Subgroups — C₂×#22608721 #33055070

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988639	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988638	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988609	C₂	1	1	yes	maximal, characteristic
4	#22608721	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988339	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988338	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988337	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924011	—	2	1	no	—
10	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
11	#17988639	—	2	2	no	—
12	#17988639	C₂²	1	4	yes	—
13	#17988638	C₄	1	4	yes	—
14	#17988638	C₂²	1	4	yes	—
15	#17988609	—	2	2	no	—
16	#17988609	C₂²	1	4	yes	—
17	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	C₂×#14556520	—	2	1	no	—
20	C₂×#14556520	C₄	1	1	yes	characteristic
21	C₂×#14556520	C₄	1	1	yes	characteristic
22	C₂×#14556503	—	2	1	no	—
23	C₂×#14556503	C₄	1	1	yes	characteristic
24	C₂×#14556503	C₄	1	1	yes	characteristic
25	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₇₃	—	2	1	no	—
27	#17988339	C₂²	1	4	yes	complemented
28	#17988338	C₂²	1	4	yes	complemented
29	#17988337	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	#335×C₇₃	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
37	C₂×#11924011	—	2	4	no	—
38	C₂×#11924011	—	4	4	no	—
39	C₂²×#10596831	—	2	1	no	—
40	C₂²×#10596831	—	2	1	no	—
41	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
42	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
46	C₄²×D₇₃	C₂×C₄	1	2	yes	—
47	C₄²×D₇₃	D₄	1	2	yes	—
48	C₄²×D₇₃	Q₈	1	2	yes	—
49	C₄²×D₇₃	C₂³	1	2	yes	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	—	2	1	no	—
52	C₄⋊#10596831	C₂³	1	2	yes	—
53	C₄⋊#10596831	C₂³	1	2	yes	—
54	#14556520	—	2	4	no	—
55	#14556520	C₂×C₄	1	4	yes	—
56	#14556520	C₂×C₄	1	4	yes	—
57	#14556503	—	2	4	no	—
58	#14556503	C₂×C₄	1	4	yes	—
59	#14556503	C₂×C₄	1	4	yes	—
60	C₄×#10596831	C₂×C₄	1	2	yes	—
61	C₄×#10596831	C₂³	1	2	yes	—
62	C₈×Dic₇₃	—	2	4	no	—
63	C₂²×C₈×C₇₃	—	2	1	no	—
64	C₂²×C₈×C₇₃	—	2	1	no	—
65	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₇₃	—	2	1	no	—
69	C₄⋊C₈×C₇₃	—	2	1	no	—
70	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
71	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₇₃	C₂³	1	2	yes	—
74	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	—	2	2	no	—
79	C₂×C₄×D₇₃	—	2	2	no	—
80	C₂×C₄×D₇₃	—	2	2	no	—
81	C₂×C₄×D₇₃	—	2	2	no	—
82	C₂×C₄×D₇₃	—	2	4	no	—
83	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
89	#11924011	—	4	8	no	—
90	C₂×#10596831	—	2	2	no	—
91	C₂×#10596831	—	2	2	no	—
92	C₂×#10596831	—	2	2	no	—
93	C₂×#10596831	—	2	2	no	—
94	C₂×#10596831	—	4	2	no	—
95	C₂×#10596831	—	4	2	no	—
96	C₂×#10596831	C₂²×C₄	1	2	yes	—
97	C₂×#10596831	C₂²×C₄	1	2	yes	—
98	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
100	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
103	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	—	2	2	no	—
109	C₂×C₈×C₇₃	—	2	2	no	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	—	4	2	no	—
112	C₂×C₈×C₇₃	—	4	2	no	—
113	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
118	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
119	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×D₇₃	—	2	2	no	—
121	C₂²×D₇₃	—	2	2	no	—
122	C₂²×D₇₃	—	2	2	no	—
123	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#131	1	2	yes	—
129	C₂×C₄×C₇₃	#131	1	2	yes	—
130	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇₃	#137	1	2	yes	—
132	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×Dic₇₃	—	2	2	no	—
136	C₂×Dic₇₃	—	2	2	no	—
137	C₂×Dic₇₃	#108	1	4	yes	—
138	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₄×D₇₃	—	2	2	no	—
142	C₄×D₇₃	—	2	2	no	—
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	4	no	—
146	C₄×D₇₃	—	2	4	no	—
147	C₄×D₇₃	—	2	4	no	—
148	C₄×D₇₃	—	2	4	no	—
149	C₈×C₇₃	—	2	2	no	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	—	4	2	no	—
152	C₈×C₇₃	—	4	2	no	—
153	#10596831	—	2	2	no	—
154	#10596831	—	2	2	no	—
155	#10596831	—	4	2	no	—
156	#10596831	—	4	2	no	—
157	C₂²×C₇₃	#424	1	2	yes	—
158	C₂²×C₇₃	#428	1	2	yes	—
159	C₂²×C₇₃	C₂×#108	1	2	yes	—
160	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂×D₇₃	—	2	2	no	—
162	C₂×D₇₃	—	2	2	no	—
163	C₂×D₇₃	—	2	2	no	—
164	C₂×D₇₃	—	4	4	no	—
165	C₄×C₇₃	—	2	1	no	—
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	#424	1	2	yes	—
168	C₄×C₇₃	#428	1	2	yes	—
169	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
173	Dic₇₃	—	2	2	no	—
174	Dic₇₃	—	2	2	no	—
175	Dic₇₃	C₂×#108	1	4	yes	—
176	C₂×#1574	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	#1574	1	4	yes	complemented
178	C₂×C₇₃	C₂×#424	1	1	yes	characteristic
179	C₂×C₇₃	C₂×#428	1	1	yes	characteristic
180	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
181	D₇₃	—	4	2	no	—
182	C₂×#431	—	73	1	no	—
183	C₂×#431	—	73	1	no	—
184	C₂×#423	—	73	1	no	—
185	#1574	—	73	8	no	—
186	C₂×#335	—	73	1	no	—
187	C₂×#335	D₇₃	1	1	yes	characteristic, complemented
188	C₂×#334	—	73	1	no	—
189	C₂×#334	—	73	1	no	—
190	C₇₃	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
191	C₂²×M₁₆	—	146	1	no	—
192	C₂²×C₄²	—	73	1	no	—
193	#431	—	73	4	no	—
194	#431	—	73	4	no	—
195	#431	—	73	4	no	—
196	#431	—	146	2	no	—
197	#423	—	73	4	no	—
198	#423	—	146	2	no	—
199	C₂×C₄⋊C₈	—	73	1	no	—
200	C₂×C₄⋊C₈	—	73	1	no	—
201	C₂×C₄⋊C₈	—	73	1	no	—
202	C₂×C₄⋊C₈	—	73	1	no	—
203	C₂×C₄⋊C₈	—	146	1	no	—
204	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
205	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
206	C₂×C₂²⋊C₈	—	73	1	no	—
207	C₂×C₂²⋊C₈	—	73	1	no	—
208	C₂×C₂²⋊C₈	—	146	1	no	—
209	C₂×C₄×C₈	—	73	1	no	—
210	C₂×C₄×C₈	—	146	1	no	—
211	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
212	#335	—	73	4	no	—
213	#335	C₂×D₇₃	1	4	yes	complemented
214	#334	—	73	4	no	—
215	#334	—	73	4	no	—
216	C₂³×C₄	—	73	1	no	—
217	C₂³×C₄	—	73	1	no	—
218	C₂³×C₄	—	73	1	no	—
219	C₂×M₁₆	—	292	4	no	—
220	C₂×M₁₆	—	146	4	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	73	1	no	—
224	C₂²×C₈	—	73	1	no	—
225	C₂²×C₈	—	146	1	no	—
226	C₂²×C₈	—	146	1	no	—
227	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
228	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	73	2	no	—
233	C₂×C₄²	—	73	2	no	—
234	C₂×C₄²	—	73	2	no	—
235	C₂×C₄²	—	73	2	no	—
236	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	73	4	no	—
240	C₄⋊C₈	—	73	4	no	—
241	C₄⋊C₈	—	73	2	no	—
242	C₄⋊C₈	—	73	2	no	—
243	C₄⋊C₈	—	146	4	no	—
244	C₄⋊C₈	—	146	1	no	—
245	C₄⋊C₈	—	146	1	no	—
246	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
247	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
248	C₂²⋊C₈	—	73	4	no	—
249	C₂²⋊C₈	—	73	4	no	—
250	C₂²⋊C₈	—	146	4	no	—
251	C₄×C₈	—	73	2	no	—
252	C₄×C₈	—	73	2	no	—
253	C₄×C₈	—	146	4	no	—
254	C₄×C₈	C₄×D₇₃	1	2	yes	—
255	C₄×C₈	C₂²×D₇₃	1	2	yes	—
256	C₂⁴	—	73	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	2	no	—
263	C₂²×C₄	—	73	2	no	—
264	C₂²×C₄	—	73	2	no	—
265	C₂²×C₄	—	73	2	no	—
266	C₂²×C₄	—	146	4	no	—
267	C₂²×C₄	—	146	2	no	—
268	C₂²×C₄	—	146	2	no	—
269	C₂²×C₄	—	146	2	no	—
270	C₂²×C₄	—	146	2	no	—
271	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
273	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
274	M₁₆	—	292	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	2	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	292	2	no	—
282	C₂×C₈	—	292	2	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	73	2	no	—
285	C₂×C₈	—	146	2	no	—
286	C₂×C₈	—	146	2	no	—
287	C₂×C₈	—	146	2	no	—
288	C₂×C₈	—	146	2	no	—
289	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
291	C₄²	—	73	2	no	—
292	C₄²	—	73	2	no	—
293	C₄²	—	73	2	no	—
294	C₄²	—	73	2	no	—
295	C₄²	—	73	2	no	—
296	C₄²	—	73	2	no	—
297	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
298	C₄²	D₄×D₇₃	1	1	yes	characteristic
299	C₄²	Q₈×D₇₃	1	1	yes	characteristic
300	C₄²	C₂³×D₇₃	1	1	yes	characteristic
301	C₂³	—	73	2	no	—
302	C₂³	—	146	2	no	—
303	C₂³	—	146	2	no	—
304	C₂³	—	146	2	no	—
305	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	73	4	no	—
308	C₂×C₄	—	73	2	no	—
309	C₂×C₄	—	73	2	no	—
310	C₂×C₄	—	73	2	no	—
311	C₂×C₄	—	146	4	no	—
312	C₂×C₄	—	146	4	no	—
313	C₂×C₄	—	146	2	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	—	146	4	no	—
316	C₂×C₄	—	146	2	no	—
317	C₂×C₄	—	146	4	no	—
318	C₂×C₄	—	146	2	no	—
319	C₂×C₄	—	146	2	no	—
320	C₂×C₄	—	146	2	no	—
321	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
322	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
323	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₄	#14556597	1	2	yes	—
325	C₂×C₄	#14556599	1	2	yes	—
326	C₂×C₄	#14556615	1	2	yes	—
327	C₂×C₄	#14556623	1	2	yes	—
328	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
330	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	292	2	no	—
336	C₈	—	292	2	no	—
337	C₈	—	146	2	no	—
338	C₈	—	146	2	no	—
339	C₂²	—	292	4	no	—
340	C₂²	—	146	2	no	—
341	C₂²	—	146	2	no	—
342	C₂²	—	146	2	no	—
343	C₂²	#17988612	1	2	yes	—
344	C₂²	#17988628	1	2	yes	—
345	C₂²	#108×D₇₃	1	2	yes	—
346	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	73	4	no	—
350	C₄	—	146	2	no	—
351	C₄	—	146	2	no	—
352	C₄	#17988612	1	2	yes	—
353	C₄	#17988628	1	2	yes	—
354	C₄	C₂×#14556597	1	1	yes	characteristic
355	C₄	C₂×#14556599	1	1	yes	characteristic
356	C₄	C₂×#14556615	1	1	yes	characteristic
357	C₄	C₂×#14556623	1	1	yes	characteristic
358	C₂	—	292	2	no	—
359	C₂	#22608721	1	4	yes	direct product factor
360	C₂	C₂×#17988612	1	1	yes	characteristic
361	C₂	C₂×#17988628	1	1	yes	characteristic
362	C₂	C₂×#108×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	C₂×#1574
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #22608721	C₂	direct (N × H)
5. C₂×#17988339	C₂	semidirect
6. C₂×#17988338	C₂	semidirect
7. C₂×#17988337	C₂	semidirect
8. C₂×#335×C₇₃	C₂	semidirect
27. #17988339	C₂²	semidirect
28. #17988338	C₂²	semidirect
29. #17988337	C₂²	semidirect
33. #335×C₇₃	C₂²	semidirect
177. C₂×C₇₃	#1574	semidirect
187. C₂×#335	D₇₃	semidirect
190. C₇₃	C₂×#1574	semidirect
213. #335	C₂×D₇₃	semidirect
359. C₂	#22608721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17988639	2
C₂×#17988638	2
C₂×#17988609	2
#22608721	2
C₂×#17988339	2
C₂×#17988338	2
C₂×#17988337	2
C₂×#335×C₇₃	2
C₂×#1574	73

Subgroups — C2×#22608721 #33055070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608721 #33055070