Subgroups — C₂×#22624757 #34002003

Order 18688.

678 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×#17989603	C₂	1	2	yes	maximal
2	#22624757	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14556501	C₂	1	1	yes	maximal, characteristic
4	#17989603	—	2	2	no	—
5	#17989603	—	2	2	no	—
6	#17989603	C₄	1	4	yes	—
7	#17989603	C₂²	1	4	yes	—
8	C₂×C₄⋊#10596833	—	2	2	no	—
9	C₂×C₂²⋊#10596833	—	2	2	no	—
10	C₂×C₄×#10596833	C₄	1	4	yes	—
11	C₂³×#10596831	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×#14556501	—	2	2	no	—
14	C₂×#14556501	—	2	2	no	—
15	C₂×#14556501	C₄	1	2	yes	—
16	C₂×#14556501	C₂²	1	2	yes	—
17	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	4	no	—
19	C₄⋊#10596833	—	2	4	no	—
20	C₄⋊#10596833	—	2	4	no	—
21	C₂²⋊#10596833	—	2	4	no	—
22	C₂²⋊#10596833	—	2	4	no	—
23	C₄×#10596833	—	2	4	no	—
24	C₄×#10596833	C₂×C₄	1	8	yes	—
25	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂³×Dic₇₃	Q₈	1	1	yes	characteristic
27	C₂²×#10596831	—	2	1	no	—
28	C₂²×#10596831	—	2	1	no	—
29	C₂²×#10596831	—	2	2	no	—
30	C₂²×#10596831	C₈	1	4	yes	complemented
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
36	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
40	#14556501	—	2	4	no	—
41	#14556501	—	2	4	no	—
42	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	—	2	1	no	—
44	C₂²×C₈×C₇₃	—	2	1	no	—
45	C₂²×C₈×C₇₃	—	2	2	no	—
46	C₂²×C₈×C₇₃	C₈	1	4	yes	complemented
47	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×#10596833	—	2	8	no	—
49	C₂×#10596833	—	4	8	no	—
50	C₂²×Dic₇₃	—	4	2	no	—
51	C₂²×Dic₇₃	—	4	2	no	—
52	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
58	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	—	2	2	no	—
64	C₂×#10596831	C₂×C₈	1	4	yes	complemented
65	C₄×Dic₇₃	C₄²	1	2	yes	—
66	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
67	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
68	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
69	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
70	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
71	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
72	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
73	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
82	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	—	2	4	no	—
86	C₂×C₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
89	C₂³×C₇₃	—	2	2	no	—
90	C₂³×C₇₃	—	2	2	no	—
91	C₂³×C₇₃	#95	1	1	yes	characteristic
92	C₂³×C₇₃	#95	1	1	yes	characteristic
93	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
94	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
95	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₇₃	#97	1	2	yes	—
101	C₂×C₄×C₇₃	#97	1	2	yes	—
102	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×Dic₇₃	—	2	4	no	—
115	C₂×Dic₇₃	—	2	4	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	—	4	1	no	—
118	C₂×Dic₇₃	—	4	1	no	—
119	C₂×Dic₇₃	—	4	2	no	—
120	C₂×Dic₇₃	—	4	1	no	—
121	C₂×Dic₇₃	—	4	1	no	—
122	C₂×Dic₇₃	#95	1	2	yes	—
123	C₂×Dic₇₃	#95	1	2	yes	—
124	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
126	#10596833	—	4	8	no	—
127	C₈×C₇₃	—	2	4	no	—
128	#10596831	—	2	4	no	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	—	2	1	no	—
133	C₂²×C₇₃	—	2	4	no	—
134	C₂²×C₇₃	—	2	4	no	—
135	C₂²×C₇₃	#336	1	2	yes	—
136	C₂²×C₇₃	#336	1	2	yes	—
137	C₂²×C₇₃	H(3,ℤ/4)	1	2	yes	—
138	C₂²×C₇₃	#338	1	2	yes	—
139	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₄×C₇₃	#336	1	2	yes	—
143	C₄×C₇₃	#336	1	2	yes	—
144	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
145	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
146	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	Dic₇₃	—	2	2	no	—
149	Dic₇₃	—	2	2	no	—
150	Dic₇₃	—	2	2	no	—
151	Dic₇₃	—	2	2	no	—
152	C₂×#1275	—	73	1	no	maximal, Sylow
153	C₂×C₇₃	—	2	2	no	—
154	C₂×C₇₃	—	2	1	no	—
155	C₂×C₇₃	—	2	1	no	—
156	C₂×C₇₃	#1275	1	4	yes	complemented
157	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
158	C₂×C₇₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
159	C₂×C₇₃	C₂×#338	1	1	yes	characteristic
160	C₂×#431	—	73	2	no	—
161	C₂²×#97	—	73	1	no	—
162	#1275	—	73	4	no	—
163	C₇₃	C₂×#1275	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₈	—	73	1	no	—
165	C₂³×C₈	#64408	1	1	yes	characteristic
166	C₂²×C₄²	—	73	1	no	—
167	#431	—	73	4	no	—
168	#431	—	73	4	no	—
169	#431	—	146	2	no	—
170	#431	—	146	2	no	—
171	C₂×C₄⋊C₈	—	146	2	no	—
172	C₂×C₂²⋊C₈	—	146	2	no	—
173	C₂×#97	—	73	2	no	—
174	C₂×#97	—	73	2	no	—
175	C₂×#97	—	146	2	no	—
176	C₂×#97	—	146	2	no	—
177	C₂×C₄×C₈	—	73	4	no	—
178	C₂³×C₄	—	73	1	no	—
179	C₂³×C₄	—	73	1	no	—
180	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	73	4	no	—
185	C₂²×C₈	—	73	2	no	—
186	C₂²×C₈	—	146	4	no	—
187	C₂²×C₈	—	146	1	no	—
188	C₂²×C₈	—	146	1	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	#10596833	1	4	yes	complemented
191	C₂²×C₈	C₂×#64408	1	2	yes	—
192	C₂×C₄²	—	73	2	no	—
193	C₂×C₄²	—	73	2	no	—
194	C₂×C₄²	—	73	2	no	—
195	C₂×C₄²	—	73	2	no	—
196	C₂×C₄²	—	73	1	no	—
197	C₂×C₄²	—	73	1	no	—
198	C₂×C₄²	—	73	1	no	—
199	C₂×C₄²	—	73	1	no	—
200	C₄⋊C₈	—	146	4	no	—
201	C₄⋊C₈	—	146	4	no	—
202	C₂²⋊C₈	—	146	4	no	—
203	C₂²⋊C₈	—	146	4	no	—
204	#97	—	146	4	no	—
205	#97	—	146	4	no	—
206	C₄×C₈	—	73	8	no	—
207	C₄×C₈	—	146	4	no	—
208	C₂⁴	C₄⋊#64408	1	1	yes	characteristic
209	C₂²×C₄	—	292	2	no	—
210	C₂²×C₄	—	292	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	1	no	—
214	C₂²×C₄	—	73	1	no	—
215	C₂²×C₄	—	73	1	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	73	2	no	—
218	C₂²×C₄	—	73	2	no	—
219	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
220	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
224	C₂²×C₄	C₂×#10596833	1	2	yes	—
225	C₂²×C₄	#11924036	1	2	yes	—
226	C₂²×C₄	#11924037	1	2	yes	—
227	C₂²×C₄	#11924038	1	2	yes	—
228	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	292	8	no	—
234	C₂×C₈	—	73	4	no	—
235	C₂×C₈	—	146	8	no	—
236	C₂×C₈	—	146	2	no	—
237	C₂×C₈	—	146	4	no	—
238	C₂×C₈	—	146	4	no	—
239	C₂×C₈	—	146	2	no	—
240	C₂×C₈	C₂×#10596833	1	4	yes	complemented
241	C₄²	—	73	2	no	—
242	C₄²	—	73	2	no	—
243	C₄²	—	73	2	no	—
244	C₄²	—	73	2	no	—
245	C₄²	—	73	2	no	—
246	C₄²	—	73	2	no	—
247	C₄²	—	73	2	no	—
248	C₄²	—	73	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
252	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
253	C₂³	C₄⋊#10596833	1	2	yes	—
254	C₂³	#14556658	1	2	yes	—
255	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
256	C₂×C₄	—	292	1	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	292	1	no	—
259	C₂×C₄	—	292	2	no	—
260	C₂×C₄	—	292	1	no	—
261	C₂×C₄	—	292	1	no	—
262	C₂×C₄	—	73	2	no	—
263	C₂×C₄	—	73	2	no	—
264	C₂×C₄	—	73	2	no	—
265	C₂×C₄	—	73	2	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	—	146	4	no	—
268	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₄×#10596833	1	2	yes	—
271	C₂×C₄	C₄×#10596833	1	2	yes	—
272	C₂×C₄	#14556646	1	2	yes	—
273	C₂×C₄	#14556646	1	2	yes	—
274	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
275	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
276	C₂×C₄	#14556648	1	2	yes	—
277	C₂×C₄	#14556648	1	2	yes	—
278	C₂×C₄	C₄⋊#10596833	1	2	yes	—
279	C₂×C₄	#14556658	1	2	yes	—
280	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
282	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
283	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
284	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
285	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	292	8	no	—
288	C₈	—	146	4	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	#17989453	1	2	yes	—
296	C₂²	#17989453	1	2	yes	—
297	C₂²	#17989455	1	2	yes	—
298	C₂²	#17989456	1	2	yes	—
299	C₂²	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
300	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
301	C₂²	C₂×#14556658	1	1	yes	characteristic
302	C₄	—	146	2	no	—
303	C₄	—	146	2	no	—
304	C₄	—	146	2	no	—
305	C₄	—	146	2	no	—
306	C₄	#17989453	1	2	yes	—
307	C₄	#17989453	1	2	yes	—
308	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
309	C₄	C₂×#14556646	1	1	yes	characteristic
310	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
311	C₄	C₂×#14556648	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	#22624757	1	4	yes	direct product factor
316	C₂	C₂×#17989453	1	1	yes	characteristic
317	C₂	C₂×#17989455	1	1	yes	characteristic
318	C₂	C₂×#17989456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	#95
prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₂×#1275
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #22624757	C₂	direct (N × H)
30. C₂²×#10596831	C₈	semidirect
46. C₂²×C₈×C₇₃	C₈	semidirect
64. C₂×#10596831	C₂×C₈	semidirect
88. C₂×C₈×C₇₃	C₂×C₈	semidirect
156. C₂×C₇₃	#1275	semidirect
163. C₇₃	C₂×#1275	semidirect
190. C₂²×C₈	#10596833	semidirect
240. C₂×C₈	C₂×#10596833	semidirect
315. C₂	#22624757	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×#17989603	2
#22624757	2
C₂²×#14556501	2
C₂×#1275	73

Subgroups — C2×#22624757 #34002003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624757 #34002003