Subgroups — C₂×#22610685 #33089400

Order 18688.

934 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	C₂²×#14556527	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988614	C₂	1	2	yes	maximal
3	C₂²×#14556501	C₂	1	1	yes	maximal, characteristic
4	#22610685	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988346	C₂	1	2	yes	maximal
6	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	C₂²	1	2	yes	—
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×#14556527	—	2	4	no	—
11	C₂×#14556527	C₂²	1	4	yes	—
12	#17988614	—	2	4	no	—
13	#17988614	C₂²	1	8	yes	—
14	C₂×C₄⋊#10596831	—	2	4	no	—
15	C₂×#14556501	—	2	4	no	—
16	C₂×#14556501	C₂²	1	4	yes	—
17	#17988346	C₂²	1	8	yes	—
18	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₇₃	C₂²	1	2	yes	—
20	#427×C₇₃	C₄	1	4	yes	complemented
21	#427×C₇₃	C₂²	1	4	yes	—
22	C₂×#107×C₇₃	C₄	1	2	yes	complemented
23	C₂×#106×C₇₃	C₄	1	2	yes	complemented
24	C₂³×Dic₇₃	—	2	1	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	4	no	—
27	C₂×#11924012	—	2	4	no	—
28	C₂×#11924012	C₂³	1	4	yes	—
29	C₂²×#10596831	—	2	2	no	—
30	C₂²×#10596831	—	2	2	no	—
31	C₂²×#10596831	—	2	2	no	—
32	C₂²×#10596831	D₄	1	2	yes	—
33	C₂²×#10596831	D₄	1	2	yes	—
34	C₂²×#10596831	C₂³	1	2	yes	—
35	C₂×C₄×Dic₇₃	—	2	2	no	—
36	C₂×C₄×Dic₇₃	—	2	1	no	—
37	C₂×C₄×Dic₇₃	Q₈	1	4	yes	—
38	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
39	#14556527	—	2	8	no	—
40	C₄⋊#10596831	—	2	16	no	—
41	#14556501	—	2	8	no	—
42	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₃	D₄	1	4	yes	—
47	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
48	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
49	#117×C₇₃	—	2	2	no	—
50	#117×C₇₃	—	2	2	no	—
51	#117×C₇₃	C₂×C₄	1	4	yes	—
52	#117×C₇₃	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
55	C₂×C₄²×C₇₃	—	2	2	no	—
56	#107×C₇₃	C₂×C₄	1	8	yes	complemented
57	#106×C₇₃	C₂×C₄	1	8	yes	complemented
58	C₂²×Dic₇₃	—	2	4	no	—
59	C₂²×Dic₇₃	—	2	2	no	—
60	C₂²×Dic₇₃	—	4	4	no	—
61	#11924012	—	2	4	no	—
62	#11924012	—	2	4	no	—
63	#11924012	—	4	4	no	—
64	C₂×#10596831	—	2	4	no	—
65	C₂×#10596831	—	2	4	no	—
66	C₂×#10596831	—	2	2	no	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	—	4	4	no	—
70	C₂×#10596831	C₂×D₄	1	4	yes	—
71	C₄×Dic₇₃	—	2	4	no	—
72	C₄×Dic₇₃	—	2	2	no	—
73	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
74	C₂⁴×C₇₃	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	—	2	4	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	#41	1	2	yes	—
85	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
89	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
93	C₄⋊C₄×C₇₃	—	2	4	no	—
94	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
95	C₂²⋊C₄×C₇₃	—	2	4	no	—
96	C₂²⋊C₄×C₇₃	—	2	4	no	—
97	C₄²×C₇₃	—	2	4	no	—
98	C₄²×C₇₃	—	2	4	no	—
99	C₂³×C₇₃	—	2	4	no	—
100	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₇₃	#125	1	2	yes	—
104	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₇₃	—	2	8	no	—
106	C₂×C₄×C₇₃	—	4	8	no	—
107	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₇₃	#121	1	2	yes	—
114	C₂×C₄×C₇₃	#121	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₃	#125	1	2	yes	—
117	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₇₃	—	2	4	no	—
126	C₂×Dic₇₃	—	4	4	no	—
127	C₂×Dic₇₃	—	4	8	no	—
128	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
130	#10596831	—	2	4	no	—
131	#10596831	—	4	4	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	4	no	—
135	C₂²×C₇₃	—	2	4	no	—
136	C₂²×C₇₃	#389	1	2	yes	—
137	C₂²×C₇₃	#389	1	2	yes	—
138	C₂²×C₇₃	#444	1	4	yes	—
139	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×#125	1	1	yes	characteristic
142	C₄×C₇₃	—	4	8	no	—
143	C₄×C₇₃	#389	1	2	yes	—
144	C₄×C₇₃	#389	1	2	yes	—
145	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	Dic₇₃	—	4	4	no	—
150	C₂×#1946	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	—	2	1	no	—
153	C₂×C₇₃	—	2	2	no	—
154	C₂×C₇₃	#1946	1	4	yes	complemented
155	C₂×C₇₃	C₂×#389	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#444	1	2	yes	—
157	C₂²×#108	—	73	1	no	—
158	C₂×#427	D₇₃	1	1	yes	characteristic
159	C₂×#424	—	73	2	no	—
160	C₂²×#97	—	73	1	no	—
161	#1946	—	73	8	no	—
162	C₂×#339	—	73	2	no	—
163	C₇₃	C₂×#1946	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	73	2	no	—
165	C₂³×C₈	—	73	1	no	—
166	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
167	C₂×#117	C₂×D₇₃	1	2	yes	—
168	C₂²×C₄²	—	73	1	no	—
169	C₂×#108	—	73	4	no	—
170	C₂×#108	—	146	4	no	—
171	#427	Dic₇₃	1	4	yes	complemented
172	#427	C₂×D₇₃	1	4	yes	—
173	C₂×#107	Dic₇₃	1	2	yes	complemented
174	C₂×#106	Dic₇₃	1	2	yes	complemented
175	#424	—	73	8	no	—
176	#424	—	146	4	no	—
177	C₂×C₄⋊C₈	—	146	4	no	—
178	C₂×#97	—	73	4	no	—
179	C₂×#97	—	146	4	no	—
180	#339	—	73	8	no	—
181	C₂³×C₄	—	146	1	no	—
182	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
183	C₂×M₁₆	—	73	4	no	—
184	C₂×M₁₆	—	146	4	no	—
185	C₂×M₁₆	—	146	4	no	—
186	C₂×M₁₆	—	146	2	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₂²×C₈	—	73	2	no	—
190	C₂²×C₈	—	146	2	no	—
191	C₂²×C₈	—	146	2	no	—
192	C₂²×C₈	—	146	2	no	—
193	C₂²×C₈	Dic₁₄₆	1	4	yes	—
194	C₂²×C₈	D₂₉₂	1	4	yes	—
195	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×Dic₇₃	1	4	yes	—
202	#117	C₂²×D₇₃	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	73	4	no	—
207	C₂×C₄²	—	73	2	no	—
208	C₂×C₄²	—	146	2	no	—
209	C₂×C₄²	—	146	1	no	—
210	#108	—	146	8	no	—
211	#107	C₂×Dic₇₃	1	8	yes	complemented
212	#106	C₂×Dic₇₃	1	8	yes	complemented
213	C₄⋊C₈	—	146	16	no	—
214	#97	—	146	8	no	—
215	C₂⁴	#11924027	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	292	4	no	—
218	C₂²×C₄	—	146	4	no	—
219	C₂²×C₄	—	146	2	no	—
220	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
225	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
226	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
227	C₂²×C₄	#11924027	1	1	yes	characteristic
228	C₂²×C₄	#11924029	1	2	yes	—
229	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
230	M₁₆	—	292	4	no	—
231	M₁₆	—	146	4	no	—
232	M₁₆	—	146	4	no	—
233	C₂×C₈	—	292	4	no	—
234	C₂×C₈	—	73	4	no	—
235	C₂×C₈	—	146	4	no	—
236	C₂×C₈	—	146	4	no	—
237	C₂×C₈	—	146	4	no	—
238	C₂×C₈	—	146	2	no	—
239	C₂×C₈	—	146	2	no	—
240	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
241	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
242	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	—
243	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
244	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
245	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
246	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×Dic₇₃	1	8	yes	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	73	4	no	—
254	C₄²	—	146	4	no	—
255	C₄²	—	146	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
258	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
259	C₂³	#14556571	1	2	yes	—
260	C₂³	#14556582	1	2	yes	—
261	C₂³	C₂×#11924027	1	1	yes	characteristic
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	292	4	no	—
265	C₂×C₄	—	292	8	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
269	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
271	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
272	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
273	C₂×C₄	#14556566	1	2	yes	—
274	C₂×C₄	#14556566	1	2	yes	—
275	C₂×C₄	#14556571	1	2	yes	—
276	C₂×C₄	#14556582	1	2	yes	—
277	C₂×C₄	#14556590	1	2	yes	—
278	C₂×C₄	#14556590	1	2	yes	—
279	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
284	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
285	C₈	—	292	4	no	—
286	C₈	—	146	4	no	—
287	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
288	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#17988486	1	2	yes	—
294	C₂²	#17988486	1	2	yes	—
295	C₂²	#17988709	1	4	yes	—
296	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
297	C₂²	C₂×#14556571	1	1	yes	characteristic
298	C₂²	C₂×#14556582	1	1	yes	characteristic
299	C₄	—	4	8	no	—
300	C₄	—	292	4	no	—
301	C₄	#17988486	1	2	yes	—
302	C₄	#17988486	1	2	yes	—
303	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
304	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
305	C₄	C₂×#14556566	1	1	yes	characteristic
306	C₄	C₂×#14556590	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	#22610685	1	4	yes	direct product factor
311	C₂	C₂×#17988486	1	1	yes	characteristic
312	C₂	C₂×#17988709	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₄×D₄
prime radical	C₂⁴×C₇₃	#41
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	C₂×#1946
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #22610685	C₂	direct (N × H)
20. #427×C₇₃	C₄	semidirect
22. C₂×#107×C₇₃	C₄	semidirect
23. C₂×#106×C₇₃	C₄	semidirect
56. #107×C₇₃	C₂×C₄	semidirect
57. #106×C₇₃	C₂×C₄	semidirect
154. C₂×C₇₃	#1946	semidirect
163. C₇₃	C₂×#1946	semidirect
171. #427	Dic₇₃	semidirect
173. C₂×#107	Dic₇₃	semidirect
174. C₂×#106	Dic₇₃	semidirect
211. #107	C₂×Dic₇₃	semidirect
212. #106	C₂×Dic₇₃	semidirect
310. C₂	#22610685	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556527	2
C₂×#17988614	2
C₂²×#14556501	2
#22610685	2
C₂×#17988346	2
C₂×#427×C₇₃	2
C₂×#1946	73

Subgroups — C2×#22610685 #33089400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22610685 #33089400