Subgroups — C₂×#22622820 #33966439

Order 18688.

1766 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989316	C₂	1	2	yes	maximal
2	#22622820	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988988	C₂	1	2	yes	maximal
5	C₂×#17988896	C₂	1	4	yes	maximal
6	C₂×#17988872	C₂	1	2	yes	maximal
7	C₂×C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₇₃	C₂	1	2	yes	maximal
9	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
10	#17989316	—	2	8	no	—
11	#17989316	C₂²	1	16	yes	—
12	C₂×#14556591	—	2	2	no	—
13	C₂×#14556591	C₂²	1	2	yes	—
14	C₂×#14556582	—	2	4	no	—
15	C₂×#14556582	C₂²	1	4	yes	—
16	C₂×Q₈×Dic₇₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₄₆	—	2	2	no	—
18	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
19	#17988992	C₂²	1	8	yes	—
20	#17988988	C₂²	1	16	yes	—
21	#17988896	C₂²	1	32	yes	—
22	#17988872	C₂²	1	16	yes	—
23	C₈×Dic₁₄₆	C₂²	1	8	yes	—
24	Q₁₆×Dic₇₃	C₂²	1	16	yes	—
25	C₂×#14556525	C₂²	1	2	yes	—
26	C₂×#14556525	C₂²	1	2	yes	—
27	C₂×#14556522	C₂²	1	4	yes	—
28	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
29	C₂×#14556520	C₂²	1	2	yes	—
30	C₂×#14556515	C₂²	1	4	yes	—
31	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
34	C₄×Q₁₆×C₇₃	C₂²	1	8	yes	—
35	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×#103×C₇₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
38	C₂²×Dic₁₄₆	—	2	2	no	—
39	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
40	C₂²×#10596831	C₂³	1	2	yes	—
41	#14556591	—	2	16	no	—
42	#14556591	—	2	4	no	—
43	#14556591	C₂³	1	8	yes	—
44	#14556582	—	2	32	no	—
45	#14556582	—	2	8	no	—
46	#14556582	C₂³	1	16	yes	—
47	Q₈×Dic₇₃	—	2	8	no	—
48	Q₈×Dic₇₃	C₂³	1	16	yes	—
49	C₄×Dic₁₄₆	—	2	16	no	—
50	C₄×Dic₁₄₆	D₄	1	4	yes	—
51	C₄×Dic₁₄₆	C₂³	1	4	yes	—
52	C₂×C₄⋊Dic₇₃	—	2	4	no	—
53	C₂×C₄⋊Dic₇₃	—	2	4	no	—
54	C₂×C₄⋊Dic₇₃	—	2	2	no	—
55	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
57	C₂×#11924007	—	2	4	no	—
58	C₂×#11924007	D₄	1	2	yes	—
59	C₂×C₄×Dic₇₃	—	2	4	no	—
60	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
61	#14556525	C₂³	1	8	yes	—
62	#14556525	C₂³	1	8	yes	—
63	#14556522	C₂³	1	16	yes	—
64	C₄⋊#10596831	C₂³	1	4	yes	—
65	#14556520	C₂³	1	8	yes	—
66	#14556515	C₂³	1	16	yes	—
67	C₈×Dic₇₃	C₂³	1	8	yes	—
68	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₇₃	Q₈	1	8	yes	complemented
70	C₂×Q₁₆×C₇₃	C₂³	1	4	yes	—
71	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₇₃	—	2	4	no	—
74	C₄×Q₈×C₇₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₇₃	—	2	2	no	—
78	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
79	#107×C₇₃	C₂³	1	4	yes	—
80	#103×C₇₃	C₂³	1	8	yes	—
81	C₄×C₈×C₇₃	C₂³	1	4	yes	—
82	C₂×Dic₁₄₆	—	2	4	no	—
83	C₂×Dic₁₄₆	—	2	8	no	—
84	C₂×Dic₁₄₆	—	4	8	no	—
85	C₂×Dic₁₄₆	C₂×D₄	1	4	yes	—
86	C₂²×Dic₇₃	—	2	4	no	—
87	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
88	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
89	C₂×#10596831	—	2	4	no	—
90	C₂×#10596831	C₂⁴	1	4	yes	—
91	C₄⋊Dic₇₃	—	2	8	no	—
92	C₄⋊Dic₇₃	—	2	4	no	—
93	C₄⋊Dic₇₃	—	2	16	no	—
94	C₄⋊Dic₇₃	—	2	16	no	—
95	C₄⋊Dic₇₃	C₂×D₄	1	4	yes	—
96	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
97	#11924007	—	2	16	no	—
98	#11924007	C₂×D₄	1	8	yes	—
99	C₄×Dic₇₃	—	2	16	no	—
100	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
101	C₄×Dic₇₃	C₂⁴	1	4	yes	—
102	C₂×Q₈×C₇₃	—	2	2	no	—
103	C₂×Q₈×C₇₃	—	2	4	no	—
104	C₂×Q₈×C₇₃	C₂×Q₈	1	8	yes	—
105	C₂×Q₈×C₇₃	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₇₃	—	2	2	no	—
107	C₂²×C₄×C₇₃	—	2	2	no	—
108	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
111	Q₁₆×C₇₃	C₂×Q₈	1	16	yes	complemented
112	C₂×C₈×C₇₃	—	2	2	no	—
113	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
114	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
115	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
116	C₄⋊C₄×C₇₃	—	2	8	no	—
117	C₄⋊C₄×C₇₃	—	2	2	no	—
118	C₄⋊C₄×C₇₃	C₂⁴	1	4	yes	—
119	C₄²×C₇₃	—	2	8	no	—
120	C₄²×C₇₃	C₂×D₄	1	2	yes	—
121	C₄²×C₇₃	C₂⁴	1	2	yes	—
122	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₇₃	—	2	8	no	—
124	Q₈×C₇₃	—	2	4	no	—
125	Q₈×C₇₃	C₂²×Q₈	1	8	yes	—
126	C₂×C₄×C₇₃	—	2	8	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	4	no	—
129	C₂×C₄×C₇₃	—	2	4	no	—
130	C₂×C₄×C₇₃	—	4	4	no	—
131	C₂×C₄×C₇₃	#135	1	4	yes	—
132	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
137	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
138	C₂×Dic₇₃	—	2	4	no	—
139	C₂×Dic₇₃	—	2	4	no	—
140	C₂×Dic₇₃	—	2	8	no	—
141	C₂×Dic₇₃	—	4	8	no	—
142	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
143	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
144	Dic₁₄₆	—	2	8	no	—
145	Dic₁₄₆	—	4	16	no	—
146	C₈×C₇₃	—	2	2	no	—
147	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
148	#10596831	—	2	4	no	—
149	C₂²×C₇₃	D₄×Q₈	1	2	yes	—
150	C₂²×C₇₃	C₂×#135	1	2	yes	—
151	C₂²×C₇₃	#578	1	2	yes	—
152	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
153	C₄×C₇₃	—	2	8	no	—
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	—	4	4	no	—
156	C₄×C₇₃	D₄×Q₈	1	2	yes	—
157	C₄×C₇₃	C₂×#135	1	4	yes	—
158	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
159	C₄×C₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₇₃	—	2	4	no	—
161	Dic₇₃	—	2	4	no	—
162	Dic₇₃	—	4	8	no	—
163	C₂×#3383	—	73	1	no	maximal, Sylow
164	C₂×C₇₃	#3383	1	4	yes	complemented
165	C₂×C₇₃	C₂×D₄×Q₈	1	1	yes	characteristic
166	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#578	1	1	yes	characteristic
168	C₂×#557	—	73	2	no	—
169	#3383	—	73	16	no	—
170	C₂×#500	—	73	1	no	—
171	C₂×#499	—	73	2	no	—
172	C₂×#479	—	73	4	no	—
173	C₂×#475	—	73	2	no	—
174	C₂×C₈×Q₈	—	73	1	no	—
175	C₂×C₄×Q₁₆	—	73	2	no	—
176	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
177	C₇₃	C₂×#3383	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
179	#557	—	73	16	no	—
180	#557	—	146	8	no	—
181	C₂×C₄⋊Q₈	—	73	2	no	—
182	C₂×C₄⋊Q₈	—	146	2	no	—
183	C₂×#125	—	73	4	no	—
184	C₂×#125	—	146	4	no	—
185	C₂×C₄×Q₈	—	73	1	no	—
186	C₂×C₄×Q₈	—	73	4	no	—
187	C₂×C₄×Q₈	—	146	2	no	—
188	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
189	#500	—	73	8	no	—
190	#499	—	73	16	no	—
191	#479	—	73	32	no	—
192	#475	—	73	16	no	—
193	C₈×Q₈	—	73	8	no	—
194	C₄×Q₁₆	—	73	16	no	—
195	C₄×Q₁₆	C₂×D₇₃	1	8	yes	—
196	C₂×#107	—	73	2	no	—
197	C₂×#107	—	73	2	no	—
198	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
199	C₂×#106	—	73	4	no	—
200	C₂×C₄⋊C₈	—	73	2	no	—
201	C₂×C₄⋊C₈	—	73	1	no	—
202	C₂×#103	—	73	4	no	—
203	C₂×#103	C₂×D₇₃	1	2	yes	—
204	C₂×C₄×C₈	—	73	2	no	—
205	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
206	C₂²×Q₈	—	73	1	no	—
207	C₂²×Q₈	—	146	2	no	—
208	C₂²×Q₈	C₂²×D₇₃	1	2	yes	—
209	C₂×Q₁₆	Dic₁₄₆	1	8	yes	complemented
210	C₂×Q₁₆	C₂²×D₇₃	1	4	yes	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	73	8	no	—
215	C₄⋊Q₈	—	146	16	no	—
216	C₄⋊Q₈	—	146	4	no	—
217	#125	—	73	16	no	—
218	#125	—	146	8	no	—
219	#125	—	146	32	no	—
220	C₄×Q₈	—	2	4	no	—
221	C₄×Q₈	—	73	4	no	—
222	C₄×Q₈	—	73	4	no	—
223	C₄×Q₈	—	73	16	no	—
224	C₄×Q₈	—	146	16	no	—
225	C₄×Q₈	—	146	8	no	—
226	C₄×Q₈	C₂²×D₇₃	1	8	yes	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	73	1	no	—
229	C₂×C₄⋊C₄	—	73	2	no	—
230	C₂×C₄⋊C₄	—	73	4	no	—
231	C₂×C₄⋊C₄	—	146	4	no	—
232	C₂×C₄⋊C₄	—	146	4	no	—
233	C₂×C₄⋊C₄	—	146	4	no	—
234	C₂×C₄⋊C₄	—	146	2	no	—
235	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	—	73	2	no	—
238	C₂×C₄²	—	146	4	no	—
239	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
240	#107	—	73	8	no	—
241	#107	—	73	8	no	—
242	#107	C₂²×D₇₃	1	4	yes	—
243	#106	—	73	16	no	—
244	C₄⋊C₈	—	73	8	no	—
245	C₄⋊C₈	—	73	4	no	—
246	#103	—	73	16	no	—
247	#103	C₂²×D₇₃	1	8	yes	—
248	C₄×C₈	—	73	8	no	—
249	C₄×C₈	C₂²×D₇₃	1	4	yes	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	292	8	no	—
253	C₂×Q₈	—	73	4	no	—
254	C₂×Q₈	—	146	4	no	—
255	C₂×Q₈	—	146	8	no	—
256	C₂×Q₈	C₂×Dic₁₄₆	1	8	yes	—
257	C₂×Q₈	C₂³×D₇₃	1	4	yes	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	146	4	no	—
263	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
264	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
266	Q₁₆	C₂×Dic₁₄₆	1	16	yes	complemented
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	73	4	no	—
269	C₂×C₈	—	146	4	no	—
270	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
271	C₂×C₈	C₂³×D₇₃	1	2	yes	—
272	C₂×C₈	C₂³×D₇₃	1	2	yes	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	73	8	no	—
276	C₄⋊C₄	—	73	4	no	—
277	C₄⋊C₄	—	73	8	no	—
278	C₄⋊C₄	—	146	16	no	—
279	C₄⋊C₄	—	146	16	no	—
280	C₄⋊C₄	—	146	16	no	—
281	C₄⋊C₄	—	146	4	no	—
282	C₄⋊C₄	—	146	8	no	—
283	C₄⋊C₄	C₂³×D₇₃	1	4	yes	—
284	C₄²	—	2	8	no	—
285	C₄²	—	73	4	no	—
286	C₄²	—	73	4	no	—
287	C₄²	—	146	16	no	—
288	C₄²	D₄×D₇₃	1	2	yes	—
289	C₄²	C₂³×D₇₃	1	2	yes	—
290	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	2	8	no	—
293	Q₈	—	292	16	no	—
294	Q₈	—	146	8	no	—
295	Q₈	C₂²×Dic₁₄₆	1	8	yes	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	292	8	no	—
302	C₂×C₄	—	73	4	no	—
303	C₂×C₄	—	73	4	no	—
304	C₂×C₄	—	146	8	no	—
305	C₂×C₄	—	146	4	no	—
306	C₂×C₄	—	146	4	no	—
307	C₂×C₄	#14556614	1	4	yes	—
308	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
309	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
311	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
312	C₂×C₄	#14556638	1	2	yes	—
313	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	146	4	no	—
316	C₈	C₂²×Dic₁₄₆	1	4	yes	—
317	C₂²	D₄×Dic₁₄₆	1	2	yes	—
318	C₂²	C₂×#14556614	1	2	yes	—
319	C₂²	#17989401	1	2	yes	—
320	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
321	C₄	—	2	8	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	292	8	no	—
325	C₄	—	146	4	no	—
326	C₄	—	146	4	no	—
327	C₄	D₄×Dic₁₄₆	1	2	yes	—
328	C₄	C₂×#14556614	1	4	yes	—
329	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
330	C₄	C₂×#14556638	1	1	yes	characteristic
331	C₂	#22622820	1	4	yes	direct product factor
332	C₂	C₂×D₄×Dic₁₄₆	1	1	yes	characteristic
333	C₂	C₂²×#14556614	1	1	yes	characteristic
334	C₂	C₂×#17989401	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	C₂×#3383
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22622820	C₂	direct (N × H)
69. C₂×Q₁₆×C₇₃	Q₈	semidirect
111. Q₁₆×C₇₃	C₂×Q₈	semidirect
164. C₂×C₇₃	#3383	semidirect
177. C₇₃	C₂×#3383	semidirect
209. C₂×Q₁₆	Dic₁₄₆	semidirect
266. Q₁₆	C₂×Dic₁₄₆	semidirect
331. C₂	#22622820	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×#17989316	2
#22622820	2
C₂×#17988992	2
C₂×#17988988	2
C₂×#17988896	2
C₂×#17988872	2
C₂×C₈×Dic₁₄₆	2
C₂×Q₁₆×Dic₇₃	2
C₂×C₄×Q₁₆×C₇₃	2
C₂×#3383	73

Subgroups — C2×#22622820 #33966439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22622820 #33966439