Subgroups — C₂×#22610909 #33090602

Order 18688.

1422 conjugacy classes of subgroups.

The subgroup lattice has 320 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	#22610909	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988346	C₂	1	4	yes	maximal
5	C₂×#533×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₇₃	—	2	2	no	—
8	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
9	#17988611	—	2	2	no	—
10	#17988611	C₂²	1	4	yes	—
11	C₂×C₄⋊#10596831	—	2	2	no	—
12	C₄²×Dic₇₃	—	2	2	no	—
13	C₄²×Dic₇₃	C₂²	1	4	yes	—
14	#17988346	C₂²	1	16	yes	—
15	#533×C₇₃	—	2	2	no	—
16	#533×C₇₃	—	2	2	no	—
17	#533×C₇₃	C₄	1	4	yes	complemented
18	#533×C₇₃	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈×C₇₃	C₄	1	2	yes	complemented
20	C₂×#125×C₇₃	C₄	1	2	yes	complemented
21	C₂×C₂²⋊Q₈×C₇₃	—	2	2	no	—
22	C₂×C₄×Q₈×C₇₃	—	2	2	no	—
23	C₂×#117×C₇₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
25	C₂³×Dic₇₃	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	8	no	—
28	C₂×#11924012	C₂³	1	4	yes	—
29	C₂²×#10596831	—	2	2	no	—
30	C₂×C₄×Dic₇₃	—	2	16	no	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	—	2	8	no	—
33	C₂×C₄×Dic₇₃	—	2	4	no	—
34	C₂×C₄×Dic₇₃	Q₈	1	8	yes	complemented
35	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
36	C₄⋊#10596831	—	2	8	no	—
37	C₂²×Q₈×C₇₃	—	2	2	no	—
38	C₂³×C₄×C₇₃	D₄	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₃	—	2	4	no	—
41	C₄⋊Q₈×C₇₃	C₂×C₄	1	8	yes	complemented
42	#125×C₇₃	—	2	4	no	—
43	#125×C₇₃	C₂×C₄	1	8	yes	complemented
44	C₂²⋊Q₈×C₇₃	—	2	8	no	—
45	C₂²⋊Q₈×C₇₃	—	2	8	no	—
46	C₄×Q₈×C₇₃	—	2	8	no	—
47	C₄×Q₈×C₇₃	—	2	8	no	—
48	#117×C₇₃	—	2	2	no	—
49	#117×C₇₃	—	2	2	no	—
50	#117×C₇₃	C₂×C₄	1	4	yes	—
51	#117×C₇₃	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
56	C₂×C₄²×C₇₃	—	2	2	no	—
57	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₇₃	D₄	1	2	yes	—
61	C₂×C₄²×C₇₃	D₄	1	2	yes	—
62	C₂×C₄²×C₇₃	D₄	1	2	yes	—
63	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
64	C₂²×Dic₇₃	—	2	16	no	—
65	C₂²×Dic₇₃	—	2	8	no	—
66	C₂²×Dic₇₃	—	2	4	no	—
67	#11924012	—	2	8	no	—
68	#11924012	—	4	4	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₂×#10596831	—	4	4	no	—
71	C₄×Dic₇₃	—	2	32	no	—
72	C₄×Dic₇₃	—	2	8	no	—
73	C₄×Dic₇₃	—	2	4	no	—
74	C₄×Dic₇₃	C₂×Q₈	1	8	yes	complemented
75	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₇₃	—	2	8	no	—
77	C₂×Q₈×C₇₃	—	4	8	no	—
78	C₂²×C₄×C₇₃	—	2	8	no	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₇₃	#41	1	4	yes	—
91	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₇₃	—	2	16	no	—
93	C₄⋊C₄×C₇₃	—	2	8	no	—
94	C₄⋊C₄×C₇₃	—	2	4	no	—
95	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
96	C₂²⋊C₄×C₇₃	—	2	4	no	—
97	C₂²⋊C₄×C₇₃	—	2	4	no	—
98	C₄²×C₇₃	—	2	4	no	—
99	C₄²×C₇₃	—	2	4	no	—
100	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
103	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₇₃	C₂×D₄	1	4	yes	—
106	C₂³×C₇₃	—	2	4	no	—
107	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₇₃	C₂²⋊Q₈	1	4	yes	—
110	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₇₃	—	4	16	no	—
112	C₂×C₄×C₇₃	—	2	4	no	—
113	C₂×C₄×C₇₃	—	2	8	no	—
114	C₂×C₄×C₇₃	—	2	8	no	—
115	C₂×C₄×C₇₃	—	2	8	no	—
116	C₂×C₄×C₇₃	—	4	8	no	—
117	C₂×C₄×C₇₃	#104	1	8	yes	—
118	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
123	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₇₃	#124	1	2	yes	—
126	C₂×C₄×C₇₃	#124	1	2	yes	—
127	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
133	C₂×Dic₇₃	—	2	16	no	—
134	C₂×Dic₇₃	—	2	32	no	—
135	C₂×Dic₇₃	—	2	8	no	—
136	#10596831	—	4	4	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	8	no	—
140	C₂²×C₇₃	#391	1	2	yes	—
141	C₂²×C₇₃	#391	1	2	yes	—
142	C₂²×C₇₃	C₂×#104	1	4	yes	—
143	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
145	C₄×C₇₃	—	2	4	no	—
146	C₄×C₇₃	—	4	8	no	—
147	C₄×C₇₃	#391	1	2	yes	—
148	C₄×C₇₃	#391	1	2	yes	—
149	C₄×C₇₃	C₂×#104	1	8	yes	—
150	C₄×C₇₃	C₂×C₄×Q₈	1	2	yes	—
151	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
152	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₇₃	—	2	16	no	—
154	C₂×#1994	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	—	2	2	no	—
157	C₂×C₇₃	#1994	1	4	yes	complemented
158	C₂×C₇₃	C₂×#391	1	1	yes	characteristic
159	C₂×C₇₃	C₂²×#104	1	2	yes	—
160	C₂×#533	D₇₃	1	1	yes	characteristic
161	C₂×#423	—	73	1	no	—
162	C₂×C₄³	—	73	1	no	—
163	#1994	—	73	8	no	—
164	C₂×#339	—	73	4	no	—
165	C₇₃	C₂×#1994	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	—	73	2	no	—
167	#533	—	2	2	no	—
168	#533	—	2	2	no	—
169	#533	Dic₇₃	1	4	yes	complemented
170	#533	C₂×D₇₃	1	4	yes	—
171	C₂×C₄⋊Q₈	Dic₇₃	1	2	yes	complemented
172	C₂×#125	Dic₇₃	1	2	yes	complemented
173	C₂×C₂²⋊Q₈	—	2	2	no	—
174	C₂×C₄×Q₈	—	2	2	no	—
175	C₂×#117	C₂×D₇₃	1	2	yes	—
176	C₂²×C₄²	—	73	2	no	—
177	C₂²×C₄²	—	146	2	no	—
178	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
179	#423	—	73	4	no	—
180	#423	—	146	2	no	—
181	C₂×C₄⋊C₈	—	146	2	no	—
182	C₄³	—	73	4	no	—
183	C₄³	—	146	2	no	—
184	#339	—	73	16	no	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂³×C₄	—	146	2	no	—
187	C₂³×C₄	C₂²⋊D₇₃	1	2	yes	—
188	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
189	C₂×M₁₆	—	73	4	no	—
190	C₂×M₁₆	—	146	8	no	—
191	C₂×M₁₆	—	146	2	no	—
192	C₂²×C₈	—	146	2	no	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄⋊Q₈	C₂×Dic₇₃	1	8	yes	complemented
195	#125	—	2	4	no	—
196	#125	C₂×Dic₇₃	1	8	yes	complemented
197	C₂²⋊Q₈	—	2	8	no	—
198	C₂²⋊Q₈	—	2	8	no	—
199	C₄×Q₈	—	2	8	no	—
200	C₄×Q₈	—	2	8	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	C₂×Dic₇₃	1	4	yes	—
204	#117	C₂²×D₇₃	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	73	8	no	—
211	C₂×C₄²	—	73	4	no	—
212	C₂×C₄²	—	146	16	no	—
213	C₂×C₄²	—	146	8	no	—
214	C₂×C₄²	—	146	2	no	—
215	C₂×C₄²	—	146	4	no	—
216	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
222	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
223	C₄⋊C₈	—	146	8	no	—
224	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	146	16	no	—
230	C₂²×C₄	—	146	8	no	—
231	C₂²×C₄	—	146	4	no	—
232	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₇₃	1	4	yes	—
234	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
235	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
241	C₂²×C₄	Q₈×D₇₃	1	4	yes	—
242	C₂²×C₄	#11924029	1	4	yes	—
243	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
244	M₁₆	—	292	4	no	—
245	M₁₆	—	146	8	no	—
246	C₂×C₈	—	292	4	no	—
247	C₂×C₈	—	146	4	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂²×Dic₇₃	1	8	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	73	8	no	—
257	C₄²	—	146	32	no	—
258	C₄²	—	146	8	no	—
259	C₄²	—	146	4	no	—
260	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
261	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
262	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
263	C₄²	C₂²×Dic₇₃	1	2	yes	—
264	C₄²	C₂²×Dic₇₃	1	2	yes	—
265	C₄²	C₂×C₂²⋊D₇₃	1	4	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
269	C₂³	#14556567	1	4	yes	—
270	C₂³	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	146	16	no	—
278	C₂×C₄	—	146	32	no	—
279	C₂×C₄	—	146	8	no	—
280	C₂×C₄	#14556518	1	8	yes	—
281	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	4	yes	—
282	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	4	yes	—
285	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
286	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
287	C₂×C₄	#14556567	1	4	yes	—
288	C₂×C₄	#14556580	1	2	yes	—
289	C₂×C₄	#14556580	1	2	yes	—
290	C₂×C₄	#14556589	1	2	yes	—
291	C₂×C₄	#14556589	1	2	yes	—
292	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
295	C₂×C₄	C₂×#11924029	1	2	yes	—
296	C₈	—	292	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	#17988494	1	2	yes	—
301	C₂²	#17988494	1	2	yes	—
302	C₂²	C₂×#14556518	1	4	yes	—
303	C₂²	C₂²×C₂²⋊Dic₇₃	1	1	yes	characteristic
304	C₂²	C₂×#14556567	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	4	8	no	—
307	C₄	—	146	16	no	—
308	C₄	#17988494	1	2	yes	—
309	C₄	#17988494	1	2	yes	—
310	C₄	C₂×#14556518	1	8	yes	—
311	C₄	C₂×Q₈×Dic₇₃	1	2	yes	—
312	C₄	C₂×#14556580	1	1	yes	characteristic
313	C₄	C₂×#14556589	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#22610909	1	4	yes	direct product factor
317	C₂	C₂×#17988494	1	1	yes	characteristic
318	C₂	C₂²×#14556518	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#533×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#533	D₇₃
hypocenter	C₇₃	C₂×#1994
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #22610909	C₂	direct (N × H)
17. #533×C₇₃	C₄	semidirect
19. C₂×C₄⋊Q₈×C₇₃	C₄	semidirect
20. C₂×#125×C₇₃	C₄	semidirect
34. C₂×C₄×Dic₇₃	Q₈	semidirect
41. C₄⋊Q₈×C₇₃	C₂×C₄	semidirect
43. #125×C₇₃	C₂×C₄	semidirect
74. C₄×Dic₇₃	C₂×Q₈	semidirect
157. C₂×C₇₃	#1994	semidirect
165. C₇₃	C₂×#1994	semidirect
169. #533	Dic₇₃	semidirect
171. C₂×C₄⋊Q₈	Dic₇₃	semidirect
172. C₂×#125	Dic₇₃	semidirect
194. C₄⋊Q₈	C₂×Dic₇₃	semidirect
196. #125	C₂×Dic₇₃	semidirect
316. C₂	#22610909	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×#17988611	2
C₂×C₄²×Dic₇₃	2
#22610909	2
C₂×#17988346	2
C₂×#533×C₇₃	2
C₂×#1994	73

Subgroups — C2×#22610909 #33090602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22610909 #33090602