Subgroups — #19146155

Order 5504.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063358	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583558	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583554	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583293	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583145	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583140	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583115	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583110	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583063	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583063	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583057	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583046	C₂	1	1	yes	maximal, characteristic, complemented
14	#14583001	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂×#10621898	C₂²	1	1	yes	characteristic
18	C₂×#10621898	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
20	#13063358	—	2	4	no	—
21	#13063347	—	2	4	no	—
22	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	C₂×#10621873	C₂²	1	1	yes	characteristic
27	C₂×#10621873	C₂²	1	1	yes	characteristic
28	C₂×#10621873	C₂²	1	1	yes	characteristic
29	C₂×#10621873	C₂²	1	1	yes	characteristic
30	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
31	#13063325	C₂²	1	1	yes	characteristic
32	#13063313	C₂²	1	1	yes	characteristic
33	#13063311	—	2	1	no	—
34	#13063307	—	2	1	no	—
35	#13063307	—	2	1	no	—
36	#13063304	C₂²	1	1	yes	characteristic
37	#13063304	C₂²	1	1	yes	characteristic
38	#13063304	C₂²	1	1	yes	characteristic
39	#13063303	—	2	1	no	—
40	#13063299	C₂²	1	1	yes	characteristic
41	#13063297	—	2	1	no	—
42	#13063296	—	2	1	no	—
43	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
44	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
45	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
46	C₄⋊C₄×D₄₃	—	2	1	no	—
47	#13063259	C₂²	1	1	yes	characteristic
48	#13063250	C₂²	1	1	yes	characteristic
49	#13063250	C₂²	1	1	yes	characteristic
50	#13063250	C₂²	1	1	yes	characteristic
51	#13063250	C₂²	1	1	yes	characteristic
52	#13063247	C₂²	1	1	yes	characteristic
53	#13063247	C₂²	1	1	yes	characteristic
54	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
60	#97×C₄₃	C₂²	1	1	yes	characteristic
61	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂×C₂²⋊D₄₃	—	2	1	no	—
67	C₂×C₂²⋊D₄₃	—	2	1	no	—
68	C₂²×Dic₄₃	—	2	1	no	—
69	#10621898	—	2	2	no	—
70	#10621898	—	2	2	no	—
71	#10621898	—	2	2	no	—
72	#10621898	—	4	1	no	—
73	#10621898	—	4	1	no	—
74	#10621898	—	4	1	no	—
75	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
76	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×Dic₈₆	—	2	2	no	—
82	C₂×Dic₈₆	—	2	1	no	—
83	C₂×Dic₈₆	—	4	1	no	—
84	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
86	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
87	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
88	M₁₆×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂³	1	1	yes	characteristic
98	C₂²⋊Dic₄₃	—	2	1	no	—
99	C₂²⋊Dic₄₃	—	2	1	no	—
100	C₂²⋊Dic₄₃	—	2	1	no	—
101	(C₂×D₄₃)⋊C₄	—	2	1	no	—
102	(C₂×D₄₃)⋊C₄	—	2	1	no	—
103	(C₂×D₄₃)⋊C₄	—	2	1	no	—
104	(C₂×D₄₃)⋊C₄	—	2	1	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
108	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
109	Dic₄₃⋊C₄	—	2	1	no	—
110	Dic₄₃⋊C₄	—	2	1	no	—
111	Dic₄₃⋊C₄	—	2	1	no	—
112	Dic₄₃⋊C₄	—	2	1	no	—
113	C₄×Dic₄₃	—	2	1	no	—
114	Dic₁₇₂	—	2	2	no	—
115	Dic₁₇₂	—	2	2	no	—
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	4	1	no	—
118	D₃₄₄	—	2	2	no	—
119	#10621873	—	2	2	no	—
120	#10621873	—	2	2	no	—
121	#10621873	—	4	1	no	—
122	#10621873	D₄	1	4	yes	complemented
123	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₄₃	—	2	1	no	—
126	C₂²×D₄₃	—	2	1	no	—
127	D₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
137	C₂²⋊D₄₃	—	4	2	no	—
138	C₂²⋊D₄₃	—	4	2	no	—
139	C₂²⋊D₄₃	—	4	2	no	—
140	C₂²⋊D₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	2	1	no	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	D₁₇₂	—	2	1	no	—
149	D₁₇₂	—	2	1	no	—
150	D₁₇₂	—	2	1	no	—
151	D₁₇₂	C₂×D₄	1	2	yes	—
152	C₄×D₄₃	—	4	2	no	—
153	C₄×D₄₃	—	4	2	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	Dic₈₆	—	2	1	no	—
157	Dic₈₆	—	2	1	no	—
158	Dic₈₆	—	2	1	no	—
159	Dic₈₆	—	2	1	no	—
160	Dic₈₆	—	2	1	no	—
161	Dic₈₆	—	2	1	no	—
162	Dic₈₆	—	2	1	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	C₂×D₄	1	2	yes	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	—	2	1	no	—
168	C₈×C₄₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄₃	—	2	1	no	—
170	C₂²×C₄₃	—	2	1	no	—
171	C₂²×C₄₃	—	2	1	no	—
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	—	2	1	no	—
174	C₂²×C₄₃	—	2	1	no	—
175	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×D₄₃	—	2	2	no	—
177	C₂×D₄₃	—	4	1	no	—
178	C₂×D₄₃	—	4	1	no	—
179	C₂×D₄₃	—	4	1	no	—
180	C₂×D₄₃	—	4	1	no	—
181	C₂×D₄₃	—	4	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	2	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
187	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
188	Dic₄₃	—	2	2	no	—
189	Dic₄₃	—	4	1	no	—
190	Dic₄₃	—	4	1	no	—
191	Dic₄₃	—	4	1	no	—
192	Dic₄₃	—	4	1	no	—
193	Dic₄₃	—	4	1	no	—
194	#3269	—	43	1	no	maximal, Sylow
195	C₂×C₄₃	—	4	1	no	—
196	C₂×C₄₃	—	4	1	no	—
197	C₂×C₄₃	D₄²	1	1	yes	characteristic
198	C₂×C₄₃	#575	1	1	yes	characteristic
199	C₂×C₄₃	#578	1	1	yes	characteristic
200	D₄₃	—	4	2	no	—
201	D₄₃	—	8	1	no	—
202	C₂×#137	—	43	1	no	—
203	C₂×#135	—	43	1	no	—
204	#548	—	43	1	no	—
205	#547	—	43	1	no	—
206	#497	—	43	1	no	—
207	#468	—	43	1	no	—
208	#467	—	43	1	no	—
209	#463	—	43	1	no	—
210	#462	—	43	1	no	—
211	#452	—	43	1	no	—
212	#452	—	43	1	no	—
213	#451	—	43	1	no	—
214	#449	—	43	1	no	—
215	#440	—	43	1	no	—
216	#435	D₄₃	1	1	yes	characteristic, complemented
217	C₄₃	#3269	1	1	yes	characteristic, complemented, Sylow
218	C₂×#41	—	43	1	no	—
219	C₂×#41	—	43	1	no	—
220	C₂×#41	—	43	1	no	—
221	C₂²×Q₈	—	43	1	no	—
222	#137	—	86	4	no	—
223	#135	—	86	4	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×Q₁₆	—	43	1	no	—
226	C₂×Q₁₆	—	43	1	no	—
227	C₂×Q₁₆	—	43	1	no	—
228	C₂×SD₁₆	—	43	1	no	—
229	C₂×SD₁₆	—	43	1	no	—
230	C₂×SD₁₆	—	43	1	no	—
231	C₂×SD₁₆	—	43	1	no	—
232	C₂×D₈	—	43	1	no	—
233	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
234	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	43	1	no	—
236	#124	—	43	1	no	—
237	#124	—	86	1	no	—
238	#123	—	86	1	no	—
239	#123	—	86	1	no	—
240	C₂²⋊Q₈	—	43	1	no	—
241	C₂²⋊Q₈	—	43	1	no	—
242	C₂²⋊Q₈	—	43	1	no	—
243	C₂²⋊Q₈	—	86	1	no	—
244	#121	—	43	1	no	—
245	#121	—	86	1	no	—
246	#121	—	86	1	no	—
247	C₄×Q₈	—	43	1	no	—
248	C₄×D₄	—	43	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
251	C₂×C₄⋊C₄	—	86	1	no	—
252	#107	—	43	1	no	—
253	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
254	#103	—	43	1	no	—
255	#103	—	43	1	no	—
256	#103	—	43	1	no	—
257	#103	—	43	1	no	—
258	#102	—	43	1	no	—
259	#102	—	43	1	no	—
260	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
261	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
262	#97	C₂×D₄₃	1	1	yes	characteristic
263	#41	—	86	2	no	—
264	#41	—	86	2	no	—
265	#41	—	86	2	no	—
266	#41	—	172	1	no	—
267	#41	—	172	1	no	—
268	#41	—	172	1	no	—
269	C₂×Q₈	—	43	1	no	—
270	C₂×Q₈	—	43	1	no	—
271	C₂×Q₈	—	43	1	no	—
272	C₂×Q₈	—	43	1	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	86	1	no	—
275	C₂×Q₈	—	172	1	no	—
276	C₂×D₄	—	43	1	no	—
277	C₂×D₄	—	43	1	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂×D₄	—	86	1	no	—
280	C₂×D₄	—	86	1	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	D₁₇₂	1	1	yes	characteristic
283	C₂²×C₄	—	86	1	no	—
284	C₂²×C₄	—	86	1	no	—
285	C₂²×C₄	—	86	1	no	—
286	C₂²×C₄	—	86	1	no	—
287	C₂²×C₄	—	86	1	no	—
288	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
290	Q₁₆	—	86	2	no	—
291	Q₁₆	—	86	2	no	—
292	Q₁₆	—	86	2	no	—
293	Q₁₆	—	172	1	no	—
294	SD₁₆	—	43	4	no	—
295	SD₁₆	—	86	2	no	—
296	SD₁₆	—	86	2	no	—
297	SD₁₆	—	172	1	no	—
298	D₈	—	86	2	no	—
299	M₁₆	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
302	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
303	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
304	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
305	C₄⋊C₄	—	43	1	no	—
306	C₄⋊C₄	—	43	1	no	—
307	C₄⋊C₄	—	86	1	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	—	86	1	no	—
310	C₄⋊C₄	—	86	1	no	—
311	C₄⋊C₄	—	86	1	no	—
312	C₄⋊C₄	—	86	1	no	—
313	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
314	C₂²⋊C₄	—	86	1	no	—
315	C₂²⋊C₄	—	86	1	no	—
316	C₂²⋊C₄	—	86	1	no	—
317	C₂²⋊C₄	—	86	1	no	—
318	C₂²⋊C₄	—	86	1	no	—
319	C₂²⋊C₄	—	86	1	no	—
320	C₂²⋊C₄	—	86	1	no	—
321	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
322	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
323	C₄²	—	86	1	no	—
324	C₄²	C₂²×D₄₃	1	1	yes	characteristic
325	C₂³	—	86	1	no	—
326	C₂³	—	86	1	no	—
327	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
328	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
329	Q₈	—	43	2	no	—
330	Q₈	—	86	1	no	—
331	Q₈	—	86	1	no	—
332	Q₈	—	86	1	no	—
333	Q₈	—	86	1	no	—
334	Q₈	—	86	1	no	—
335	Q₈	—	86	1	no	—
336	Q₈	—	86	1	no	—
337	Q₈	—	172	2	no	—
338	D₄	—	4	1	no	—
339	D₄	—	43	2	no	—
340	D₄	—	86	1	no	—
341	D₄	—	86	1	no	—
342	D₄	—	86	1	no	—
343	D₄	—	172	2	no	—
344	D₄	—	172	2	no	—
345	D₄	—	172	2	no	—
346	D₄	—	172	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	86	1	no	—
352	C₂×C₄	—	86	1	no	—
353	C₂×C₄	—	86	1	no	—
354	C₂×C₄	—	86	1	no	—
355	C₂×C₄	—	86	1	no	—
356	C₂×C₄	—	86	1	no	—
357	C₂×C₄	—	172	2	no	—
358	C₂×C₄	—	172	2	no	—
359	C₂×C₄	—	172	2	no	—
360	C₂×C₄	—	172	2	no	—
361	C₂×C₄	—	172	2	no	—
362	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
363	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
364	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
365	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
366	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	D₄×D₄₃	1	2	yes	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	86	2	no	—
378	C₂²	—	172	1	no	—
379	C₂²	—	172	1	no	—
380	C₂²	—	172	1	no	—
381	C₂²	—	172	1	no	—
382	C₂²	—	172	1	no	—
383	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	86	2	no	—
389	C₄	—	172	1	no	—
390	C₄	—	172	1	no	—
391	C₄	—	172	1	no	—
392	C₄	—	172	1	no	—
393	C₄	—	172	1	no	—
394	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
395	C₄	#13063371	1	1	yes	characteristic
396	C₂	—	4	1	no	—
397	C₂	—	4	1	no	—
398	C₂	—	344	1	no	—
399	C₂	—	172	2	no	—
400	C₂	D₄×D₁₇₂	1	1	yes	characteristic
401	C₂	#14583699	1	1	yes	characteristic
402	C₂	#14583716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#435	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#435	D₄₃
hypocenter	C₄₃	#3269
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. C₂×#13063358	C₂	semidirect
2. C₂×#13063347	C₂	semidirect
3. #14583558	C₂	semidirect
4. #14583554	C₂	semidirect
5. #14583293	C₂	semidirect
6. #14583145	C₂	semidirect
7. #14583140	C₂	semidirect
8. #14583115	C₂	semidirect
9. #14583110	C₂	semidirect
10. #14583063	C₂	semidirect
11. #14583063	C₂	semidirect
12. #14583057	C₂	semidirect
13. #14583046	C₂	semidirect
14. #14583001	C₂	semidirect
15. C₄₃×#435	C₂	semidirect
122. #10621873	D₄	semidirect
216. #435	D₄₃	semidirect
217. C₄₃	#3269	semidirect