Subgroups — #19148508

Order 5504.

1104 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583623	C₂	1	2	yes	maximal, complemented
2	#14583615	C₂	1	2	yes	maximal, complemented
3	#14583610	C₂	1	2	yes	maximal, complemented
4	#14583597	C₂	1	4	yes	maximal, complemented
5	#14583560	C₂	1	4	yes	maximal, complemented
6	#14583552	C₂	1	2	yes	maximal, complemented
7	#14583523	C₂	1	4	yes	maximal, complemented
8	#14583480	C₂	1	2	yes	maximal, complemented
9	#14583454	C₂	1	2	yes	maximal, complemented
10	#14583450	C₂	1	2	yes	maximal, complemented
11	#14583449	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583380	C₂	1	2	yes	maximal, complemented
13	C₄×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#10621902	C₂²	1	4	yes	—
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂×#10621898	C₂²	1	2	yes	—
18	#13063325	C₂²	1	1	yes	characteristic
19	#13063322	C₂²	1	1	yes	characteristic
20	#13063320	C₂²	1	8	yes	—
21	#13063316	C₂²	1	4	yes	—
22	#13063314	C₂²	1	8	yes	—
23	#13063314	C₂²	1	4	yes	—
24	#13063313	C₂²	1	2	yes	—
25	#13063310	C₂²	1	8	yes	—
26	#13063305	C₂²	1	4	yes	—
27	#13063305	C₂²	1	4	yes	—
28	#13063301	C₂²	1	4	yes	—
29	#13063301	C₂²	1	8	yes	—
30	#13063300	C₂²	1	4	yes	—
31	#13063300	C₂²	1	4	yes	—
32	#13063300	C₂²	1	4	yes	—
33	#13063299	C₂²	1	2	yes	—
34	#13063299	C₂²	1	8	yes	—
35	#13063299	C₂²	1	2	yes	—
36	Q₈×Dic₄₃	C₂²	1	4	yes	—
37	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
39	#13063290	C₂²	1	4	yes	—
40	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
41	C₄×D₁₇₂	C₂²	1	2	yes	—
42	C₄×D₁₇₂	C₂²	1	2	yes	—
43	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
44	#13063287	C₂²	1	4	yes	—
45	#13063287	C₂²	1	4	yes	—
46	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
47	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	—
48	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
49	#13063284	C₂²	1	1	yes	characteristic
50	#13063283	C₂²	1	4	yes	—
51	#13063283	C₂²	1	4	yes	—
52	#13063282	C₂²	1	2	yes	—
53	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
54	C₄²×D₄₃	C₂²	1	2	yes	—
55	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
56	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
57	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
58	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
59	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
60	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
62	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
63	C₂×Q₈×C₄₃	—	4	4	no	—
64	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
65	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
66	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
69	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
71	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
72	#10621902	—	4	8	no	—
73	#10621898	—	2	4	no	—
74	#10621898	—	2	8	no	—
75	C₂×D₁₇₂	C₂³	1	2	yes	—
76	C₂×D₁₇₂	C₂³	1	2	yes	—
77	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
78	C₂×D₁₇₂	C₂³	1	2	yes	—
79	C₂×D₁₇₂	C₂³	1	8	yes	—
80	C₂×C₄×D₄₃	C₂³	1	2	yes	—
81	C₂×C₄×D₄₃	C₂³	1	2	yes	—
82	C₂×C₄×D₄₃	C₂³	1	2	yes	—
83	C₂×C₄×D₄₃	C₂³	1	4	yes	—
84	C₂×C₄×D₄₃	C₂³	1	8	yes	—
85	C₂×Dic₈₆	C₂³	1	2	yes	—
86	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
88	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
89	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
90	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
91	C₄²×C₄₃	C₂³	1	1	yes	characteristic
92	C₄²×C₄₃	C₂³	1	1	yes	characteristic
93	C₄²×C₄₃	C₂³	1	1	yes	characteristic
94	C₄²×C₄₃	C₂³	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
96	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
97	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
98	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
99	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
100	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
101	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
102	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
103	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₄₃	C₂³	1	2	yes	—
106	C₄⋊Dic₄₃	C₂³	1	2	yes	—
107	C₄⋊Dic₄₃	C₂³	1	8	yes	—
108	Dic₄₃⋊C₄	C₂³	1	8	yes	—
109	Dic₄₃⋊C₄	C₂³	1	2	yes	—
110	Dic₄₃⋊C₄	C₂³	1	2	yes	—
111	C₄×Dic₄₃	C₂³	1	2	yes	—
112	C₄×Dic₄₃	C₂³	1	2	yes	—
113	C₄×Dic₄₃	C₂³	1	2	yes	—
114	C₄×Dic₄₃	C₂³	1	8	yes	—
115	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂²×D₄₃	C₂⁴	1	2	yes	—
117	C₂²×D₄₃	C₂⁴	1	4	yes	—
118	Q₈×C₄₃	—	2	8	no	—
119	Q₈×C₄₃	—	4	4	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	4	4	no	—
123	C₂×C₄×C₄₃	#41	1	4	yes	complemented
124	C₂×C₄×C₄₃	C₂⁴	1	8	yes	—
125	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₄₃	—	2	4	no	—
132	C₂²⋊D₄₃	—	2	8	no	—
133	C₂×Dic₄₃	C₂⁴	1	2	yes	—
134	C₂×Dic₄₃	C₂⁴	1	2	yes	—
135	C₂×Dic₄₃	C₂⁴	1	2	yes	—
136	C₂×Dic₄₃	C₂⁴	1	4	yes	—
137	D₁₇₂	—	2	4	no	—
138	D₁₇₂	—	2	2	no	—
139	D₁₇₂	—	2	4	no	—
140	D₁₇₂	—	2	4	no	—
141	D₁₇₂	—	4	8	no	—
142	C₄×D₄₃	—	2	4	no	—
143	C₄×D₄₃	—	2	8	no	—
144	C₄×D₄₃	—	2	4	no	—
145	C₄×D₄₃	—	2	4	no	—
146	C₄×D₄₃	—	4	8	no	—
147	Dic₈₆	—	2	4	no	—
148	Dic₈₆	—	2	2	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	C₂×#41	1	2	yes	—
152	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
153	C₂×D₄₃	—	2	2	no	—
154	C₂×D₄₃	—	2	4	no	—
155	C₂×D₄₃	—	2	2	no	—
156	C₂×D₄₃	—	2	4	no	—
157	C₂×D₄₃	—	2	2	no	—
158	C₂×D₄₃	—	2	4	no	—
159	C₄×C₄₃	—	2	8	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	C₂×#41	1	2	yes	—
163	C₄×C₄₃	C₂×#41	1	2	yes	—
164	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
165	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
166	Dic₄₃	—	2	4	no	—
167	Dic₄₃	—	2	2	no	—
168	Dic₄₃	—	2	2	no	—
169	Dic₄₃	—	2	2	no	—
170	#3511	—	43	1	no	maximal, Sylow
171	C₂×C₄₃	—	2	2	no	—
172	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
173	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
175	D₄₃	—	4	2	no	—
176	D₄₃	—	4	4	no	—
177	#556	—	43	2	no	—
178	#555	—	43	2	no	—
179	#554	—	43	2	no	—
180	#553	—	43	4	no	—
181	#548	—	43	4	no	—
182	#547	—	43	2	no	—
183	#542	—	43	4	no	—
184	#537	—	43	2	no	—
185	#533	—	43	2	no	—
186	#532	—	43	2	no	—
187	#532	—	43	1	no	—
188	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
189	#520	—	43	2	no	—
190	C₄×#41	—	43	1	no	—
191	C₄₃	#3511	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	43	4	no	—
193	C₂×#41	—	43	1	no	—
194	C₂×#41	—	43	2	no	—
195	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
196	C₄⋊Q₈	—	43	1	no	—
197	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
198	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
199	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
200	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
201	#127	—	43	1	no	—
202	#126	—	43	8	no	—
203	#125	—	43	4	no	—
204	#124	—	43	4	no	—
205	#124	—	43	8	no	—
206	#124	—	43	2	no	—
207	#123	—	43	8	no	—
208	C₂²⋊Q₈	—	43	4	no	—
209	C₂²⋊Q₈	—	43	8	no	—
210	C₂²⋊Q₈	—	43	4	no	—
211	C₂²⋊Q₈	—	43	4	no	—
212	#121	—	43	2	no	—
213	#121	—	43	2	no	—
214	#121	—	43	8	no	—
215	#121	—	43	4	no	—
216	#121	—	43	4	no	—
217	#121	—	43	4	no	—
218	C₄×Q₈	—	43	4	no	—
219	C₄×Q₈	—	43	4	no	—
220	C₄×Q₈	—	43	1	no	—
221	C₄×Q₈	—	43	1	no	—
222	C₄×D₄	—	43	2	no	—
223	C₄×D₄	—	43	2	no	—
224	C₄×D₄	—	43	8	no	—
225	C₄×D₄	—	43	4	no	—
226	C₄×D₄	—	43	4	no	—
227	C₄×D₄	—	43	1	no	—
228	C₄×D₄	—	43	1	no	—
229	C₄×D₄	—	43	2	no	—
230	C₄×D₄	—	43	2	no	—
231	#117	—	43	1	no	—
232	#117	—	43	4	no	—
233	#117	—	43	4	no	—
234	#117	—	43	2	no	—
235	C₂×C₄⋊C₄	—	43	4	no	—
236	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
238	C₂×C₄²	—	43	2	no	—
239	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
240	#41	—	86	4	no	—
241	#41	—	86	8	no	—
242	#41	—	172	8	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	43	1	no	—
245	C₂×Q₈	—	43	2	no	—
246	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
247	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
248	C₂×D₄	—	43	2	no	—
249	C₂×D₄	—	43	4	no	—
250	C₂×D₄	—	43	2	no	—
251	C₂×D₄	—	43	1	no	—
252	C₂×D₄	—	43	2	no	—
253	C₂×D₄	—	43	2	no	—
254	C₂×D₄	—	43	8	no	—
255	C₂²×C₄	—	43	2	no	—
256	C₂²×C₄	—	43	2	no	—
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	43	4	no	—
259	C₂²×C₄	—	43	8	no	—
260	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
263	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	—	43	1	no	—
265	C₄⋊C₄	—	43	8	no	—
266	C₄⋊C₄	—	43	1	no	—
267	C₄⋊C₄	—	43	2	no	—
268	C₄⋊C₄	—	43	2	no	—
269	C₄⋊C₄	—	43	2	no	—
270	C₄⋊C₄	—	43	2	no	—
271	C₄⋊C₄	—	43	8	no	—
272	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
273	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
274	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
275	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
276	C₂²⋊C₄	—	43	2	no	—
277	C₂²⋊C₄	—	43	8	no	—
278	C₂²⋊C₄	—	43	2	no	—
279	C₂²⋊C₄	—	43	2	no	—
280	C₂²⋊C₄	—	43	8	no	—
281	C₂²⋊C₄	—	43	8	no	—
282	C₂²⋊C₄	—	43	4	no	—
283	C₂²⋊C₄	—	43	4	no	—
284	C₄²	—	43	2	no	—
285	C₄²	—	43	2	no	—
286	C₄²	—	43	2	no	—
287	C₄²	—	43	8	no	—
288	C₄²	C₂²×D₄₃	1	1	yes	characteristic
289	C₄²	C₂²×D₄₃	1	1	yes	characteristic
290	C₄²	C₂²×D₄₃	1	1	yes	characteristic
291	C₄²	C₂²×D₄₃	1	1	yes	characteristic
292	C₂³	—	43	2	no	—
293	C₂³	—	43	4	no	—
294	C₂³	C₂³×D₄₃	1	1	yes	characteristic
295	Q₈	—	2	8	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	86	2	no	—
298	Q₈	—	86	4	no	—
299	D₄	—	86	4	no	—
300	D₄	—	86	8	no	—
301	D₄	—	86	2	no	—
302	D₄	—	86	4	no	—
303	D₄	—	86	4	no	—
304	D₄	—	86	4	no	—
305	D₄	—	172	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	43	2	no	—
310	C₂×C₄	—	43	2	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	43	2	no	—
313	C₂×C₄	—	86	8	no	—
314	C₂×C₄	—	86	4	no	—
315	C₂×C₄	—	86	4	no	—
316	C₂×C₄	—	86	4	no	—
317	C₂×C₄	—	172	8	no	—
318	C₂×C₄	#10621902	1	4	yes	complemented
319	C₂×C₄	C₂³×D₄₃	1	8	yes	—
320	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	86	2	no	—
329	C₂²	—	86	4	no	—
330	C₂²	—	86	2	no	—
331	C₂²	—	86	4	no	—
332	C₂²	—	86	2	no	—
333	C₂²	—	86	4	no	—
334	C₂²	C₂×#10621902	1	2	yes	—
335	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	8	no	—
338	C₄	—	2	1	no	—
339	C₄	—	86	2	no	—
340	C₄	—	86	4	no	—
341	C₄	—	86	2	no	—
342	C₄	—	86	2	no	—
343	C₄	C₂×#10621902	1	2	yes	—
344	C₄	C₂×#10621902	1	2	yes	—
345	C₄	#13063370	1	2	yes	—
346	C₄	#13063373	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	172	2	no	—
349	C₂	—	172	4	no	—
350	C₂	C₂²×#10621902	1	1	yes	characteristic
351	C₂	C₂×#13063370	1	1	yes	characteristic
352	C₂	C₂×#13063373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#3511
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14583623	C₂	semidirect
2. #14583615	C₂	semidirect
3. #14583610	C₂	semidirect
4. #14583597	C₂	semidirect
5. #14583560	C₂	semidirect
6. #14583552	C₂	semidirect
7. #14583523	C₂	semidirect
8. #14583480	C₂	semidirect
9. #14583454	C₂	semidirect
10. #14583450	C₂	semidirect
11. #14583449	C₂	semidirect
12. #14583380	C₂	semidirect
13. C₄×#10621898	C₂	semidirect
14. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
123. C₂×C₄×C₄₃	#41	semidirect
188. C₂×C₄⋊Q₈	D₄₃	semidirect
191. C₄₃	#3511	semidirect
318. C₂×C₄	#10621902	semidirect