Subgroups — C₂×#22627008 #34034057

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989706	C₂	1	2	yes	maximal
2	#22627008	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14556645	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556688	—	2	2	no	—
5	#17989706	—	2	2	no	—
6	#17989706	—	2	2	no	—
7	#17989706	C₄	1	4	yes	complemented
8	#17989706	C₂²	1	4	yes	—
9	C₂×#14556687	C₄	1	4	yes	complemented
10	C₂×#14556686	—	2	2	no	—
11	C₂²×#11924036	C₂²	1	1	yes	characteristic
12	C₂²×C₄×#64408	C₄	1	1	yes	characteristic
13	C₂×#14556645	—	2	2	no	—
14	C₂×#14556645	—	2	2	no	—
15	C₂×#14556645	C₄	1	2	yes	—
16	C₂×#14556645	C₂²	1	2	yes	—
17	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
18	#14556688	—	2	4	no	—
19	#14556688	—	2	4	no	—
20	#14556687	—	2	4	no	—
21	#14556687	C₂×C₄	1	8	yes	complemented
22	#14556686	—	2	4	no	—
23	#14556686	—	2	4	no	—
24	C₂×#11924040	—	2	4	no	—
25	C₂³×#64408	C₈	1	2	yes	—
26	C₂×#11924036	C₂×C₄	1	2	yes	—
27	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924036	D₄	1	2	yes	—
29	C₂×#11924036	D₄	1	2	yes	—
30	C₂×#11924036	D₄	1	1	yes	characteristic
31	C₂×#11924036	D₄	1	1	yes	characteristic
32	C₂×#11924036	Q₈	1	1	yes	characteristic
33	C₂×#11924036	Q₈	1	1	yes	characteristic
34	C₂×#11924036	C₂³	1	1	yes	characteristic
35	C₂²×#10596833	C₂×C₄	1	2	yes	—
36	C₂×C₄×#64408	—	2	1	no	—
37	C₂×C₄×#64408	—	2	1	no	—
38	C₂×C₄×#64408	—	2	1	no	—
39	C₂×C₄×#64408	—	2	1	no	—
40	C₂×C₄×#64408	C₈	1	2	yes	—
41	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
42	#14556645	—	2	4	no	—
43	#14556645	—	2	4	no	—
44	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×#10596831	C₈	1	1	yes	characteristic
46	C₂×C₈×D₇₃	—	2	1	no	—
47	C₂×C₈×D₇₃	—	2	1	no	—
48	C₂×C₈×D₇₃	—	2	1	no	—
49	C₂×C₈×D₇₃	—	2	1	no	—
50	C₂×C₈×D₇₃	C₈	1	2	yes	—
51	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
53	#11924040	—	2	8	no	—
54	C₂²×#64408	—	2	2	no	—
55	C₂²×#64408	—	2	2	no	—
56	C₂²×#64408	—	4	4	no	—
57	C₂²×#64408	C₂×C₈	1	4	yes	—
58	#11924036	C₂²⋊C₄	1	2	yes	—
59	#11924036	C₂²⋊C₄	1	2	yes	—
60	#11924036	C₂²⋊C₄	1	2	yes	—
61	#11924036	C₄⋊C₄	1	1	yes	characteristic
62	#11924036	C₄⋊C₄	1	1	yes	characteristic
63	#11924036	C₄⋊C₄	1	2	yes	—
64	#11924036	C₄⋊C₄	1	2	yes	—
65	#11924036	C₂×D₄	1	1	yes	characteristic
66	#11924036	C₂×D₄	1	2	yes	—
67	#11924036	C₂×Q₈	1	1	yes	characteristic
68	C₂×#10596833	C₄²	1	2	yes	—
69	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
70	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
71	C₂×#10596833	C₄⋊C₄	1	2	yes	—
72	C₂×#10596833	C₄⋊C₄	1	2	yes	—
73	C₂×#10596833	C₂²×C₄	1	2	yes	—
74	C₄×#64408	—	2	2	no	—
75	C₄×#64408	—	2	2	no	—
76	C₄×#64408	—	2	2	no	—
77	C₄×#64408	—	2	2	no	—
78	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
88	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
89	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
90	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596831	—	2	1	no	—
93	C₂×#10596831	—	2	1	no	—
94	C₂×#10596831	C₂×C₈	1	2	yes	—
95	C₈×D₇₃	—	2	2	no	—
96	C₈×D₇₃	—	2	2	no	—
97	C₈×D₇₃	—	2	2	no	—
98	C₈×D₇₃	—	2	2	no	—
99	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂×C₈×C₇₃	—	2	1	no	—
101	C₂×C₈×C₇₃	—	2	1	no	—
102	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
103	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
104	C₂²×D₇₃	—	2	4	no	—
105	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
106	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
109	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
110	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×#64408	—	2	4	no	—
112	C₂×#64408	—	4	4	no	—
113	C₂×#64408	—	4	4	no	—
114	C₂×#64408	—	4	4	no	—
115	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	#95	1	1	yes	characteristic
122	C₂×Dic₇₃	#95	1	1	yes	characteristic
123	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	#97	1	1	yes	characteristic
125	C₂×Dic₇₃	#97	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
127	C₄×D₇₃	#95	1	1	yes	characteristic
128	C₄×D₇₃	#95	1	1	yes	characteristic
129	C₄×D₇₃	#97	1	1	yes	characteristic
130	C₄×D₇₃	#97	1	1	yes	characteristic
131	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
132	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
133	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
134	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
135	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
136	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
137	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
138	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
139	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
142	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
143	#10596833	#95	1	2	yes	—
144	#10596833	#95	1	2	yes	—
145	#10596833	C₂×C₂²⋊C₄	1	2	yes	—
146	#10596833	C₂×C₄⋊C₄	1	2	yes	—
147	C₈×C₇₃	—	2	2	no	—
148	#10596831	—	2	2	no	—
149	C₂²×C₇₃	—	2	2	no	—
150	C₂²×C₇₃	#336	1	1	yes	characteristic
151	C₂²×C₇₃	#336	1	1	yes	characteristic
152	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂×D₇₃	—	2	2	no	—
154	C₂×D₇₃	—	2	2	no	—
155	C₂×D₇₃	—	2	2	no	—
156	C₂×D₇₃	—	2	2	no	—
157	C₂×D₇₃	—	2	2	no	—
158	C₂×D₇₃	#336	1	1	yes	characteristic
159	C₂×D₇₃	#336	1	1	yes	characteristic
160	C₂×D₇₃	#349	1	4	yes	—
161	C₂×D₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
163	#64408	—	4	4	no	—
164	C₄×C₇₃	#336	1	1	yes	characteristic
165	C₄×C₇₃	#336	1	1	yes	characteristic
166	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
168	Dic₇₃	#336	1	1	yes	characteristic
169	Dic₇₃	#336	1	1	yes	characteristic
170	Dic₇₃	C₂×#95	1	1	yes	characteristic
171	Dic₇₃	C₂×#97	1	1	yes	characteristic
172	C₂×#1377	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	—	2	1	no	—
175	C₂×C₇₃	#1377	1	2	yes	complemented
176	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
177	D₇₃	—	2	1	no	—
178	D₇₃	—	2	1	no	—
179	D₇₃	#1377	1	2	yes	complemented
180	D₇₃	C₂×#349	1	2	yes	—
181	C₂²×M₃₂	—	73	2	no	—
182	C₂²×#97	—	73	1	no	—
183	#1377	—	73	4	no	—
184	C₇₃	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₈	—	73	1	no	—
186	C₂³×C₈	—	73	1	no	—
187	C₂²×C₄²	—	73	1	no	—
188	C₂×M₃₂	—	73	4	no	—
189	C₂×M₃₂	—	73	4	no	—
190	C₂×M₃₂	—	73	4	no	—
191	C₂×M₃₂	—	146	2	no	—
192	C₂×M₃₂	—	146	2	no	—
193	C₂×M₃₂	—	146	2	no	—
194	C₂²×C₁₆	—	146	2	no	—
195	C₂×#97	—	73	2	no	—
196	C₂×#97	—	73	2	no	—
197	C₂×#97	—	146	2	no	—
198	C₂×#97	—	146	2	no	—
199	C₂³×C₄	—	73	2	no	—
200	C₂³×C₄	—	73	1	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₂²×C₈	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₂²×C₈	—	73	1	no	—
212	C₂²×C₈	—	73	2	no	—
213	C₂²×C₈	—	73	2	no	—
214	C₂²×C₈	—	146	1	no	—
215	C₂²×C₈	—	146	1	no	—
216	C₂²×C₈	—	146	1	no	—
217	C₂²×C₈	—	146	1	no	—
218	C₂²×C₈	#10596842	1	1	yes	characteristic
219	C₂×C₄²	—	73	2	no	—
220	C₂×C₄²	—	73	2	no	—
221	C₂×C₄²	—	146	1	no	—
222	C₂×C₄²	—	146	1	no	—
223	C₂×C₄²	—	146	1	no	—
224	C₂×C₄²	—	146	1	no	—
225	M₃₂	—	73	8	no	—
226	M₃₂	—	146	4	no	—
227	M₃₂	—	146	4	no	—
228	M₃₂	—	146	4	no	—
229	C₂×C₁₆	—	146	4	no	—
230	C₂×C₁₆	—	146	4	no	—
231	C₂×C₁₆	—	146	4	no	—
232	#97	—	146	4	no	—
233	#97	—	146	4	no	—
234	C₂⁴	—	73	1	no	—
235	C₂²×C₄	—	292	4	no	—
236	C₂²×C₄	—	73	4	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	146	2	no	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	73	2	no	—
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	2	no	—
268	C₂×C₈	—	73	2	no	—
269	C₂×C₈	—	73	1	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	2	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	—	146	2	no	—
274	C₂×C₈	—	146	2	no	—
275	C₂×C₈	—	146	2	no	—
276	C₂×C₈	—	146	1	no	—
277	C₂×C₈	—	146	1	no	—
278	C₂×C₈	C₂×#10596842	1	2	yes	—
279	C₄²	—	146	2	no	—
280	C₄²	—	146	2	no	—
281	C₄²	—	146	2	no	—
282	C₄²	—	146	2	no	—
283	C₁₆	—	146	8	no	—
284	C₂³	—	73	1	no	—
285	C₂³	—	73	1	no	—
286	C₂³	—	73	1	no	—
287	C₂³	—	73	1	no	—
288	C₂³	—	73	1	no	—
289	C₂³	—	73	1	no	—
290	C₂³	—	146	4	no	—
291	C₂³	C₄×#10596842	1	1	yes	characteristic
292	C₂×C₄	—	292	4	no	—
293	C₂×C₄	—	292	4	no	—
294	C₂×C₄	—	292	4	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	1	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	73	1	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	—	73	1	no	—
311	C₂×C₄	—	73	1	no	—
312	C₂×C₄	—	73	1	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	73	1	no	—
315	C₂×C₄	—	73	1	no	—
316	C₂×C₄	—	73	1	no	—
317	C₂×C₄	—	146	4	no	—
318	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
319	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
320	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
321	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
322	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
323	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	73	2	no	—
326	C₈	—	73	2	no	—
327	C₈	—	73	2	no	—
328	C₈	—	73	2	no	—
329	C₈	—	146	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	73	4	no	—
332	C₂²	—	73	1	no	—
333	C₂²	—	73	1	no	—
334	C₂²	—	73	1	no	—
335	C₂²	—	73	1	no	—
336	C₂²	—	146	2	no	—
337	C₂²	—	146	2	no	—
338	C₂²	—	146	2	no	—
339	C₂²	—	146	2	no	—
340	C₂²	—	146	2	no	—
341	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
342	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
343	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
344	C₄	—	292	4	no	—
345	C₄	—	73	1	no	—
346	C₄	—	73	1	no	—
347	C₄	—	73	1	no	—
348	C₄	—	73	1	no	—
349	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
350	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
351	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
352	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	73	2	no	—
356	C₂	—	73	2	no	—
357	C₂	—	146	1	no	—
358	C₂	—	146	1	no	—
359	C₂	#22627008	1	2	yes	direct product factor
360	C₂	C₂×(C₂×#10596842)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10596842
commutator	C₂²×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂×#1377
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #22627008	C₂	direct (N × H)
7. #17989706	C₄	semidirect
9. C₂×#14556687	C₄	semidirect
21. #14556687	C₂×C₄	semidirect
175. C₂×C₇₃	#1377	semidirect
179. D₇₃	#1377	semidirect
184. C₇₃	C₂×#1377	semidirect
359. C₂	#22627008	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×#17989706	2
#22627008	2
C₂²×#14556645	2
C₂×#1377	73

Subgroups — C2×#22627008 #34034057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22627008 #34034057