Subgroups — #32991794

Order 18688.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 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×C₄²×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988346	C₂	1	4	yes	maximal
4	C₇₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₇₃	—	2	2	no	—
7	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
8	#17988611	—	2	2	no	—
9	#17988611	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10596831	—	2	2	no	—
11	C₄²×Dic₇₃	—	2	2	no	—
12	C₄²×Dic₇₃	C₄	1	4	yes	complemented
13	#17988346	—	2	8	no	—
14	C₂×#117×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	#384×C₇₃	C₄	1	2	yes	complemented
17	#382×C₇₃	C₄	1	2	yes	complemented
18	#380×C₇₃	—	2	2	no	—
19	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
20	C₂³×Dic₇₃	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	4	no	—
23	C₂×#11924012	—	2	4	no	—
24	C₂×#11924012	C₂×C₄	1	4	yes	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂×C₄×Dic₇₃	—	2	16	no	—
27	C₂×C₄×Dic₇₃	—	2	8	no	—
28	C₂×C₄×Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	—	2	2	no	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₇₃	D₄	1	4	yes	—
33	C₂×C₄×Dic₇₃	Q₈	1	4	yes	—
34	C₄⋊#10596831	—	2	8	no	—
35	C₂³×C₄×C₇₃	D₄	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	#117×C₇₃	—	2	4	no	—
38	#117×C₇₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
42	C₂×C₄²×C₇₃	—	2	2	no	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	D₄	1	2	yes	—
48	C₂×C₄²×C₇₃	D₄	1	2	yes	—
49	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
50	#95×C₇₃	—	2	4	no	—
51	C₂²×Dic₇₃	—	2	16	no	—
52	C₂²×Dic₇₃	—	2	4	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	2	4	no	—
55	#11924012	—	2	8	no	—
56	#11924012	—	4	4	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	4	2	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₄×Dic₇₃	—	2	32	no	—
61	C₄×Dic₇₃	—	2	4	no	—
62	C₄×Dic₇₃	—	2	4	no	—
63	C₄×Dic₇₃	—	2	4	no	—
64	C₄×Dic₇₃	C₄⋊C₄	1	8	yes	complemented
65	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	—	2	8	no	—
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
81	C₂²×C₄×C₇₃	#41	1	2	yes	—
82	C₄⋊C₄×C₇₃	—	2	8	no	—
83	C₄⋊C₄×C₇₃	—	4	8	no	—
84	C₂²⋊C₄×C₇₃	—	4	4	no	—
85	C₄²×C₇₃	—	4	2	no	—
86	C₄²×C₇₃	—	4	2	no	—
87	C₄²×C₇₃	C₄²	1	2	yes	—
88	C₄²×C₇₃	C₄²	1	2	yes	—
89	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
91	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
93	C₂³×C₇₃	—	2	4	no	—
94	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₇₃	#121	1	2	yes	—
98	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₇₃	—	2	4	no	—
100	C₂×C₄×C₇₃	—	2	8	no	—
101	C₂×C₄×C₇₃	—	2	8	no	—
102	C₂×C₄×C₇₃	—	4	4	no	—
103	C₂×C₄×C₇₃	—	4	8	no	—
104	C₂×C₄×C₇₃	#95	1	4	yes	—
105	C₂×C₄×C₇₃	#95	1	4	yes	—
106	C₂×C₄×C₇₃	#104	1	8	yes	—
107	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#117	1	2	yes	—
113	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	#121	1	2	yes	—
118	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#125	1	2	yes	—
122	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
125	C₂×Dic₇₃	—	2	16	no	—
126	C₂×Dic₇₃	—	2	16	no	—
127	C₂×Dic₇₃	—	2	16	no	—
128	C₂×Dic₇₃	—	2	8	no	—
129	#10596831	—	4	4	no	—
130	C₂²×C₇₃	—	2	4	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	4	no	—
134	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
135	C₂²×C₇₃	#380	1	2	yes	—
136	C₂²×C₇₃	#389	1	2	yes	—
137	C₂²×C₇₃	#390	1	1	yes	characteristic
138	C₂²×C₇₃	#391	1	1	yes	characteristic
139	C₂²×C₇₃	C₂×#104	1	4	yes	—
140	C₄×C₇₃	—	2	4	no	—
141	C₄×C₇₃	—	8	4	no	—
142	C₄×C₇₃	#339	1	8	yes	complemented
143	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
144	C₄×C₇₃	#382	1	1	yes	characteristic
145	C₄×C₇₃	#384	1	1	yes	characteristic
146	C₄×C₇₃	#389	1	2	yes	—
147	C₄×C₇₃	#390	1	1	yes	characteristic
148	C₄×C₇₃	#391	1	1	yes	characteristic
149	Dic₇₃	—	2	16	no	—
150	#10839	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	2	2	no	—
152	C₂×C₇₃	—	2	2	no	—
153	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
154	C₂×C₇₃	C₂×#339	1	2	yes	—
155	C₂×C₇₃	#1962	1	2	yes	—
156	C₂×C₇₃	#1994	1	2	yes	—
157	C₂×#423	—	73	1	no	—
158	#2285	D₇₃	1	1	yes	characteristic
159	C₂×C₄³	—	73	1	no	—
160	C₂×#339	—	73	4	no	—
161	C₇₃	#10839	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	73	2	no	—
163	C₂×#117	C₂×D₇₃	1	2	yes	—
164	C₂²×C₄²	—	73	2	no	—
165	C₂²×C₄²	—	146	2	no	—
166	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
167	#423	—	73	4	no	—
168	#423	—	146	2	no	—
169	C₂×C₄⋊C₈	—	146	2	no	—
170	#384	Dic₇₃	1	2	yes	complemented
171	#382	Dic₇₃	1	2	yes	complemented
172	#380	—	2	2	no	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄³	—	73	4	no	—
175	C₄³	—	146	2	no	—
176	#339	—	146	8	no	—
177	C₂³×C₄	—	146	2	no	—
178	C₂³×C₄	C₂²⋊D₇₃	1	2	yes	—
179	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
180	C₂×M₁₆	—	73	4	no	—
181	C₂×M₁₆	—	146	4	no	—
182	C₂×M₁₆	—	146	4	no	—
183	C₂×M₁₆	—	146	2	no	—
184	C₂²×C₈	—	146	2	no	—
185	#117	—	2	4	no	—
186	#117	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	73	4	no	—
192	C₂×C₄²	—	73	4	no	—
193	C₂×C₄²	—	73	4	no	—
194	C₂×C₄²	—	146	16	no	—
195	C₂×C₄²	—	146	8	no	—
196	C₂×C₄²	—	146	2	no	—
197	C₂×C₄²	—	146	2	no	—
198	C₂×C₄²	—	146	2	no	—
199	C₂×C₄²	Dic₁₄₆	1	2	yes	—
200	C₂×C₄²	C₄×D₇₃	1	2	yes	—
201	C₂×C₄²	C₄×D₇₃	1	2	yes	—
202	C₂×C₄²	D₂₉₂	1	2	yes	—
203	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
206	C₄⋊C₈	—	146	8	no	—
207	#95	—	2	4	no	—
208	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	146	16	no	—
212	C₂²×C₄	—	146	4	no	—
213	C₂²×C₄	—	146	4	no	—
214	C₂²×C₄	—	146	4	no	—
215	C₂²×C₄	#11924005	1	4	yes	—
216	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
218	C₂²×C₄	#11924007	1	4	yes	—
219	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₇₃	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
227	C₂²×C₄	#11924027	1	2	yes	—
228	C₂²×C₄	#11924029	1	2	yes	—
229	M₁₆	—	292	4	no	—
230	M₁₆	—	146	8	no	—
231	C₂×C₈	—	292	2	no	—
232	C₂×C₈	—	292	2	no	—
233	C₂×C₈	—	146	4	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	2	no	—
239	C₄²	—	73	8	no	—
240	C₄²	—	146	32	no	—
241	C₄²	—	146	4	no	—
242	C₄²	—	146	4	no	—
243	C₄²	—	146	4	no	—
244	C₄²	C₄×Dic₇₃	1	2	yes	—
245	C₄²	C₄×Dic₇₃	1	2	yes	—
246	C₄²	#11924005	1	4	yes	—
247	C₄²	#11924007	1	4	yes	—
248	C₄²	C₄⋊Dic₇₃	1	2	yes	—
249	C₄²	C₄⋊Dic₇₃	1	2	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	C₂×#11924005	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
253	C₂³	C₂×#11924007	1	1	yes	characteristic
254	C₂³	#14556562	1	2	yes	—
255	C₂³	#14556567	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	146	16	no	—
262	C₂×C₄	—	146	16	no	—
263	C₂×C₄	—	146	16	no	—
264	C₂×C₄	—	146	8	no	—
265	C₂×C₄	#14556498	1	4	yes	—
266	C₂×C₄	#14556498	1	4	yes	—
267	C₂×C₄	#14556518	1	8	yes	—
268	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
271	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
273	C₂×C₄	#14556549	1	2	yes	—
274	C₂×C₄	#14556553	1	2	yes	—
275	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
276	C₂×C₄	#14556556	1	2	yes	—
277	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
278	C₂×C₄	#14556566	1	2	yes	—
279	C₂×C₄	#14556571	1	2	yes	—
280	C₂×C₄	#14556578	1	1	yes	characteristic
281	C₂×C₄	#14556580	1	1	yes	characteristic
282	C₂×C₄	#14556582	1	2	yes	—
283	C₂×C₄	#14556587	1	1	yes	characteristic
284	C₂×C₄	#14556589	1	1	yes	characteristic
285	C₂×C₄	#14556590	1	2	yes	—
286	C₈	—	292	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	C₂×#14556498	1	1	yes	characteristic
292	C₂²	#17988444	1	2	yes	—
293	C₂²	#17988483	1	2	yes	—
294	C₂²	#17988490	1	1	yes	characteristic
295	C₂²	#17988494	1	1	yes	characteristic
296	C₂²	C₂×#14556518	1	4	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	8	4	no	—
299	C₄	—	146	16	no	—
300	C₄	#17988346	1	8	yes	complemented
301	C₄	C₄⋊C₄×Dic₇₃	1	2	yes	—
302	C₄	#17988452	1	1	yes	characteristic
303	C₄	#17988456	1	1	yes	characteristic
304	C₄	#17988486	1	2	yes	—
305	C₄	#17988488	1	1	yes	characteristic
306	C₄	#17988492	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#22608160	1	1	yes	characteristic
310	C₂	C₂×#17988346	1	2	yes	—
311	C₂	#22610760	1	2	yes	—
312	C₂	#22610909	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#2285	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#2285	D₇₃
hypocenter	C₇₃	#10839
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
9. #17988611	C₄	semidirect
12. C₄²×Dic₇₃	C₄	semidirect
16. #384×C₇₃	C₄	semidirect
17. #382×C₇₃	C₄	semidirect
64. C₄×Dic₇₃	C₄⋊C₄	semidirect
142. C₄×C₇₃	#339	semidirect
161. C₇₃	#10839	semidirect
170. #384	Dic₇₃	semidirect
171. #382	Dic₇₃	semidirect
300. C₄	#17988346	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×C₄²×Dic₇₃	2
C₂×#17988346	2
C₇₃×#2285	2
#10839	73