Subgroups — #32979105

Order 18688.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608355	C₂	1	2	yes	maximal
4	#22607826	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₃	—	2	1	no	—
7	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
8	C₈×D₂₉₂	—	2	1	no	—
9	C₈×D₂₉₂	—	2	1	no	—
10	C₈×D₂₉₂	C₄	1	2	yes	—
11	C₈×D₂₉₂	C₄	1	2	yes	—
12	#17988640	C₂²	1	2	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×#14556503	—	2	1	no	—
15	#17988340	—	2	1	no	—
16	#17988338	—	2	1	no	—
17	#17988314	—	2	4	no	—
18	C₂×#108×C₇₃	—	2	1	no	—
19	C₂×#108×C₇₃	—	2	1	no	—
20	#423×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂²×D₂₉₂	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₇₃	—	2	1	no	—
24	C₂²×#10596831	—	2	1	no	—
25	C₂×C₈×D₇₃	—	2	2	no	—
26	C₂×C₈×D₇₃	—	2	2	no	—
27	C₂×C₈×D₇₃	—	4	2	no	—
28	C₂×C₈×D₇₃	—	4	2	no	—
29	C₄×D₂₉₂	—	2	1	no	—
30	C₄×D₂₉₂	—	2	1	no	—
31	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
32	C₄×D₂₉₂	D₄	1	2	yes	—
33	C₂×C₄⋊Dic₇₃	Q₈	1	1	yes	characteristic
34	C₂×#11924005	—	2	1	no	—
35	C₄⋊#10596831	—	2	1	no	—
36	C₄⋊#10596831	—	2	2	no	—
37	C₄⋊#10596831	C₂×C₄	1	2	yes	—
38	C₂²⋊#10596831	—	2	2	no	—
39	#14556503	—	2	2	no	—
40	#14556503	—	2	2	no	—
41	C₄×#10596831	—	2	2	no	—
42	C₂×M₁₆×C₇₃	—	2	2	no	—
43	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
46	#108×C₇₃	—	4	2	no	—
47	#108×C₇₃	—	4	2	no	—
48	C₄⋊C₈×C₇₃	—	2	2	no	—
49	C₄⋊C₈×C₇₃	—	2	2	no	—
50	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
54	C₂³×D₇₃	—	2	1	no	—
55	C₂×D₂₉₂	—	2	1	no	—
56	C₂×D₂₉₂	—	2	4	no	—
57	C₂×D₂₉₂	C₄⋊C₄	1	2	yes	—
58	C₂²×Dic₇₃	—	2	1	no	—
59	C₂×C₄×D₇₃	—	2	2	no	—
60	C₂×C₄×D₇₃	—	2	2	no	—
61	C₂×C₄×D₇₃	—	4	1	no	—
62	C₂×C₄×D₇₃	—	4	1	no	—
63	C₂×C₄×D₇₃	—	4	1	no	—
64	C₂×C₄×D₇₃	—	4	1	no	—
65	C₂×#10596831	—	2	2	no	—
66	C₂×#10596831	—	4	1	no	—
67	C₂×#10596831	—	4	1	no	—
68	C₂×#10596831	—	4	2	no	—
69	C₈×D₇₃	—	4	4	no	—
70	C₈×D₇₃	—	4	4	no	—
71	C₄⋊Dic₇₃	—	2	1	no	—
72	C₄⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
73	#11924005	—	2	2	no	—
74	#11924005	—	2	2	no	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
78	M₁₆×C₇₃	—	4	2	no	—
79	M₁₆×C₇₃	—	4	2	no	—
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	—	4	2	no	—
83	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	D₈	1	2	yes	—
88	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
89	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₃	#41	1	1	yes	characteristic
93	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂²×D₇₃	—	2	4	no	—
95	C₂²×D₇₃	—	2	2	no	—
96	C₂²×D₇₃	—	4	2	no	—
97	C₂²×D₇₃	—	4	1	no	—
98	C₂²×D₇₃	—	4	1	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	4	1	no	—
101	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
111	C₂×Dic₇₃	—	2	2	no	—
112	C₂×Dic₇₃	—	4	1	no	—
113	C₂×Dic₇₃	—	4	1	no	—
114	D₂₉₂	—	2	4	no	—
115	D₂₉₂	—	2	2	no	—
116	D₂₉₂	#108	1	4	yes	complemented
117	C₄×D₇₃	—	4	2	no	—
118	C₄×D₇₃	—	4	2	no	—
119	C₄×D₇₃	—	4	2	no	—
120	C₄×D₇₃	—	4	2	no	—
121	C₈×C₇₃	—	2	1	no	—
122	C₈×C₇₃	—	2	1	no	—
123	C₈×C₇₃	—	8	2	no	—
124	C₈×C₇₃	#102	1	2	yes	—
125	C₈×C₇₃	#102	1	2	yes	—
126	#10596831	—	4	2	no	—
127	#10596831	—	8	2	no	—
128	C₂²×C₇₃	—	2	1	no	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	#380	1	1	yes	characteristic
131	C₂²×C₇₃	#474	1	1	yes	characteristic
132	C₂²×C₇₃	#476	1	1	yes	characteristic
133	C₂×D₇₃	—	2	4	no	—
134	C₂×D₇₃	—	4	4	no	—
135	C₂×D₇₃	—	4	4	no	—
136	C₂×D₇₃	—	4	2	no	—
137	C₂×D₇₃	—	4	2	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	—	2	1	no	—
140	C₄×C₇₃	—	2	1	no	—
141	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
142	C₄×C₇₃	#417	1	1	yes	characteristic
143	C₄×C₇₃	C₂×#108	1	2	yes	—
144	C₄×C₇₃	#459	1	1	yes	characteristic
145	C₄×C₇₃	#461	1	1	yes	characteristic
146	Dic₇₃	—	4	2	no	—
147	#8658	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	1	no	—
149	C₂×C₇₃	—	2	1	no	—
150	C₂×C₇₃	#1802	1	1	yes	characteristic
151	C₂×C₇₃	#1810	1	1	yes	characteristic
152	C₂×C₇₃	#1814	1	1	yes	characteristic
153	D₇₃	—	4	4	no	—
154	C₂×C₈×D₄	—	73	1	no	—
155	#1848	D₇₃	1	1	yes	characteristic, complemented
156	#1564	—	73	1	no	—
157	#1486	—	73	2	no	—
158	#1301	—	73	2	no	—
159	C₇₃	#8658	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	146	1	no	—
161	C₂×C₄×D₄	—	73	1	no	—
162	C₈×D₄	—	73	2	no	—
163	C₈×D₄	—	73	2	no	—
164	C₈×D₄	—	146	1	no	—
165	C₈×D₄	—	146	1	no	—
166	#431	—	73	2	no	—
167	C₂×#108	—	2	1	no	—
168	C₂×#108	—	2	1	no	—
169	#423	C₂×D₇₃	1	2	yes	—
170	C₂×C₄⋊C₈	—	73	1	no	—
171	C₂×C₂²⋊C₈	—	146	1	no	—
172	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
173	#335	—	146	1	no	—
174	#334	—	146	1	no	—
175	D₄⋊C₈	—	146	4	no	—
176	C₂²×D₄	—	73	1	no	—
177	C₂³×C₄	—	146	1	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂²×C₈	—	292	2	no	—
180	C₂²×C₈	—	292	2	no	—
181	C₂²×C₈	—	146	1	no	—
182	C₂²×C₈	—	146	2	no	—
183	C₂²×C₈	—	146	2	no	—
184	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
186	C₄×D₄	—	73	2	no	—
187	C₄×D₄	—	73	2	no	—
188	C₄×D₄	—	146	1	no	—
189	C₄×D₄	—	146	1	no	—
190	C₂×C₄⋊C₄	—	73	1	no	—
191	C₂×C₂²⋊C₄	—	146	1	no	—
192	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
193	#108	—	4	2	no	—
194	#108	—	4	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	73	2	no	—
198	C₄⋊C₈	—	146	2	no	—
199	C₄⋊C₈	—	146	1	no	—
200	C₂²⋊C₈	—	146	2	no	—
201	C₂²⋊C₈	—	146	2	no	—
202	C₂²⋊C₈	—	146	2	no	—
203	C₄×C₈	—	146	2	no	—
204	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
205	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₄×C₈	D₂₉₂	1	1	yes	characteristic
207	C₄×C₈	D₂₉₂	1	1	yes	characteristic
208	C₂⁴	—	146	1	no	—
209	C₂×D₄	—	73	2	no	—
210	C₂×D₄	—	146	4	no	—
211	C₂×D₄	—	146	1	no	—
212	C₂²×C₄	—	292	1	no	—
213	C₂²×C₄	—	292	1	no	—
214	C₂²×C₄	—	292	1	no	—
215	C₂²×C₄	—	292	1	no	—
216	C₂²×C₄	—	146	1	no	—
217	C₂²×C₄	—	146	2	no	—
218	C₂²×C₄	—	146	2	no	—
219	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
221	C₂²×C₄	#11924029	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	292	1	no	—
228	C₂×C₈	—	292	1	no	—
229	C₂×C₈	—	292	4	no	—
230	C₂×C₈	—	292	4	no	—
231	C₂×C₈	—	292	2	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	#11924005	1	1	yes	characteristic
234	C₂×C₈	#11924005	1	1	yes	characteristic
235	C₂×C₈	#11924005	1	1	yes	characteristic
236	C₂×C₈	#11924005	1	1	yes	characteristic
237	C₂×C₈	#11924013	1	2	yes	—
238	C₂×C₈	#11924015	1	2	yes	—
239	C₄⋊C₄	—	73	2	no	—
240	C₄⋊C₄	—	146	1	no	—
241	C₂²⋊C₄	—	146	2	no	—
242	C₂²⋊C₄	—	146	2	no	—
243	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₄²	D₄×D₇₃	1	1	yes	characteristic
245	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
246	C₄²	#11924027	1	1	yes	characteristic
247	C₂³	—	292	2	no	—
248	C₂³	—	292	1	no	—
249	C₂³	—	292	1	no	—
250	C₂³	—	146	4	no	—
251	C₂³	—	146	2	no	—
252	C₂³	#14556567	1	1	yes	characteristic
253	D₄	—	73	4	no	—
254	D₄	—	146	4	no	—
255	D₄	—	146	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	292	1	no	—
259	C₂×C₄	—	292	1	no	—
260	C₂×C₄	—	292	2	no	—
261	C₂×C₄	—	292	2	no	—
262	C₂×C₄	—	292	2	no	—
263	C₂×C₄	—	292	2	no	—
264	C₂×C₄	—	146	2	no	—
265	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₄	#14556549	1	1	yes	characteristic
268	C₂×C₄	#14556562	1	1	yes	characteristic
269	C₂×C₄	#14556566	1	1	yes	characteristic
270	C₂×C₄	#14556571	1	1	yes	characteristic
271	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
272	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
273	C₂×C₄	#14556617	1	1	yes	characteristic
274	C₂×C₄	#14556622	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	292	2	no	—
279	C₈	—	584	2	no	—
280	C₈	#14556511	1	2	yes	—
281	C₈	#14556511	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	292	4	no	—
285	C₂²	—	292	4	no	—
286	C₂²	—	292	2	no	—
287	C₂²	—	292	2	no	—
288	C₂²	—	146	4	no	—
289	C₂²	#17988444	1	1	yes	characteristic
290	C₂²	#17988864	1	1	yes	characteristic
291	C₂²	#17988874	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	292	2	no	—
296	C₄	C₂×#14556511	1	1	yes	characteristic
297	C₄	#17988586	1	1	yes	characteristic
298	C₄	#108×D₇₃	1	2	yes	—
299	C₄	#17988779	1	1	yes	characteristic
300	C₄	#17988789	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	292	4	no	—
304	C₂	#22609928	1	1	yes	characteristic
305	C₂	#22609966	1	1	yes	characteristic
306	C₂	#22609986	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556562
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂²×D₂₉₂	Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#8658
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #22608679	C₂	semidirect
4. #22607826	C₂	semidirect
5. C₇₃×#1848	C₂	semidirect
116. D₂₉₂	#108	semidirect
155. #1848	D₇₃	semidirect
159. C₇₃	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₉₂	2
#22608679	2
#22608355	2
#22607826	2
C₇₃×#1848	2
#8658	73