Subgroups — C₂×#22626941 #34033243

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989700	C₂	1	2	yes	maximal
2	#22626941	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14556645	C₂	1	1	yes	maximal, characteristic
4	#17989700	—	2	2	no	—
5	#17989700	—	2	2	no	—
6	#17989700	C₄	1	4	yes	—
7	#17989700	C₂²	1	4	yes	—
8	C₂×C₄⋊#10596842	—	2	2	no	—
9	C₂×C₂²⋊#10596842	C₄	1	4	yes	—
10	C₂×C₄×#10596842	—	2	2	no	—
11	C₂²×#11924036	C₄	1	1	yes	characteristic
12	C₂²×C₄×#64408	C₂²	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	C₂²×#10596842	—	2	4	no	—
19	C₄⋊#10596842	—	2	4	no	—
20	C₄⋊#10596842	—	2	4	no	—
21	C₂²⋊#10596842	—	2	4	no	—
22	C₂²⋊#10596842	C₂×C₄	1	8	yes	—
23	C₄×#10596842	—	2	4	no	—
24	C₄×#10596842	—	2	4	no	—
25	C₂³×#64408	C₂×C₄	1	2	yes	—
26	C₂×#11924036	—	2	1	no	—
27	C₂×#11924036	—	2	1	no	—
28	C₂×#11924036	—	2	1	no	—
29	C₂×#11924036	—	2	1	no	—
30	C₂×#11924036	C₈	1	2	yes	complemented
31	C₂×#11924036	C₂×C₄	1	2	yes	—
32	C₂²×#10596833	C₈	1	2	yes	complemented
33	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
34	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×#64408	D₄	1	2	yes	—
36	C₂×C₄×#64408	D₄	1	2	yes	—
37	C₂×C₄×#64408	D₄	1	1	yes	characteristic
38	C₂×C₄×#64408	D₄	1	1	yes	characteristic
39	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
40	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
41	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
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, complemented
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	complemented
51	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
53	C₂×#10596842	—	2	8	no	—
54	C₂×#10596842	—	4	8	no	—
55	C₂²×#64408	—	4	4	no	—
56	C₂²×#64408	C₄²	1	2	yes	—
57	C₂²×#64408	C₂²⋊C₄	1	2	yes	—
58	C₂²×#64408	C₂²⋊C₄	1	2	yes	—
59	C₂²×#64408	C₄⋊C₄	1	2	yes	—
60	C₂²×#64408	C₄⋊C₄	1	2	yes	—
61	C₂²×#64408	C₂²×C₄	1	2	yes	—
62	#11924036	—	2	2	no	—
63	#11924036	—	2	2	no	—
64	#11924036	—	2	2	no	—
65	#11924036	—	2	2	no	—
66	C₂×#10596833	—	2	2	no	—
67	C₂×#10596833	—	2	2	no	—
68	C₂×#10596833	C₂×C₈	1	4	yes	complemented
69	C₄×#64408	C₂²⋊C₄	1	2	yes	—
70	C₄×#64408	C₂²⋊C₄	1	2	yes	—
71	C₄×#64408	C₂²⋊C₄	1	2	yes	—
72	C₄×#64408	C₄⋊C₄	1	2	yes	—
73	C₄×#64408	C₄⋊C₄	1	2	yes	—
74	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
75	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
76	C₄×#64408	C₂×D₄	1	2	yes	—
77	C₄×#64408	C₂×D₄	1	1	yes	characteristic
78	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
79	C₂³×D₇₃	C₄²	1	1	yes	characteristic
80	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
82	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₂×C₈	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	M₁₆	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596831	—	2	1	no	—
94	C₂×#10596831	—	2	1	no	—
95	C₂×#10596831	C₂×C₈	1	2	yes	complemented
96	C₈×D₇₃	—	2	2	no	—
97	C₈×D₇₃	—	2	2	no	—
98	C₈×D₇₃	—	2	2	no	—
99	C₈×D₇₃	—	2	2	no	—
100	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₂×C₈×C₇₃	—	2	1	no	—
102	C₂×C₈×C₇₃	—	2	1	no	—
103	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	complemented
104	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
105	C₂²×D₇₃	—	2	4	no	—
106	C₂²×D₇₃	#95	1	1	yes	characteristic
107	C₂²×D₇₃	#95	1	1	yes	characteristic
108	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
109	C₂²×D₇₃	#97	1	1	yes	characteristic
110	C₂²×D₇₃	#97	1	1	yes	characteristic
111	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×#64408	—	2	4	no	—
113	C₂×#64408	—	2	4	no	—
114	C₂×#64408	—	4	2	no	—
115	C₂×#64408	—	4	2	no	—
116	C₂×#64408	—	4	2	no	—
117	C₂×#64408	—	4	2	no	—
118	C₂×#64408	#95	1	2	yes	—
119	C₂×#64408	#95	1	2	yes	—
120	C₂×#64408	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×#64408	C₂×C₄⋊C₄	1	2	yes	—
122	#10596842	—	4	8	no	—
123	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
133	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
134	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
135	C₄×D₇₃	#95	1	1	yes	characteristic
136	C₄×D₇₃	#95	1	1	yes	characteristic
137	C₄×D₇₃	#97	1	1	yes	characteristic
138	C₄×D₇₃	#97	1	1	yes	characteristic
139	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
140	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
141	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
142	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
143	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
144	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
145	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
146	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
147	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
150	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
151	#10596833	—	2	4	no	—
152	C₈×C₇₃	—	2	2	no	—
153	#10596831	—	2	2	no	—
154	C₂²×C₇₃	—	2	2	no	—
155	C₂²×C₇₃	#336	1	1	yes	characteristic
156	C₂²×C₇₃	#336	1	1	yes	characteristic
157	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
158	C₂×D₇₃	—	2	2	no	—
159	C₂×D₇₃	—	2	2	no	—
160	C₂×D₇₃	—	2	2	no	—
161	C₂×D₇₃	—	2	2	no	—
162	C₂×D₇₃	—	2	2	no	—
163	C₂×D₇₃	#336	1	1	yes	characteristic
164	C₂×D₇₃	#336	1	1	yes	characteristic
165	C₂×D₇₃	#344	1	4	yes	—
166	C₂×D₇₃	C₂×#95	1	1	yes	characteristic
167	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
168	#64408	—	2	2	no	—
169	#64408	—	2	2	no	—
170	#64408	—	2	2	no	—
171	#64408	—	2	2	no	—
172	C₄×C₇₃	#336	1	1	yes	characteristic
173	C₄×C₇₃	#336	1	1	yes	characteristic
174	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
176	Dic₇₃	#336	1	1	yes	characteristic
177	Dic₇₃	#336	1	1	yes	characteristic
178	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
180	C₂×#1281	—	73	1	no	maximal, Sylow
181	C₂×C₇₃	—	2	1	no	—
182	C₂×C₇₃	—	2	1	no	—
183	C₂×C₇₃	#1281	1	2	yes	complemented
184	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
185	D₇₃	—	2	1	no	—
186	D₇₃	—	2	1	no	—
187	D₇₃	#1281	1	2	yes	complemented
188	D₇₃	C₂×#344	1	2	yes	—
189	C₂×#431	—	73	2	no	—
190	C₂²×#97	—	73	1	no	—
191	#1281	—	73	4	no	—
192	C₇₃	C₂×#1281	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₈	—	73	1	no	—
194	C₂³×C₈	—	73	1	no	—
195	C₂²×C₄²	—	73	1	no	—
196	#431	—	73	4	no	—
197	#431	—	73	4	no	—
198	#431	—	146	2	no	—
199	#431	—	146	2	no	—
200	C₂×C₄⋊C₈	—	146	2	no	—
201	C₂×C₂²⋊C₈	—	73	4	no	—
202	C₂×#97	—	73	2	no	—
203	C₂×#97	—	73	2	no	—
204	C₂×#97	—	146	2	no	—
205	C₂×#97	—	146	2	no	—
206	C₂×C₄×C₈	—	146	2	no	—
207	C₂³×C₄	—	73	2	no	—
208	C₂³×C₄	—	73	1	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	—	73	2	no	—
211	C₂²×C₈	—	73	2	no	—
212	C₂²×C₈	—	73	1	no	—
213	C₂²×C₈	—	73	2	no	—
214	C₂²×C₈	—	73	2	no	—
215	C₂²×C₈	—	146	1	no	—
216	C₂²×C₈	—	146	1	no	—
217	C₂²×C₈	—	146	1	no	—
218	C₂²×C₈	—	146	1	no	—
219	C₂²×C₈	—	146	4	no	—
220	C₂²×C₈	—	146	1	no	—
221	C₂²×C₈	—	146	1	no	—
222	C₂²×C₈	—	146	1	no	—
223	C₂²×C₈	—	146	1	no	—
224	C₂²×C₈	#10596842	1	1	yes	characteristic, complemented
225	C₂×C₄²	—	73	2	no	—
226	C₂×C₄²	—	73	1	no	—
227	C₂×C₄²	—	73	1	no	—
228	C₂×C₄²	—	73	2	no	—
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	2	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	73	1	no	—
233	C₂×C₄²	—	73	1	no	—
234	C₄⋊C₈	—	146	4	no	—
235	C₄⋊C₈	—	146	4	no	—
236	C₂²⋊C₈	—	73	8	no	—
237	C₂²⋊C₈	—	146	4	no	—
238	#97	—	146	4	no	—
239	#97	—	146	4	no	—
240	C₄×C₈	—	146	4	no	—
241	C₄×C₈	—	146	4	no	—
242	C₂⁴	—	73	1	no	—
243	C₂²×C₄	—	292	4	no	—
244	C₂²×C₄	—	73	2	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	2	no	—
247	C₂²×C₄	—	73	2	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	1	no	—
251	C₂²×C₄	—	73	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	73	1	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	2	no	—
263	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	292	8	no	—
267	C₂×C₈	—	73	4	no	—
268	C₂×C₈	—	73	2	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	—	146	2	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	—	146	8	no	—
274	C₂×C₈	—	146	2	no	—
275	C₂×C₈	—	146	2	no	—
276	C₂×C₈	—	146	2	no	—
277	C₂×C₈	—	146	2	no	—
278	C₂×C₈	—	146	2	no	—
279	C₂×C₈	—	146	2	no	—
280	C₂×C₈	—	146	1	no	—
281	C₂×C₈	—	146	1	no	—
282	C₂×C₈	C₂×#10596842	1	2	yes	complemented
283	C₄²	—	73	2	no	—
284	C₄²	—	73	2	no	—
285	C₄²	—	73	2	no	—
286	C₄²	—	73	2	no	—
287	C₄²	—	73	2	no	—
288	C₄²	—	73	1	no	—
289	C₄²	—	73	1	no	—
290	C₄²	—	73	2	no	—
291	C₄²	—	73	1	no	—
292	C₄²	—	73	1	no	—
293	C₂³	—	73	1	no	—
294	C₂³	—	73	1	no	—
295	C₂³	—	73	1	no	—
296	C₂³	—	73	1	no	—
297	C₂³	—	73	1	no	—
298	C₂³	—	73	1	no	—
299	C₂³	—	146	4	no	—
300	C₂³	C₄×#10596842	1	1	yes	characteristic
301	C₂×C₄	—	292	2	no	—
302	C₂×C₄	—	292	2	no	—
303	C₂×C₄	—	292	2	no	—
304	C₂×C₄	—	292	2	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	—	73	1	no	—
318	C₂×C₄	—	73	1	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	1	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	73	1	no	—
327	C₂×C₄	—	73	2	no	—
328	C₂×C₄	—	73	2	no	—
329	C₂×C₄	—	73	2	no	—
330	C₂×C₄	—	73	2	no	—
331	C₂×C₄	—	146	4	no	—
332	C₂×C₄	—	146	4	no	—
333	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
335	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
336	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
337	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
338	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	292	8	no	—
341	C₈	—	146	4	no	—
342	C₈	—	146	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	73	4	no	—
345	C₂²	—	73	1	no	—
346	C₂²	—	73	1	no	—
347	C₂²	—	73	1	no	—
348	C₂²	—	73	1	no	—
349	C₂²	—	146	2	no	—
350	C₂²	—	146	2	no	—
351	C₂²	—	146	2	no	—
352	C₂²	—	146	2	no	—
353	C₂²	—	146	2	no	—
354	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
355	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
356	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
357	C₄	—	73	1	no	—
358	C₄	—	73	1	no	—
359	C₄	—	73	1	no	—
360	C₄	—	73	1	no	—
361	C₄	—	146	2	no	—
362	C₄	—	146	2	no	—
363	C₄	—	146	2	no	—
364	C₄	—	146	2	no	—
365	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
366	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
367	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
368	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	73	2	no	—
372	C₂	—	73	2	no	—
373	C₂	—	146	1	no	—
374	C₂	—	146	1	no	—
375	C₂	#22626941	1	2	yes	direct product factor
376	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	C₄²
cosocle	C₂×C₄×#64408	C₂³
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂×#1281
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. #22626941	C₂	direct (N × H)
30. C₂×#11924036	C₈	semidirect
32. C₂²×#10596833	C₈	semidirect
45. C₂²×#10596831	C₈	semidirect
50. C₂×C₈×D₇₃	C₈	semidirect
52. C₂²×C₈×C₇₃	C₈	semidirect
68. C₂×#10596833	C₂×C₈	semidirect
95. C₂×#10596831	C₂×C₈	semidirect
103. C₂×C₈×C₇₃	C₂×C₈	semidirect
183. C₂×C₇₃	#1281	semidirect
187. D₇₃	#1281	semidirect
192. C₇₃	C₂×#1281	semidirect
224. C₂²×C₈	#10596842	semidirect
282. C₂×C₈	C₂×#10596842	semidirect
375. C₂	#22626941	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×#17989700	2
#22626941	2
C₂²×#14556645	2
C₂×#1281	73

Subgroups — C2×#22626941 #34033243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626941 #34033243