Subgroups — C₂×#19147577 #25977499

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×#14583628	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583571	C₂	1	1	yes	maximal, characteristic
3	#19147577	C₂	1	16	yes	maximal, direct product factor
4	C₂×#14583272	C₂	1	3	yes	maximal, complemented
5	C₂×#14583100	C₂	1	3	yes	maximal, complemented
6	C₂×#14583105	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#10621873	C₂	1	3	yes	maximal, complemented
8	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621902	—	2	1	no	—
10	C₂²×#10621873	C₂²	1	3	yes	—
11	C₂²×#10621873	C₂²	1	6	yes	—
12	#14583628	—	2	4	no	—
13	#14583628	C₂²	1	8	yes	complemented
14	#14583571	—	2	4	no	—
15	#14583571	C₂²	1	8	yes	—
16	C₂×#13063325	C₂²	1	3	yes	—
17	C₂×#13063322	C₂²	1	3	yes	—
18	C₂×#13063316	—	2	3	no	—
19	C₂×#13063300	—	2	3	no	—
20	C₂×Q₈×Dic₄₃	—	2	1	no	—
21	C₂×C₄×Dic₈₆	C₂²	1	3	yes	—
22	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
23	#14583272	C₂²	1	24	yes	complemented
24	#14583105	C₂²	1	24	yes	complemented
25	#14583100	C₂²	1	24	yes	complemented
26	C₄×#10621873	C₂²	1	24	yes	complemented
27	C₂×#13063257	C₂²	1	1	yes	characteristic
28	C₂×#13063250	C₂²	1	3	yes	—
29	C₂×#13063247	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₄₃	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
34	C₂×#10621902	—	2	8	no	—
35	C₂×#10621902	—	4	8	no	—
36	C₂²×Dic₈₆	C₂³	1	3	yes	—
37	C₂²×D₁₇₂	—	2	3	no	—
38	C₂²×D₁₇₂	—	2	3	no	—
39	C₂²×D₁₇₂	C₂³	1	3	yes	—
40	C₂²×C₄×D₄₃	—	2	3	no	—
41	C₂×#10621873	—	2	12	no	—
42	C₂×#10621873	—	4	12	no	—
43	C₂×#10621873	C₂³	1	12	yes	—
44	C₂×#10621873	C₂³	1	24	yes	—
45	#13063325	—	2	6	no	—
46	#13063325	C₂³	1	12	yes	—
47	#13063322	—	2	6	no	—
48	#13063322	C₂³	1	12	yes	—
49	#13063316	—	2	24	no	—
50	#13063300	—	2	24	no	—
51	Q₈×Dic₄₃	—	2	8	no	—
52	C₄×Dic₈₆	—	2	6	no	—
53	C₄×Dic₈₆	C₂³	1	12	yes	—
54	C₄×D₁₇₂	—	2	6	no	—
55	C₄×D₁₇₂	C₂³	1	12	yes	—
56	C₂×C₄⋊Dic₄₃	—	2	3	no	—
57	C₂×C₄⋊Dic₄₃	—	2	3	no	—
58	C₂×C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
60	C₂×Dic₄₃⋊C₄	—	2	3	no	—
61	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
62	C₂×C₄×Dic₄₃	—	2	3	no	—
63	#13063257	C₂³	1	4	yes	—
64	#13063250	C₂³	1	12	yes	—
65	#13063247	C₂³	1	12	yes	—
66	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₄₃	D₄	1	4	yes	—
70	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
72	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
73	C₄⋊C₈×C₄₃	C₂³	1	12	yes	—
74	C₄×C₈×C₄₃	C₂³	1	12	yes	—
75	C₂³×D₄₃	—	2	3	no	—
76	C₂×Q₈×C₄₃	SD₁₆	1	8	yes	complemented
77	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₃	—	2	1	no	—
82	C₂²×Dic₄₃	—	2	3	no	—
83	C₂²×Dic₄₃	—	2	1	no	—
84	#10621902	—	4	32	no	—
85	C₂×D₁₇₂	—	2	3	no	—
86	C₂×D₁₇₂	—	2	12	no	—
87	C₂×D₁₇₂	—	2	12	no	—
88	C₂×D₁₇₂	—	2	6	no	—
89	C₂×D₁₇₂	—	2	6	no	—
90	C₂×D₁₇₂	—	4	12	no	—
91	C₂×D₁₇₂	—	8	6	no	—
92	C₂×D₁₇₂	C₂⁴	1	6	yes	—
93	C₂×C₄×D₄₃	—	2	12	no	—
94	C₂×C₄×D₄₃	—	4	12	no	—
95	C₂×Dic₈₆	—	2	6	no	—
96	C₂×Dic₈₆	—	2	6	no	—
97	C₂×Dic₈₆	—	2	3	no	—
98	C₂×Dic₈₆	C₂⁴	1	6	yes	—
99	C₂×C₈×C₄₃	—	2	6	no	—
100	C₂×C₈×C₄₃	#41	1	4	yes	complemented
101	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
102	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
103	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
104	C₄²×C₄₃	C₂×D₄	1	6	yes	—
105	C₄²×C₄₃	C₂⁴	1	6	yes	—
106	(C₂×D₄₃)⋊C₄	—	2	12	no	—
107	C₄⋊Dic₄₃	—	2	12	no	—
108	C₄⋊Dic₄₃	—	2	12	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	C₂⁴	1	2	yes	—
112	C₄⋊Dic₄₃	C₂⁴	1	2	yes	—
113	Dic₄₃⋊C₄	—	2	12	no	—
114	C₄×Dic₄₃	—	2	12	no	—
115	#10621873	—	2	24	no	—
116	#10621873	—	4	24	no	—
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	C₂×SD₁₆	1	16	yes	complemented
123	C₂×C₄×C₄₃	—	2	6	no	—
124	C₂×C₄×C₄₃	C₂×SD₁₆	1	12	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
128	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	6	no	—
133	C₂×Dic₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	4	6	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	D₁₇₂	—	2	6	no	—
138	D₁₇₂	—	2	6	no	—
139	D₁₇₂	—	4	6	no	—
140	D₁₇₂	—	4	24	no	—
141	D₁₇₂	—	8	12	no	—
142	C₄×D₄₃	—	4	24	no	—
143	Dic₈₆	—	2	6	no	—
144	Dic₈₆	—	2	6	no	—
145	Dic₈₆	—	4	6	no	—
146	C₈×C₄₃	—	2	6	no	—
147	C₈×C₄₃	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	#550	1	2	yes	—
159	C₄×C₄₃	C₂²×SD₁₆	1	12	yes	—
160	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
161	C₄×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
162	Dic₄₃	—	4	6	no	—
163	Dic₄₃	—	4	2	no	—
164	Dic₄₃	—	4	2	no	—
165	C₂×#557	—	43	1	no	—
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	D₄₃	1	1	yes	characteristic, complemented
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	—	43	8	no	—
182	#557	—	86	4	no	—
183	#550	—	43	8	no	—
184	#550	—	86	4	no	—
185	C₂×C₄⋊Q₈	—	43	3	no	—
186	C₂×#127	—	43	3	no	—
187	C₂×#125	—	86	3	no	—
188	C₂×#121	—	86	3	no	—
189	C₂×C₄×Q₈	—	43	3	no	—
190	C₂×C₄×Q₈	—	86	1	no	—
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	C₂×D₄₃	1	8	yes	complemented
197	C₄×SD₁₆	—	43	24	no	—
198	C₂×#106	—	43	1	no	—
199	C₂×C₄⋊C₈	C₂×D₄₃	1	3	yes	—
200	C₂×#103	—	43	3	no	—
201	C₂×#102	—	43	3	no	—
202	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
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	—	43	3	no	—
207	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
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	C₂²×D₄₃	1	3	yes	—
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	43	12	no	—
219	C₄⋊Q₈	—	86	6	no	—
220	#127	—	43	12	no	—
221	#127	—	86	6	no	—
222	#125	—	86	24	no	—
223	#121	—	86	24	no	—
224	C₄×Q₈	—	43	12	no	—
225	C₄×Q₈	—	86	8	no	—
226	C₄×Q₈	—	86	6	no	—
227	C₄×Q₈	D₁₇₂	1	4	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	—	43	1	no	—
232	C₂×C₄⋊C₄	—	43	1	no	—
233	C₂×C₄⋊C₄	—	86	3	no	—
234	C₂×C₄⋊C₄	—	86	3	no	—
235	C₂×C₄⋊C₄	—	86	3	no	—
236	C₂×C₄⋊C₄	D₁₇₂	1	3	yes	—
237	C₂×C₂²⋊C₄	—	86	3	no	—
238	C₂×C₄²	—	86	3	no	—
239	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
240	#106	—	43	4	no	—
241	C₄⋊C₈	C₂²×D₄₃	1	12	yes	—
242	#103	—	43	12	no	—
243	#102	—	43	12	no	—
244	C₄×C₈	C₂²×D₄₃	1	12	yes	—
245	C₂⁴	—	86	3	no	—
246	#41	—	172	32	no	—
247	C₂×Q₈	—	43	6	no	—
248	C₂×Q₈	—	86	3	no	—
249	C₂×Q₈	—	86	6	no	—
250	C₂×Q₈	—	86	6	no	—
251	C₂×Q₈	#10621873	1	8	yes	complemented
252	C₂×Q₈	C₂×D₁₇₂	1	4	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	—	86	1	no	—
262	C₂²×C₄	—	86	3	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	12	no	—
265	C₂²×C₄	—	172	12	no	—
266	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
267	C₂²×C₄	C₂×D₁₇₂	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	—	2	6	no	—
272	C₂×C₈	#10621902	1	4	yes	complemented
273	C₂×C₈	C₂³×D₄₃	1	6	yes	—
274	C₂×C₈	C₂³×D₄₃	1	2	yes	—
275	C₄⋊C₄	—	43	2	no	—
276	C₄⋊C₄	—	43	2	no	—
277	C₄⋊C₄	—	86	1	no	—
278	C₄⋊C₄	—	86	1	no	—
279	C₄⋊C₄	—	86	12	no	—
280	C₄⋊C₄	—	86	12	no	—
281	C₄⋊C₄	—	86	12	no	—
282	C₄⋊C₄	C₂×D₁₇₂	1	12	yes	—
283	C₂²⋊C₄	—	86	12	no	—
284	C₄²	—	86	12	no	—
285	C₄²	C₂×D₁₇₂	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×D₁₇₂	1	1	yes	characteristic
292	Q₈	—	86	6	no	—
293	Q₈	—	86	6	no	—
294	Q₈	—	172	6	no	—
295	Q₈	C₂×#10621873	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	—	86	2	no	—
303	C₂×C₄	—	86	6	no	—
304	C₂×C₄	—	86	2	no	—
305	C₂×C₄	—	172	6	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	24	no	—
309	C₂×C₄	C₂×#10621873	1	12	yes	—
310	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
311	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
313	C₂×C₄	C₂×#10621902	1	2	yes	—
314	C₂×C₄	#13063371	1	2	yes	—
315	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
316	C₈	—	2	6	no	—
317	C₈	C₂×#10621902	1	4	yes	complemented
318	C₂²	—	344	12	no	—
319	C₂²	—	172	6	no	—
320	C₂²	—	172	6	no	—
321	C₂²	—	172	6	no	—
322	C₂²	#14583571	1	2	yes	—
323	C₂²	C₂²×#10621873	1	2	yes	—
324	C₂²	#14583711	1	2	yes	—
325	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
326	C₄	—	2	6	no	—
327	C₄	—	172	2	no	—
328	C₄	—	172	6	no	—
329	C₄	—	172	2	no	—
330	C₄	#14583571	1	2	yes	—
331	C₄	C₂²×#10621873	1	12	yes	—
332	C₄	C₂²×#10621902	1	1	yes	characteristic
333	C₄	C₂×#13063371	1	1	yes	characteristic
334	C₂	—	344	6	no	—
335	C₂	#19147577	1	4	yes	direct product factor
336	C₂	C₂×#14583571	1	1	yes	characteristic
337	C₂	C₂³×#10621873	1	1	yes	characteristic
338	C₂	C₂×#14583711	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#14583571	C₂
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	C₂×#3392
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14583628	C₂	semidirect
3. #19147577	C₂	direct (N × H)
4. C₂×#14583272	C₂	semidirect
5. C₂×#14583100	C₂	semidirect
6. C₂×#14583105	C₂	semidirect
7. C₂×C₄×#10621873	C₂	semidirect
8. C₂×C₈×Q₈×C₄₃	C₂	semidirect
13. #14583628	C₂²	semidirect
23. #14583272	C₂²	semidirect
24. #14583105	C₂²	semidirect
25. #14583100	C₂²	semidirect
26. C₄×#10621873	C₂²	semidirect
31. C₈×Q₈×C₄₃	C₂²	semidirect
76. C₂×Q₈×C₄₃	SD₁₆	semidirect
100. C₂×C₈×C₄₃	#41	semidirect
122. Q₈×C₄₃	C₂×SD₁₆	semidirect
147. C₈×C₄₃	C₂×#41	semidirect
171. C₂×C₈×Q₈	D₄₃	semidirect
173. C₂×C₄₃	#3392	semidirect
196. C₈×Q₈	C₂×D₄₃	semidirect
203. C₄₃	C₂×#3392	semidirect
251. C₂×Q₈	#10621873	semidirect
272. C₂×C₈	#10621902	semidirect
295. Q₈	C₂×#10621873	semidirect
317. C₈	C₂×#10621902	semidirect
335. C₂	#19147577	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×#14583628	2
C₂×#14583571	2
#19147577	2
C₂×#14583272	2
C₂×#14583100	2
C₂×#14583105	2
C₂×C₄×#10621873	2
C₂×C₈×Q₈×C₄₃	2
C₂×#3392	43

Subgroups — C2×#19147577 #25977499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147577 #25977499