127	#10596831	—	4	4	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	4	4	no	—
132	C₂²×C₇₃	#382	1	2	yes	—
133	C₂²×C₇₃	#382	1	2	yes	—
134	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×#125	1	2	yes	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	#382	1	2	yes	—
140	C₄×C₇₃	#382	1	2	yes	—
141	C₄×C₇₃	#430	1	4	yes	—
142	C₄×C₇₃	C₂×#117	1	2	yes	—
143	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
144	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
145	#20769	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	2	no	—
147	C₂×C₇₃	—	2	2	no	—
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	C₂×#382	1	1	yes	characteristic
150	C₂×C₇₃	C₂×#430	1	2	yes	—
151	C₂×#423	—	73	1	no	—
152	C₂×#423	D₇₃	1	1	yes	characteristic
153	C₂×C₄×M₁₆	—	73	1	no	—
154	#1838	—	73	4	no	—
155	#1837	—	73	4	no	—
156	#1738	—	73	4	no	—
157	C₇₃	#20769	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	73	2	no	—
159	C₂²×M₁₆	—	73	2	no	—
160	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
161	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
162	#423	—	2	2	no	—
163	#423	—	73	2	no	—
164	#423	—	73	2	no	—
165	#423	—	146	2	no	—
166	#423	C₂×D₇₃	1	2	yes	—
167	#423	C₂×D₇₃	1	2	yes	—
168	C₂×C₄⋊C₈	—	2	2	no	—
169	C₂×C₄⋊C₈	—	146	2	no	—
170	C₄×M₁₆	—	73	2	no	—
171	C₄×M₁₆	—	73	2	no	—
172	C₄×M₁₆	—	73	2	no	—
173	C₄×M₁₆	—	73	2	no	—
174	C₂×#97	—	73	2	no	—
175	C₂×C₄×C₈	—	73	2	no	—
176	#341	—	73	16	no	—
177	C₂³×C₄	Dic₁₄₆	1	2	yes	—
178	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	8	no	—
181	C₂×M₁₆	—	73	4	no	—
182	C₂×M₁₆	—	73	4	no	—
183	C₂×M₁₆	—	73	4	no	—
184	C₂×M₁₆	—	146	8	no	—
185	C₂×M₁₆	—	146	2	no	—
186	C₂×M₁₆	—	146	4	no	—
187	C₂×M₁₆	—	146	4	no	—
188	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	73	4	no	—
191	C₂²×C₈	—	146	2	no	—
192	C₂×C₄²	Dic₁₄₆	1	2	yes	—
193	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
199	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	146	4	no	—
204	C₄⋊C₈	—	146	4	no	—
205	#97	—	73	4	no	—
206	#97	—	73	4	no	—
207	C₄×C₈	—	73	4	no	—
208	C₄×C₈	—	73	4	no	—
209	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	#11924007	1	2	yes	—
212	C₂²×C₄	#11924007	1	2	yes	—
213	C₂²×C₄	#11924007	1	2	yes	—
214	C₂²×C₄	#11924007	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
222	C₂²×C₄	#11924027	1	4	yes	—
223	C₂²×C₄	#11924029	1	4	yes	—
224	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	M₁₆	—	4	4	no	—
227	M₁₆	—	292	4	no	—
228	M₁₆	—	146	8	no	—
229	M₁₆	—	146	8	no	—
230	M₁₆	—	146	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	292	4	no	—
234	C₂×C₈	—	73	8	no	—
235	C₂×C₈	—	146	4	no	—
236	C₂×C₈	—	146	4	no	—
237	C₂×C₈	—	146	4	no	—
238	C₄²	#11924007	1	2	yes	—
239	C₄²	#11924007	1	2	yes	—
240	C₄²	#11924007	1	2	yes	—
241	C₄²	#11924007	1	2	yes	—
242	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
245	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
247	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	C₂×#11924007	1	1	yes	characteristic
250	C₂³	C₂×#11924007	1	1	yes	characteristic
251	C₂³	#14556582	1	4	yes	—
252	C₂³	C₂²×Dic₁₄₆	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×#11924007	1	2	yes	—
260	C₂×C₄	C₂×#11924007	1	2	yes	—
261	C₂×C₄	C₂×#11924007	1	2	yes	—
262	C₂×C₄	C₂×#11924007	1	2	yes	—
263	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
264	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
265	C₂×C₄	#14556549	1	4	yes	—
266	C₂×C₄	#14556549	1	4	yes	—
267	C₂×C₄	#14556578	1	2	yes	—
268	C₂×C₄	#14556578	1	2	yes	—
269	C₂×C₄	#14556580	1	2	yes	—
270	C₂×C₄	#14556580	1	2	yes	—
271	C₂×C₄	#14556582	1	4	yes	—
272	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#11924027	1	2	yes	—
275	C₂×C₄	C₂×#11924029	1	2	yes	—
276	C₈	—	4	4	no	—
277	C₈	—	292	4	no	—
278	C₈	—	146	8	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	4	no	—
283	C₂²	#17988452	1	2	yes	—
284	C₂²	#17988452	1	2	yes	—
285	C₂²	C₂²×#11924007	1	1	yes	characteristic
286	C₂²	C₂×#14556582	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	#17988452	1	2	yes	—
291	C₄	#17988452	1	2	yes	—
292	C₄	#17988635	1	4	yes	—
293	C₄	C₂×#14556549	1	2	yes	—
294	C₄	C₂×#14556578	1	1	yes	characteristic
295	C₄	C₂×#14556580	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	C₂×#17988452	1	1	yes	characteristic
300	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	#20769
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
3. #22610071	C₂	semidirect
4. #22610068	C₂	semidirect
157. C₇₃	#20769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
C₂×C₄×#11924012	2
#22610071	2
#22610068	2
#22609657	2
C₂×#423×C₇₃	2
#20769	73