Subgroups — C₂×#19147572 #25977494

Order 11008.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583571	C₂	1	1	yes	maximal, characteristic
2	#19147572	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583273	C₂	1	3	yes	maximal, complemented
4	C₂×#14583099	C₂	1	3	yes	maximal, complemented
5	C₂×#14583101	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10621884	C₂	1	3	yes	maximal, complemented
8	C₂×C₄₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621902	—	2	1	no	—
10	C₂²×#10621884	C₂²	1	3	yes	—
11	C₂²×#10621884	C₂²	1	6	yes	—
12	#14583571	—	2	4	no	—
13	#14583571	C₂²	1	8	yes	—
14	C₂×#13063322	C₂²	1	3	yes	—
15	C₂×#13063300	—	2	3	no	—
16	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
17	#14583273	C₂²	1	24	yes	complemented
18	#14583101	C₂²	1	24	yes	complemented
19	#14583099	C₂²	1	24	yes	complemented
20	Q₈×C₄₃⋊C₈	C₂²	1	8	yes	complemented
21	C₄×#10621884	C₂²	1	24	yes	complemented
22	C₂×#13063256	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
24	C₂×#13063249	C₂²	1	3	yes	—
25	C₂×#13063245	C₂²	1	3	yes	—
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
27	C₄₃×#557	—	2	4	no	—
28	C₄₃×#557	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Q₈×C₄₃	C₂²	1	3	yes	—
30	C₂×#125×C₄₃	—	2	3	no	—
31	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#10621902	—	2	8	no	—
35	C₂×#10621902	—	4	8	no	—
36	C₂²×D₁₇₂	—	2	3	no	—
37	C₂²×D₁₇₂	—	2	3	no	—
38	C₂²×D₁₇₂	C₂³	1	3	yes	—
39	C₂²×C₄×D₄₃	—	2	3	no	—
40	C₂×#10621884	—	2	12	no	—
41	C₂×#10621884	—	4	12	no	—
42	C₂×#10621884	C₂³	1	12	yes	—
43	C₂×#10621884	C₂³	1	24	yes	—
44	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
45	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
46	#13063322	—	2	6	no	—
47	#13063322	C₂³	1	12	yes	—
48	#13063300	—	2	24	no	—
49	C₄×D₁₇₂	—	2	6	no	—
50	C₄×D₁₇₂	C₂³	1	12	yes	—
51	C₂×C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
52	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
53	#13063256	C₂³	1	4	yes	—
54	C₄⋊(C₄₃⋊C₈)	C₂³	1	12	yes	—
55	#13063249	C₂³	1	12	yes	—
56	#13063245	C₂³	1	12	yes	—
57	C₄×C₄₃⋊C₈	C₂³	1	12	yes	—
58	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₄₃	C₂³	1	3	yes	—
60	C₄⋊Q₈×C₄₃	—	2	6	no	—
61	C₄⋊Q₈×C₄₃	C₂³	1	12	yes	—
62	#125×C₄₃	—	2	24	no	—
63	C₄×Q₈×C₄₃	—	2	8	no	—
64	C₄×Q₈×C₄₃	—	2	6	no	—
65	C₄×Q₈×C₄₃	D₄	1	4	yes	—
66	C₄×Q₈×C₄₃	C₂³	1	12	yes	—
67	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
71	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
72	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₃	—	2	3	no	—
74	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
75	C₂³×D₄₃	—	2	3	no	—
76	C₂×Q₈×C₄₃	—	2	3	no	—
77	C₂×Q₈×C₄₃	—	2	6	no	—
78	C₂×Q₈×C₄₃	—	2	6	no	—
79	C₂×Q₈×C₄₃	SD₁₆	1	8	yes	complemented
80	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
81	C₂×Q₈×C₄₃	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₄₃	—	2	1	no	—
83	C₂²×C₄×C₄₃	—	2	3	no	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₃	—	2	1	no	—
88	#10621902	—	4	32	no	—
89	C₂×D₁₇₂	—	2	3	no	—
90	C₂×D₁₇₂	—	2	12	no	—
91	C₂×D₁₇₂	—	2	12	no	—
92	C₂×D₁₇₂	—	2	6	no	—
93	C₂×D₁₇₂	—	2	6	no	—
94	C₂×D₁₇₂	—	4	12	no	—
95	C₂×D₁₇₂	—	8	6	no	—
96	C₂×D₁₇₂	C₂⁴	1	6	yes	—
97	C₂×C₄×D₄₃	—	2	12	no	—
98	C₂×C₄×D₄₃	—	4	12	no	—
99	C₄⋊C₄×C₄₃	—	2	12	no	—
100	C₄⋊C₄×C₄₃	—	2	12	no	—
101	C₄⋊C₄×C₄₃	—	2	12	no	—
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
104	C₄⋊C₄×C₄₃	C₂⁴	1	2	yes	—
105	C₄²×C₄₃	—	2	12	no	—
106	C₄²×C₄₃	C₂×D₄	1	6	yes	—
107	C₄²×C₄₃	C₂⁴	1	6	yes	—
108	#10621884	—	2	24	no	—
109	#10621884	—	4	24	no	—
110	(C₂×D₄₃)⋊C₄	—	2	12	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	C₂⁴	1	2	yes	—
113	C₂×C₄₃⋊C₈	—	2	6	no	—
114	C₂×C₄₃⋊C₈	#41	1	4	yes	complemented
115	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
116	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
117	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²×D₄₃	—	2	6	no	—
119	C₂²×D₄₃	—	4	6	no	—
120	C₂²×D₄₃	—	4	6	no	—
121	C₂²×D₄₃	—	4	6	no	—
122	Q₈×C₄₃	—	2	6	no	—
123	Q₈×C₄₃	—	2	6	no	—
124	Q₈×C₄₃	—	4	6	no	—
125	Q₈×C₄₃	C₂×SD₁₆	1	16	yes	complemented
126	C₂×C₄×C₄₃	—	2	6	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	6	no	—
129	C₂×C₄×C₄₃	—	4	6	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	C₂×SD₁₆	1	12	yes	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
135	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	D₁₇₂	—	2	6	no	—
141	D₁₇₂	—	2	6	no	—
142	D₁₇₂	—	4	6	no	—
143	D₁₇₂	—	4	24	no	—
144	D₁₇₂	—	8	12	no	—
145	C₄×D₄₃	—	4	24	no	—
146	C₄₃⋊C₈	—	2	6	no	—
147	C₄₃⋊C₈	C₂×#41	1	4	yes	complemented
148	C₂×#3392	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	#550	1	2	yes	—
150	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
151	C₂²×C₄₃	#577	1	2	yes	—
152	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
153	C₂×D₄₃	—	4	6	no	—
154	C₂×D₄₃	—	4	6	no	—
155	C₂×D₄₃	—	4	6	no	—
156	C₂×D₄₃	—	8	12	no	—
157	C₄×C₄₃	—	2	6	no	—
158	C₄×C₄₃	—	4	6	no	—
159	C₄×C₄₃	—	4	2	no	—
160	C₄×C₄₃	#550	1	2	yes	—
161	C₄×C₄₃	C₂²×SD₁₆	1	12	yes	—
162	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
163	C₄×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₄₃	—	4	2	no	—
165	C₂×#557	D₄₃	1	1	yes	characteristic, complemented
166	C₂×#550	—	43	1	no	—
167	#3392	—	43	16	no	—
168	C₂×#492	—	43	3	no	—
169	C₂×#460	—	43	3	no	—
170	C₂×#461	—	43	3	no	—
171	C₂×C₈×Q₈	—	43	1	no	—
172	C₂×C₄×SD₁₆	—	43	3	no	—
173	C₂×C₄₃	#3392	1	4	yes	complemented
174	C₂×C₄₃	C₂×#550	1	1	yes	characteristic
175	C₂×C₄₃	C₂³×SD₁₆	1	1	yes	characteristic
176	C₂×C₄₃	C₂×#577	1	1	yes	characteristic
177	D₄₃	—	8	6	no	—
178	C₂²×#41	—	86	1	no	—
179	C₂²×SD₁₆	—	43	3	no	—
180	C₂²×SD₁₆	—	43	6	no	—
181	#557	—	2	4	no	—
182	#557	C₂×D₄₃	1	8	yes	complemented
183	#550	—	43	8	no	—
184	#550	—	86	4	no	—
185	C₂×C₄⋊Q₈	C₂×D₄₃	1	3	yes	—
186	C₂×#127	—	43	3	no	—
187	C₂×#125	—	2	3	no	—
188	C₂×#121	—	86	3	no	—
189	C₂×C₄×Q₈	—	2	1	no	—
190	C₂×C₄×Q₈	C₂×D₄₃	1	3	yes	—
191	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₄×D₄	—	43	3	no	—
193	#492	—	43	24	no	—
194	#461	—	43	24	no	—
195	#460	—	43	24	no	—
196	C₈×Q₈	—	43	8	no	—
197	C₄×SD₁₆	—	43	24	no	—
198	C₂×#106	—	43	1	no	—
199	C₂×C₄⋊C₈	—	43	3	no	—
200	C₂×#103	—	43	3	no	—
201	C₂×#102	—	43	3	no	—
202	C₂×C₄×C₈	—	43	3	no	—
203	C₄₃	C₂×#3392	1	1	yes	characteristic, complemented, Sylow
204	C₂×#41	—	86	8	no	—
205	C₂×#41	—	172	8	no	—
206	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
207	C₂²×Q₈	C₂²×D₄₃	1	3	yes	—
208	C₂²×D₄	—	43	3	no	—
209	C₂²×D₄	—	86	3	no	—
210	C₂²×D₄	—	86	3	no	—
211	C₂³×C₄	—	86	3	no	—
212	C₂×SD₁₆	—	43	12	no	—
213	C₂×SD₁₆	—	43	24	no	—
214	C₂×SD₁₆	—	86	12	no	—
215	C₂×SD₁₆	—	172	12	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	3	no	—
218	C₄⋊Q₈	—	2	6	no	—
219	C₄⋊Q₈	C₂²×D₄₃	1	12	yes	—
220	#127	—	43	12	no	—
221	#127	—	86	6	no	—
222	#125	—	2	24	no	—
223	#121	—	86	24	no	—
224	C₄×Q₈	—	2	8	no	—
225	C₄×Q₈	—	2	6	no	—
226	C₄×Q₈	C₂²⋊D₄₃	1	4	yes	—
227	C₄×Q₈	C₂²×D₄₃	1	12	yes	—
228	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
229	C₄×D₄	—	43	12	no	—
230	C₄×D₄	—	86	6	no	—
231	C₂×C₄⋊C₄	—	2	3	no	—
232	C₂×C₄⋊C₄	—	2	3	no	—
233	C₂×C₄⋊C₄	—	2	3	no	—
234	C₂×C₄⋊C₄	—	43	1	no	—
235	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
236	C₂×C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₂²⋊C₄	—	86	3	no	—
238	C₂×C₄²	—	2	3	no	—
239	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
240	#106	—	43	4	no	—
241	C₄⋊C₈	—	43	12	no	—
242	#103	—	43	12	no	—
243	#102	—	43	12	no	—
244	C₄×C₈	—	43	12	no	—
245	C₂⁴	—	86	3	no	—
246	#41	—	172	32	no	—
247	C₂×Q₈	—	2	3	no	—
248	C₂×Q₈	—	2	6	no	—
249	C₂×Q₈	—	2	6	no	—
250	C₂×Q₈	#10621884	1	8	yes	complemented
251	C₂×Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
252	C₂×Q₈	C₂³×D₄₃	1	6	yes	—
253	C₂×D₄	—	43	6	no	—
254	C₂×D₄	—	86	12	no	—
255	C₂×D₄	—	86	3	no	—
256	C₂×D₄	—	86	12	no	—
257	C₂×D₄	—	86	6	no	—
258	C₂×D₄	—	86	6	no	—
259	C₂×D₄	—	344	6	no	—
260	C₂×D₄	—	172	12	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	3	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	12	no	—
265	C₂²×C₄	—	172	12	no	—
266	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
268	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
269	SD₁₆	—	86	24	no	—
270	SD₁₆	—	172	24	no	—
271	C₂×C₈	—	43	4	no	—
272	C₂×C₈	—	43	2	no	—
273	C₂×C₈	—	43	6	no	—
274	C₂×C₈	—	86	6	no	—
275	C₄⋊C₄	—	2	12	no	—
276	C₄⋊C₄	—	2	12	no	—
277	C₄⋊C₄	—	2	12	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	43	2	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
282	C₄⋊C₄	C₂³×D₄₃	1	2	yes	—
283	C₂²⋊C₄	—	86	12	no	—
284	C₄²	—	2	12	no	—
285	C₄²	C₂×C₂²⋊D₄₃	1	6	yes	—
286	C₄²	C₂³×D₄₃	1	6	yes	—
287	C₂³	—	86	6	no	—
288	C₂³	—	172	6	no	—
289	C₂³	—	172	6	no	—
290	C₂³	—	172	6	no	—
291	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
292	Q₈	—	2	6	no	—
293	Q₈	—	2	6	no	—
294	Q₈	—	4	6	no	—
295	Q₈	C₂×#10621884	1	16	yes	complemented
296	D₄	—	86	6	no	—
297	D₄	—	86	6	no	—
298	D₄	—	344	12	no	—
299	D₄	—	172	6	no	—
300	D₄	—	172	24	no	—
301	C₂×C₄	—	2	6	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	6	no	—
304	C₂×C₄	—	4	6	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	24	no	—
309	C₂×C₄	C₂×#10621884	1	12	yes	—
310	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
311	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
313	C₂×C₄	#41×D₄₃	1	2	yes	—
314	C₂×C₄	#13063371	1	2	yes	—
315	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
316	C₈	—	43	4	no	—
317	C₈	—	86	6	no	—
318	C₂²	—	344	12	no	—
319	C₂²	—	172	6	no	—
320	C₂²	—	172	6	no	—
321	C₂²	—	172	6	no	—
322	C₂²	#14583568	1	2	yes	—
323	C₂²	C₂²×#10621884	1	2	yes	—
324	C₂²	#14583707	1	2	yes	—
325	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
326	C₄	—	2	6	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	6	no	—
329	C₄	—	172	2	no	—
330	C₄	#14583568	1	2	yes	—
331	C₄	C₂²×#10621884	1	12	yes	—
332	C₄	C₂×#41×D₄₃	1	1	yes	characteristic
333	C₄	C₂×#13063371	1	1	yes	characteristic
334	C₂	—	344	6	no	—
335	C₂	#19147572	1	4	yes	direct product factor
336	C₂	C₂×#14583568	1	1	yes	characteristic
337	C₂	C₂³×#10621884	1	1	yes	characteristic
338	C₂	C₂×#14583707	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#557	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#14583571	C₂
hypercenter	C₂×#557	D₄₃
hypocenter	C₄₃	C₂×#3392
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
2. #19147572	C₂	direct (N × H)
3. C₂×#14583273	C₂	semidirect
4. C₂×#14583099	C₂	semidirect
5. C₂×#14583101	C₂	semidirect
6. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
7. C₂×C₄×#10621884	C₂	semidirect
8. C₂×C₄₃×#557	C₂	semidirect
17. #14583273	C₂²	semidirect
18. #14583101	C₂²	semidirect
19. #14583099	C₂²	semidirect
20. Q₈×C₄₃⋊C₈	C₂²	semidirect
21. C₄×#10621884	C₂²	semidirect
28. C₄₃×#557	C₂²	semidirect
79. C₂×Q₈×C₄₃	SD₁₆	semidirect
114. C₂×C₄₃⋊C₈	#41	semidirect
125. Q₈×C₄₃	C₂×SD₁₆	semidirect
147. C₄₃⋊C₈	C₂×#41	semidirect
165. C₂×#557	D₄₃	semidirect
173. C₂×C₄₃	#3392	semidirect
182. #557	C₂×D₄₃	semidirect
203. C₄₃	C₂×#3392	semidirect
250. C₂×Q₈	#10621884	semidirect
295. Q₈	C₂×#10621884	semidirect
335. C₂	#19147572	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×#14583571	2
#19147572	2
C₂×#14583273	2
C₂×#14583099	2
C₂×#14583101	2
C₂×Q₈×C₄₃⋊C₈	2
C₂×C₄×#10621884	2
C₂×C₄₃×#557	2
C₂×#3392	43

Subgroups — C2×#19147572 #25977494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147572 #25977494