Subgroups — #34004918

Order 18688.

598 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	#22625785	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	2	yes	maximal, complemented
3	#22624761	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
6	C₂×C₂²⋊#10596833	—	2	2	no	—
7	#17989453	—	2	2	no	—
8	#17989450	—	2	4	no	—
9	C₂³×#10596831	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	—	2	4	no	—
12	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂²×#10596833	—	2	4	no	—
18	C₄⋊#10596833	—	2	4	no	—
19	C₄⋊#10596833	—	2	4	no	—
20	C₂²⋊#10596833	—	4	4	no	—
21	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
22	C₂³×Dic₇₃	Q₈	1	1	yes	characteristic
23	C₂²×#10596831	—	2	2	no	—
24	C₂²×#10596831	—	2	2	no	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	—	2	2	no	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	—	2	2	no	—
36	C₈×Dic₇₃	—	2	2	no	—
37	C₈×Dic₇₃	D₄	1	4	yes	—
38	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂²×C₈×C₇₃	—	2	2	no	—
41	C₂²×C₈×C₇₃	C₈	1	4	yes	complemented
42	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×#10596833	—	4	4	no	—
45	C₂×#10596833	—	4	4	no	—
46	C₂×#10596833	—	4	4	no	—
47	C₂²×Dic₇₃	—	2	2	no	—
48	C₂²×Dic₇₃	—	2	2	no	—
49	C₂²×Dic₇₃	—	2	2	no	—
50	C₂²×Dic₇₃	—	2	2	no	—
51	C₂²×Dic₇₃	—	2	2	no	—
52	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₇₃	D₈	1	2	yes	—
55	C₂²×Dic₇₃	Q₁₆	1	2	yes	—
56	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×#10596831	—	2	8	no	—
59	C₂×#10596831	—	2	1	no	—
60	C₂×#10596831	—	2	1	no	—
61	C₂×#10596831	—	2	1	no	—
62	C₂×#10596831	—	2	1	no	—
63	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
64	C₂×#10596831	M₁₆	1	2	yes	—
65	C₄×Dic₇₃	—	2	1	no	—
66	C₄×Dic₇₃	—	2	1	no	—
67	C₄×Dic₇₃	—	2	2	no	—
68	C₄×Dic₇₃	—	2	2	no	—
69	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
70	C₄×Dic₇₃	#41	1	2	yes	—
71	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	—	2	1	no	—
73	C₂²×C₄×C₇₃	—	2	1	no	—
74	C₂²×C₄×C₇₃	—	2	1	no	—
75	C₂²×C₄×C₇₃	—	2	1	no	—
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	—	2	8	no	—
83	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
87	C₂³×C₇₃	—	2	1	no	—
88	C₂³×C₇₃	—	2	1	no	—
89	C₂³×C₇₃	—	2	1	no	—
90	C₂³×C₇₃	—	2	1	no	—
91	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
92	C₂³×C₇₃	#107	1	2	yes	—
93	C₂³×C₇₃	#108	1	2	yes	—
94	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	—	2	1	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×Dic₇₃	—	2	8	no	—
109	C₂×Dic₇₃	—	2	1	no	—
110	C₂×Dic₇₃	—	2	1	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	2	1	no	—
113	C₂×Dic₇₃	—	2	1	no	—
114	C₂×Dic₇₃	—	2	1	no	—
115	C₂×Dic₇₃	—	2	4	no	—
116	C₂×Dic₇₃	—	2	4	no	—
117	C₂×Dic₇₃	#107	1	2	yes	—
118	C₂×Dic₇₃	#108	1	2	yes	—
119	C₂×Dic₇₃	#121	1	2	yes	—
120	C₂×Dic₇₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×Dic₇₃	#135	1	2	yes	—
124	#10596833	—	8	4	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	C₂²⋊C₈	1	4	yes	complemented
127	#10596831	—	2	2	no	—
128	#10596831	—	2	2	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	#335	1	4	yes	complemented
136	C₂²×C₇₃	#380	1	1	yes	characteristic
137	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₇₃	#423	1	1	yes	characteristic
139	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
140	C₂²×C₇₃	#427	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
143	C₄×C₇₃	—	2	2	no	—
144	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₇₃	#407	1	1	yes	characteristic
146	C₄×C₇₃	#431	1	1	yes	characteristic
147	C₄×C₇₃	#432	1	1	yes	characteristic
148	Dic₇₃	—	2	2	no	—
149	Dic₇₃	—	2	2	no	—
150	Dic₇₃	—	2	2	no	—
151	Dic₇₃	#466	1	2	yes	—
152	Dic₇₃	#467	1	2	yes	—
153	#8596	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	—	2	4	no	—
155	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
156	C₂×C₇₃	#1564	1	1	yes	characteristic
157	C₂×C₇₃	#1567	1	1	yes	characteristic
158	C₂×C₇₃	#1568	1	1	yes	characteristic
159	C₂×C₇₃	#1797	1	1	yes	characteristic
160	C₂×C₇₃	#1808	1	1	yes	characteristic
161	C₂×C₇₃	#1810	1	1	yes	characteristic
162	C₂²×C₄×C₈	—	73	1	no	—
163	#1797	—	73	2	no	—
164	C₂×#335	—	73	2	no	—
165	#1277	—	73	2	no	—
166	C₇₃	#8596	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₈	—	73	1	no	—
168	C₂³×C₈	#64408	1	1	yes	characteristic
169	C₂²×C₄²	—	73	1	no	—
170	C₂×C₄⋊C₈	—	73	4	no	—
171	C₂×C₂²⋊C₈	—	146	2	no	—
172	C₂×C₄×C₈	—	73	1	no	—
173	C₂×C₄×C₈	—	73	1	no	—
174	C₂×C₄×C₈	—	73	1	no	—
175	C₂×C₄×C₈	—	73	1	no	—
176	C₂×C₄×C₈	—	146	4	no	—
177	#336	—	146	2	no	—
178	#335	—	146	4	no	—
179	C₂³×C₄	—	73	1	no	—
180	C₂³×C₄	—	73	1	no	—
181	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	73	1	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	146	4	no	—
187	C₂²×C₈	—	146	2	no	—
188	C₂²×C₈	—	146	2	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	#10596833	1	4	yes	complemented
191	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
192	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
193	C₂×C₄²	—	73	1	no	—
194	C₂×C₄²	—	73	1	no	—
195	C₂×C₄²	—	73	1	no	—
196	C₂×C₄²	—	73	1	no	—
197	C₂×C₄²	—	146	2	no	—
198	C₂×C₄²	—	146	2	no	—
199	C₄⋊C₈	—	146	4	no	—
200	C₄⋊C₈	—	146	4	no	—
201	C₂²⋊C₈	—	292	4	no	—
202	C₄×C₈	—	73	4	no	—
203	C₄×C₈	—	146	2	no	—
204	C₄×C₈	—	146	2	no	—
205	C₄×C₈	—	146	2	no	—
206	C₂⁴	C₄⋊#64408	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	73	1	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	2	no	—
214	C₂²×C₄	—	73	1	no	—
215	C₂²×C₄	—	73	1	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	146	2	no	—
218	C₂²×C₄	—	146	2	no	—
219	C₂²×C₄	—	146	2	no	—
220	C₂²×C₄	—	146	2	no	—
221	C₂²×C₄	—	146	2	no	—
222	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
223	C₂²×C₄	C₂×#10596833	1	2	yes	—
224	C₂²×C₄	#11924038	1	2	yes	—
225	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	292	4	no	—
230	C₂×C₈	—	292	4	no	—
231	C₂×C₈	—	292	4	no	—
232	C₂×C₈	—	73	2	no	—
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	146	8	no	—
235	C₂×C₈	—	146	1	no	—
236	C₂×C₈	—	146	1	no	—
237	C₂×C₈	—	146	1	no	—
238	C₂×C₈	—	146	1	no	—
239	C₂×C₈	C₂²⋊#64408	1	2	yes	—
240	C₂×C₈	C₂×#10596833	1	2	yes	—
241	C₂×C₈	C₂×#10596833	1	2	yes	—
242	C₂×C₈	#11924037	1	2	yes	—
243	C₄²	—	73	2	no	—
244	C₄²	—	73	2	no	—
245	C₄²	—	146	2	no	—
246	C₄²	—	146	2	no	—
247	C₄²	—	146	1	no	—
248	C₄²	—	146	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	C₄⋊#10596833	1	2	yes	—
254	C₂³	#14556660	1	2	yes	—
255	C₂³	#14556662	1	2	yes	—
256	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	73	2	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	73	2	no	—
267	C₂×C₄	—	73	2	no	—
268	C₂×C₄	—	73	2	no	—
269	C₂×C₄	—	73	2	no	—
270	C₂×C₄	—	146	4	no	—
271	C₂×C₄	—	146	4	no	—
272	C₂×C₄	—	146	8	no	—
273	C₂×C₄	—	146	1	no	—
274	C₂×C₄	—	146	1	no	—
275	C₂×C₄	—	146	1	no	—
276	C₂×C₄	—	146	1	no	—
277	C₂×C₄	—	146	1	no	—
278	C₂×C₄	—	146	1	no	—
279	C₂×C₄	C₄⋊#10596833	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
281	C₂×C₄	#14556670	1	1	yes	characteristic
282	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
283	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
284	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	584	4	no	—
288	C₈	—	146	2	no	—
289	C₈	—	146	2	no	—
290	C₈	C₂²⋊#10596833	1	4	yes	complemented
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	#17989450	1	4	yes	complemented
298	C₂²	#17989529	1	1	yes	characteristic
299	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
300	C₂²	#17989590	1	1	yes	characteristic
301	C₂²	C₂×#14556660	1	1	yes	characteristic
302	C₂²	#17989597	1	1	yes	characteristic
303	C₂²	D₇₃⋊#108	1	1	yes	characteristic
304	C₂²	C₂×#14556662	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	73	2	no	—
307	C₄	—	73	2	no	—
308	C₄	—	146	2	no	—
309	C₄	—	146	2	no	—
310	C₄	—	146	2	no	—
311	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
312	C₄	#17989558	1	1	yes	characteristic
313	C₄	#17989603	1	1	yes	characteristic
314	C₄	#17989608	1	1	yes	characteristic
315	C₂	—	2	4	no	—
316	C₂	C₂×#17989450	1	1	yes	characteristic
317	C₂	#22625478	1	1	yes	characteristic
318	C₂	#22625488	1	1	yes	characteristic
319	C₂	#22625493	1	1	yes	characteristic
320	C₂	#22625785	1	1	yes	characteristic
321	C₂	#22625808	1	1	yes	characteristic
322	C₂	#22625811	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	#8596
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17989450	C₂	semidirect
3. #22624761	C₂	semidirect
41. C₂²×C₈×C₇₃	C₈	semidirect
126. C₈×C₇₃	C₂²⋊C₈	semidirect
135. C₂²×C₇₃	#335	semidirect
166. C₇₃	#8596	semidirect
190. C₂²×C₈	#10596833	semidirect
290. C₈	C₂²⋊#10596833	semidirect
297. C₂²	#17989450	semidirect

Maximal subgroups

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

subgroup	index
#22625785	2
C₂×#17989450	2
#22624761	2
C₂²×C₈×Dic₇₃	2
#8596	73