Subgroups — #19146170

Order 5504.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 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	C₂²×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583558	C₂	1	1	yes	maximal, characteristic
5	#14583293	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583157	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583122	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	#14583057	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583057	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583040	C₂	1	1	yes	maximal, characteristic, complemented
14	#14583007	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	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
20	#13063358	—	2	4	no	—
21	C₂×Dic₁₇₂	—	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×Dic₁₇₂	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
28	C₂×#10621873	C₂²	1	1	yes	characteristic
29	C₂×#10621873	C₂²	1	1	yes	characteristic
30	C₂×#10621873	C₂²	1	1	yes	characteristic
31	#13063325	C₂²	1	1	yes	characteristic
32	#13063324	—	2	1	no	—
33	#13063313	C₂²	1	1	yes	characteristic
34	#13063311	—	2	1	no	—
35	#13063304	C₂²	1	1	yes	characteristic
36	#13063304	C₂²	1	1	yes	characteristic
37	#13063304	C₂²	1	1	yes	characteristic
38	#13063303	—	2	1	no	—
39	#13063302	—	2	1	no	—
40	#13063302	—	2	1	no	—
41	#13063299	C₂²	1	1	yes	characteristic
42	#13063297	—	2	1	no	—
43	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
44	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
45	D₄×Dic₄₃	—	2	1	no	—
46	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
47	#13063257	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	#13063250	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	C₂²×Dic₄₃	—	2	1	no	—
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂²×Dic₄₃	—	2	1	no	—
69	#10621898	—	2	2	no	—
70	#10621898	—	4	1	no	—
71	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×Dic₈₆	—	2	2	no	—
74	C₂×Dic₈₆	—	2	2	no	—
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	2	1	no	—
77	C₂×Dic₈₆	—	2	1	no	—
78	C₂×Dic₈₆	—	2	1	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×Dic₈₆	—	4	1	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
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	M₁₆×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
94	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂³	1	1	yes	characteristic
97	C₂²⋊Dic₄₃	—	2	1	no	—
98	C₂²⋊Dic₄₃	—	2	1	no	—
99	C₂²⋊Dic₄₃	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	—	2	1	no	—
101	(C₂×D₄₃)⋊C₄	—	2	1	no	—
102	(C₂×D₄₃)⋊C₄	—	2	1	no	—
103	C₄⋊Dic₄₃	—	2	1	no	—
104	C₄⋊Dic₄₃	—	2	1	no	—
105	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
107	Dic₄₃⋊C₄	—	2	1	no	—
108	Dic₄₃⋊C₄	—	2	1	no	—
109	Dic₄₃⋊C₄	—	2	1	no	—
110	Dic₄₃⋊C₄	—	2	1	no	—
111	Dic₄₃⋊C₄	—	2	1	no	—
112	C₄×Dic₄₃	—	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	—	2	2	no	—
118	Dic₁₇₂	—	4	1	no	—
119	Dic₁₇₂	D₄	1	4	yes	complemented
120	#10621873	—	2	2	no	—
121	#10621873	—	2	2	no	—
122	#10621873	—	4	1	no	—
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	D₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	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	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₄₃	—	4	2	no	—
137	C₂²⋊D₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	2	1	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	—	4	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	D₁₇₂	—	2	1	no	—
149	D₁₇₂	—	2	1	no	—
150	C₄×D₄₃	—	4	2	no	—
151	Dic₈₆	—	2	1	no	—
152	Dic₈₆	—	2	1	no	—
153	Dic₈₆	—	2	1	no	—
154	Dic₈₆	—	2	1	no	—
155	Dic₈₆	—	2	1	no	—
156	Dic₈₆	—	2	1	no	—
157	Dic₈₆	—	2	1	no	—
158	Dic₈₆	—	2	1	no	—
159	Dic₈₆	—	4	2	no	—
160	Dic₈₆	—	4	2	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	C₂×D₄	1	2	yes	—
163	Dic₈₆	C₂×D₄	1	2	yes	—
164	C₈×C₄₃	—	2	1	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄₃	—	2	1	no	—
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	#137	1	2	yes	complemented
174	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	1	no	—
177	C₂×D₄₃	—	4	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₄₃	—	2	2	no	—
185	Dic₄₃	—	2	2	no	—
186	Dic₄₃	—	4	1	no	—
187	Dic₄₃	—	4	1	no	—
188	Dic₄₃	—	4	1	no	—
189	Dic₄₃	—	4	1	no	—
190	Dic₄₃	—	4	1	no	—
191	#3270	—	43	1	no	maximal, Sylow
192	C₂×C₄₃	—	2	2	no	—
193	C₂×C₄₃	—	4	1	no	—
194	C₂×C₄₃	D₄²	1	1	yes	characteristic
195	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
196	C₂×C₄₃	#578	1	1	yes	characteristic
197	D₄₃	—	8	1	no	—
198	C₂×#137	—	43	1	no	—
199	C₂²×Q₁₆	—	43	1	no	—
200	D₄×Q₈	—	43	1	no	—
201	#548	—	43	1	no	—
202	#497	—	43	1	no	—
203	#470	—	43	1	no	—
204	#465	—	43	1	no	—
205	#464	—	43	1	no	—
206	#462	—	43	1	no	—
207	#452	—	43	1	no	—
208	#451	—	43	1	no	—
209	#451	—	43	1	no	—
210	#448	—	43	1	no	—
211	#441	—	43	1	no	—
212	#435	D₄₃	1	1	yes	characteristic, complemented
213	C₄₃	#3270	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	43	1	no	—
215	C₂²×Q₈	—	43	1	no	—
216	C₂²×Q₈	—	43	1	no	—
217	C₂²×Q₈	—	43	1	no	—
218	#137	—	86	4	no	—
219	C₂×Q₁₆	—	43	1	no	—
220	C₂×Q₁₆	—	43	1	no	—
221	C₂×Q₁₆	—	43	1	no	—
222	C₂×Q₁₆	—	43	1	no	—
223	C₂×Q₁₆	—	43	1	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×Q₁₆	—	86	4	no	—
226	C₂×SD₁₆	—	43	1	no	—
227	C₂×SD₁₆	—	43	1	no	—
228	C₂×SD₁₆	—	43	1	no	—
229	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
230	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
231	C₄⋊Q₈	—	43	1	no	—
232	C₄⋊Q₈	—	86	1	no	—
233	#124	—	43	1	no	—
234	#124	—	86	1	no	—
235	C₂²⋊Q₈	—	43	1	no	—
236	C₂²⋊Q₈	—	43	1	no	—
237	C₂²⋊Q₈	—	43	1	no	—
238	C₂²⋊Q₈	—	86	1	no	—
239	C₂²⋊Q₈	—	86	1	no	—
240	C₂²⋊Q₈	—	86	1	no	—
241	#121	—	43	1	no	—
242	#121	—	86	1	no	—
243	C₄×Q₈	—	43	1	no	—
244	C₄×Q₈	—	43	1	no	—
245	C₄×D₄	—	86	1	no	—
246	C₄×D₄	—	86	1	no	—
247	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
248	#106	—	43	1	no	—
249	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
250	#103	—	43	1	no	—
251	#103	—	43	1	no	—
252	#103	—	43	1	no	—
253	#103	—	43	1	no	—
254	#103	—	43	1	no	—
255	#102	—	43	1	no	—
256	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
257	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
258	#97	C₂×D₄₃	1	1	yes	characteristic
259	#41	—	86	2	no	—
260	#41	—	172	1	no	—
261	C₂×Q₈	—	43	1	no	—
262	C₂×Q₈	—	43	1	no	—
263	C₂×Q₈	—	43	1	no	—
264	C₂×Q₈	—	43	1	no	—
265	C₂×Q₈	—	43	1	no	—
266	C₂×Q₈	—	86	2	no	—
267	C₂×Q₈	—	86	2	no	—
268	C₂×Q₈	—	86	2	no	—
269	C₂×Q₈	—	86	1	no	—
270	C₂×Q₈	—	86	1	no	—
271	C₂×Q₈	—	86	1	no	—
272	C₂×Q₈	—	172	1	no	—
273	C₂×Q₈	—	172	1	no	—
274	C₂×Q₈	—	172	1	no	—
275	C₂×D₄	—	43	1	no	—
276	C₂×D₄	—	86	1	no	—
277	C₂×D₄	—	86	1	no	—
278	C₂×D₄	D₁₇₂	1	1	yes	characteristic
279	C₂²×C₄	—	86	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
285	Q₁₆	—	43	4	no	—
286	Q₁₆	—	86	2	no	—
287	Q₁₆	—	86	2	no	—
288	Q₁₆	—	86	2	no	—
289	Q₁₆	—	86	2	no	—
290	Q₁₆	—	172	1	no	—
291	SD₁₆	—	86	2	no	—
292	SD₁₆	—	86	2	no	—
293	SD₁₆	—	172	1	no	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
298	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
300	C₄⋊C₄	—	43	1	no	—
301	C₄⋊C₄	—	43	1	no	—
302	C₄⋊C₄	—	86	1	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	86	1	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₄⋊C₄	—	86	1	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
310	C₂²⋊C₄	—	86	1	no	—
311	C₂²⋊C₄	—	86	1	no	—
312	C₂²⋊C₄	—	86	1	no	—
313	C₂²⋊C₄	—	86	1	no	—
314	C₂²⋊C₄	—	86	1	no	—
315	C₂²⋊C₄	—	86	1	no	—
316	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
317	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
318	C₄²	—	86	1	no	—
319	C₄²	—	86	1	no	—
320	C₄²	C₂²×D₄₃	1	1	yes	characteristic
321	C₂³	—	86	1	no	—
322	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
323	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
324	Q₈	—	43	2	no	—
325	Q₈	—	43	2	no	—
326	Q₈	—	86	1	no	—
327	Q₈	—	86	1	no	—
328	Q₈	—	86	1	no	—
329	Q₈	—	86	1	no	—
330	Q₈	—	86	1	no	—
331	Q₈	—	86	1	no	—
332	Q₈	—	86	1	no	—
333	Q₈	—	86	1	no	—
334	Q₈	—	172	2	no	—
335	Q₈	—	172	2	no	—
336	Q₈	—	172	2	no	—
337	D₄	—	2	2	no	—
338	D₄	—	86	1	no	—
339	D₄	—	86	1	no	—
340	D₄	—	172	2	no	—
341	D₄	—	172	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	86	1	no	—
349	C₂×C₄	—	86	1	no	—
350	C₂×C₄	—	86	1	no	—
351	C₂×C₄	—	86	1	no	—
352	C₂×C₄	—	86	1	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	2	no	—
356	C₂×C₄	—	172	2	no	—
357	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
358	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
359	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
360	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	D₄×D₄₃	1	2	yes	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	172	1	no	—
372	C₂²	—	172	1	no	—
373	C₂²	—	172	1	no	—
374	C₂²	#13063358	1	2	yes	complemented
375	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	86	2	no	—
381	C₄	—	86	2	no	—
382	C₄	—	172	1	no	—
383	C₄	—	172	1	no	—
384	C₄	—	172	1	no	—
385	C₄	—	172	1	no	—
386	C₄	—	172	1	no	—
387	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
388	C₄	#13063371	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	4	1	no	—
391	C₂	—	344	1	no	—
392	C₂	D₄×D₁₇₂	1	1	yes	characteristic
393	C₂	C₂×#13063358	1	1	yes	characteristic
394	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	#3270
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. C₂²×Dic₁₇₂	C₂	semidirect
3. D₄×Dic₈₆	C₂	semidirect
5. #14583293	C₂	semidirect
6. #14583157	C₂	semidirect
7. #14583128	C₂	semidirect
8. #14583122	C₂	semidirect
9. #14583110	C₂	semidirect
10. #14583063	C₂	semidirect
11. #14583057	C₂	semidirect
12. #14583057	C₂	semidirect
13. #14583040	C₂	semidirect
14. #14583007	C₂	semidirect
15. C₄₃×#435	C₂	semidirect
119. Dic₁₇₂	D₄	semidirect
173. C₂²×C₄₃	#137	semidirect
212. #435	D₄₃	semidirect
213. C₄₃	#3270	semidirect
374. C₂²	#13063358	semidirect