Subgroups — C₂×#21437240 #29495205

Order 15104.

2074 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838207	C₂	1	1	yes	maximal, characteristic, complemented
2	#21437240	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16837843	C₂	1	3	yes	maximal, complemented
4	C₂×#16837837	C₂	1	3	yes	maximal, complemented
5	C₂×#16837832	C₂	1	3	yes	maximal, complemented
6	C₂²×#14273670	C₂	1	3	yes	maximal, complemented
7	C₂×#16837819	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#578	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273773	—	2	1	no	—
10	#16838207	—	2	4	no	—
11	#16838207	C₂²	1	8	yes	complemented
12	C₂×#14273755	—	2	3	no	—
13	C₂×#14273749	C₂²	1	3	yes	—
14	C₂²×D₄₇₂	C₂²	1	3	yes	—
15	#16837843	C₂²	1	24	yes	complemented
16	#16837837	C₂²	1	24	yes	complemented
17	#16837832	C₂²	1	24	yes	complemented
18	C₂×#14273673	C₂²	1	3	yes	complemented
19	C₂×#14273671	C₂²	1	1	yes	characteristic, complemented
20	C₂×#14273670	C₂²	1	3	yes	complemented
21	C₂×#14273670	C₂²	1	6	yes	complemented
22	C₂×#14273670	C₂²	1	24	yes	complemented
23	C₂×#14273668	C₂²	1	3	yes	complemented
24	#16837819	C₂²	1	8	yes	complemented
25	C₂×#14273667	C₂²	1	3	yes	complemented
26	C₂²×C₅₉⋊C₁₆	C₂²	1	3	yes	complemented
27	C₂×2₋¹⁺⁴×C₅₉	—	2	1	no	—
28	C₅₉×#578	—	2	4	no	—
29	C₅₉×#578	C₂²	1	8	yes	complemented
30	C₂×#137×C₅₉	—	2	3	no	—
31	C₂×#135×C₅₉	C₂²	1	3	yes	complemented
32	C₂²×Q₁₆×C₅₉	C₂²	1	3	yes	complemented
33	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
34	#14273773	—	2	8	no	—
35	#14273773	—	4	4	no	—
36	C₂×#11745689	—	4	1	no	—
37	C₂×#11745685	—	2	3	no	—
38	C₂²×D₂₃₆	—	2	3	no	—
39	C₂×D₄×D₅₉	—	4	3	no	—
40	#14273755	—	2	24	no	—
41	#14273749	—	2	6	no	—
42	#14273749	C₂³	1	12	yes	—
43	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
44	C₂×#11745660	—	2	3	no	—
45	C₂×D₄₇₂	—	2	3	no	—
46	C₂×D₄₇₂	—	2	6	no	—
47	C₂×D₄₇₂	C₂³	1	3	yes	—
48	C₂×D₄₇₂	C₂³	1	12	yes	—
49	#14273673	C₂³	1	12	yes	complemented
50	#14273671	C₂³	1	4	yes	complemented
51	#14273670	C₂³	1	12	yes	complemented
52	#14273670	C₂³	1	24	yes	complemented
53	#14273668	C₂³	1	12	yes	complemented
54	#14273667	C₂³	1	12	yes	complemented
55	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₅₉⋊C₁₆	C₂³	1	3	yes	—
57	C₂×C₅₉⋊C₁₆	C₂³	1	12	yes	complemented
58	2₋¹⁺⁴×C₅₉	—	2	8	no	—
59	2₋¹⁺⁴×C₅₉	—	4	4	no	—
60	C₂×#41×C₅₉	—	2	3	no	—
61	C₂×#41×C₅₉	—	4	3	no	—
62	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
63	C₂²×Q₈×C₅₉	—	2	3	no	—
64	C₂²×Q₈×C₅₉	—	4	1	no	—
65	#137×C₅₉	—	2	24	no	—
66	#135×C₅₉	—	2	6	no	—
67	#135×C₅₉	C₂³	1	12	yes	complemented
68	C₂×Q₁₆×C₅₉	—	2	3	no	—
69	C₂×Q₁₆×C₅₉	—	2	6	no	—
70	C₂×Q₁₆×C₅₉	C₂³	1	3	yes	—
71	C₂×Q₁₆×C₅₉	C₂³	1	12	yes	complemented
72	C₂×SD₁₆×C₅₉	—	2	3	no	—
73	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
74	#131×C₅₉	D₄	1	4	yes	—
75	#131×C₅₉	C₂³	1	4	yes	—
76	C₂×M₁₆×C₅₉	D₄	1	3	yes	—
77	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
78	C₂³×D₅₉	—	4	3	no	—
79	#41×C₅₉	—	2	12	no	—
80	#41×C₅₉	—	4	24	no	—
81	#41×C₅₉	—	4	6	no	—
82	#41×C₅₉	D₈	1	4	yes	—
83	#41×C₅₉	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₅₉	—	2	3	no	—
85	C₂×Q₈×C₅₉	—	2	12	no	—
86	C₂×Q₈×C₅₉	—	4	8	no	—
87	C₂×Q₈×C₅₉	—	4	3	no	—
88	C₂×Q₈×C₅₉	—	4	6	no	—
89	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
90	C₂×D₄×C₅₉	—	2	1	no	—
91	C₂×D₄×C₅₉	—	4	3	no	—
92	C₂×D₄×C₅₉	D₈	1	3	yes	—
93	C₂²×C₄×C₅₉	—	4	3	no	—
94	C₂²×C₄×C₅₉	—	4	3	no	—
95	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
96	C₂×C₂²⋊D₅₉	—	4	3	no	—
97	#11745689	—	4	8	no	—
98	D₄×D₅₉	—	4	24	no	—
99	#11745685	—	2	12	no	—
100	#11745685	—	4	6	no	—
101	C₂×D₂₃₆	—	2	3	no	—
102	C₂×D₂₃₆	—	2	12	no	—
103	C₂×D₂₃₆	—	4	3	no	—
104	C₂×D₂₃₆	—	4	6	no	—
105	C₂×C₄×D₅₉	—	4	3	no	—
106	C₂×Dic₁₁₈	—	2	1	no	—
107	Q₁₆×C₅₉	—	2	3	no	—
108	Q₁₆×C₅₉	—	2	12	no	—
109	Q₁₆×C₅₉	C₂⁴	1	6	yes	—
110	SD₁₆×C₅₉	—	2	12	no	—
111	D₈×C₅₉	—	2	1	no	—
112	D₈×C₅₉	C₂⁴	1	2	yes	—
113	M₁₆×C₅₉	C₂×D₄	1	12	yes	—
114	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
115	C₂×C₈×C₅₉	C₂×D₄	1	6	yes	—
116	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₅₉	C₂⁴	1	6	yes	—
118	Dic₂₃₆	—	2	1	no	—
119	Dic₂₃₆	C₂⁴	1	2	yes	—
120	D₄₇₂	—	2	3	no	—
121	D₄₇₂	—	2	12	no	—
122	D₄₇₂	C₂⁴	1	6	yes	—
123	#11745660	—	2	12	no	—
124	C₅₉⋊C₁₆	C₂⁴	1	2	yes	—
125	C₅₉⋊C₁₆	C₂⁴	1	6	yes	—
126	C₂³×C₅₉	—	4	1	no	—
127	C₂³×C₅₉	C₂×D₈	1	3	yes	—
128	C₂²×D₅₉	—	4	3	no	—
129	C₂²×D₅₉	—	4	12	no	—
130	C₂²×D₅₉	—	16	6	no	—
131	Q₈×C₅₉	—	2	6	no	—
132	Q₈×C₅₉	—	4	3	no	—
133	Q₈×C₅₉	—	4	12	no	—
134	Q₈×C₅₉	C₂×D₈	1	4	yes	—
135	D₄×C₅₉	—	2	2	no	—
136	D₄×C₅₉	—	4	12	no	—
137	D₄×C₅₉	—	4	1	no	—
138	D₄×C₅₉	C₂×D₈	1	12	yes	—
139	C₂×C₄×C₅₉	—	4	3	no	—
140	C₂×C₄×C₅₉	—	4	12	no	—
141	C₂×C₄×C₅₉	—	4	12	no	—
142	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
143	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
146	C₂²⋊D₅₉	—	4	12	no	—
147	C₂×Dic₅₉	—	4	1	no	—
148	D₂₃₆	—	2	6	no	—
149	D₂₃₆	—	4	3	no	—
150	D₂₃₆	—	4	12	no	—
151	C₄×D₅₉	—	4	12	no	—
152	Dic₁₁₈	—	2	2	no	—
153	Dic₁₁₈	—	4	1	no	—
154	C₈×C₅₉	C₂²×D₄	1	6	yes	—
155	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
156	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
157	C₂×#3416	—	59	1	no	maximal, Sylow
158	C₂²×C₅₉	—	2	6	no	—
159	C₂²×C₅₉	—	4	2	no	—
160	C₂²×C₅₉	—	8	2	no	—
161	C₂²×C₅₉	C₂²×D₈	1	6	yes	—
162	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₅₉	—	4	6	no	—
164	C₂×D₅₉	—	8	6	no	—
165	C₂×D₅₉	—	16	12	no	—
166	C₄×C₅₉	—	4	6	no	—
167	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₅₉	C₂²×D₈	1	6	yes	—
169	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
170	Dic₅₉	—	4	2	no	—
171	C₂×#578	D₅₉	1	1	yes	characteristic, complemented
172	C₂×#576	—	59	1	no	—
173	#3416	—	59	16	no	—
174	C₂×#510	—	59	3	no	—
175	C₂×#509	—	59	3	no	—
176	C₂×#508	—	59	3	no	—
177	C₂²×SD₃₂	—	59	3	no	—
178	C₂×#504	—	59	1	no	—
179	C₂×C₅₉	—	2	6	no	—
180	C₂×C₅₉	—	8	2	no	—
181	C₂×C₅₉	#3416	1	2	yes	complemented
182	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
183	D₅₉	—	8	6	no	—
184	C₂×2₋¹⁺⁴	—	2	1	no	—
185	C₂×H(5,𝔽₂)	—	118	1	no	—
186	#578	—	2	4	no	—
187	#578	C₂×D₅₉	1	8	yes	complemented
188	#576	—	59	8	no	—
189	#576	—	118	4	no	—
190	C₂×#137	—	2	3	no	—
191	C₂×AGL(1,ℤ/8)	—	118	3	no	—
192	C₂×#135	—	59	3	no	—
193	C₂×#135	C₂×D₅₉	1	3	yes	complemented
194	C₂²×Q₁₆	C₂×D₅₉	1	3	yes	complemented
195	C₂²×D₈	—	59	3	no	—
196	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
197	#510	—	59	24	no	—
198	#509	—	59	24	no	—
199	#508	—	59	24	no	—
200	C₂×Q₃₂	—	59	3	no	—
201	C₂×SD₃₂	—	59	3	no	—
202	C₂×SD₃₂	—	59	6	no	—
203	C₂×SD₃₂	—	59	24	no	—
204	C₂×SD₃₂	—	59	1	no	—
205	C₂×D₁₆	—	59	3	no	—
206	#504	—	59	8	no	—
207	C₂×M₃₂	—	59	3	no	—
208	C₂²×C₁₆	—	59	3	no	—
209	C₅₉	C₂×#3416	1	1	yes	characteristic, complemented, Sylow
210	2₋¹⁺⁴	—	2	8	no	—
211	2₋¹⁺⁴	—	4	4	no	—
212	H(5,𝔽₂)	—	118	8	no	—
213	H(5,𝔽₂)	—	236	4	no	—
214	C₂×#41	—	2	3	no	—
215	C₂×#41	—	4	3	no	—
216	C₂×#41	—	118	3	no	—
217	C₂×#41	—	236	1	no	—
218	C₂×#41	C₂²⋊D₅₉	1	1	yes	characteristic
219	C₂²×Q₈	—	2	3	no	—
220	C₂²×Q₈	—	4	1	no	—
221	C₂²×D₄	—	118	3	no	—
222	C₂²×D₄	—	236	3	no	—
223	#137	—	2	24	no	—
224	AGL(1,ℤ/8)	—	118	24	no	—
225	#135	—	2	6	no	—
226	#135	—	59	12	no	—
227	#135	—	118	6	no	—
228	#135	C₂²×D₅₉	1	12	yes	complemented
229	C₂×Q₁₆	—	2	3	no	—
230	C₂×Q₁₆	—	2	6	no	—
231	C₂×Q₁₆	—	59	1	no	—
232	C₂×Q₁₆	C₂²×D₅₉	1	3	yes	—
233	C₂×Q₁₆	C₂²×D₅₉	1	12	yes	complemented
234	C₂×SD₁₆	—	2	3	no	—
235	C₂×SD₁₆	—	118	3	no	—
236	C₂×D₈	—	59	3	no	—
237	C₂×D₈	—	59	12	no	—
238	C₂×D₈	—	118	3	no	—
239	C₂×D₈	—	118	6	no	—
240	C₂×D₈	C₂²×D₅₉	1	1	yes	characteristic
241	#131	C₂²⋊D₅₉	1	4	yes	—
242	#131	C₂²×D₅₉	1	4	yes	—
243	C₂×M₁₆	C₂²⋊D₅₉	1	3	yes	—
244	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
245	Q₃₂	—	59	12	no	—
246	SD₃₂	—	59	12	no	—
247	SD₃₂	—	59	24	no	—
248	SD₃₂	—	59	4	no	—
249	D₁₆	—	59	12	no	—
250	M₃₂	—	59	12	no	—
251	C₂×C₁₆	—	59	1	no	—
252	C₂×C₁₆	—	59	3	no	—
253	C₂×C₁₆	—	59	12	no	—
254	C₂⁴	—	236	3	no	—
255	#41	—	2	12	no	—
256	#41	—	4	24	no	—
257	#41	—	4	6	no	—
258	#41	—	118	12	no	—
259	#41	—	236	8	no	—
260	#41	—	236	6	no	—
261	#41	#11745669	1	4	yes	—
262	#41	C₂×C₂²⋊D₅₉	1	4	yes	—
263	C₂×Q₈	—	2	3	no	—
264	C₂×Q₈	—	2	12	no	—
265	C₂×Q₈	—	4	8	no	—
266	C₂×Q₈	—	4	3	no	—
267	C₂×Q₈	—	4	6	no	—
268	C₂×Q₈	—	118	1	no	—
269	C₂×Q₈	#11745669	1	1	yes	characteristic
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	3	no	—
272	C₂×D₄	—	118	3	no	—
273	C₂×D₄	—	118	12	no	—
274	C₂×D₄	—	236	3	no	—
275	C₂×D₄	—	236	24	no	—
276	C₂×D₄	—	236	3	no	—
277	C₂×D₄	—	236	6	no	—
278	C₂×D₄	#11745669	1	3	yes	—
279	C₂²×C₄	—	4	3	no	—
280	C₂²×C₄	—	4	3	no	—
281	C₂²×C₄	—	236	3	no	—
282	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
283	Q₁₆	—	2	3	no	—
284	Q₁₆	—	2	12	no	—
285	Q₁₆	—	59	2	no	—
286	Q₁₆	—	118	1	no	—
287	Q₁₆	C₂³×D₅₉	1	6	yes	—
288	SD₁₆	—	2	12	no	—
289	SD₁₆	—	118	12	no	—
290	D₈	—	2	1	no	—
291	D₈	—	59	6	no	—
292	D₈	—	118	3	no	—
293	D₈	—	118	12	no	—
294	D₈	C₂³×D₅₉	1	2	yes	—
295	M₁₆	C₂×C₂²⋊D₅₉	1	12	yes	—
296	C₂×C₈	C₂×C₂²⋊D₅₉	1	3	yes	—
297	C₂×C₈	C₂×C₂²⋊D₅₉	1	6	yes	—
298	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₅₉	1	6	yes	—
300	C₁₆	—	59	2	no	—
301	C₁₆	—	59	6	no	—
302	C₂³	—	4	1	no	—
303	C₂³	—	944	6	no	—
304	C₂³	—	236	12	no	—
305	C₂³	—	236	3	no	—
306	C₂³	C₂×#11745669	1	3	yes	—
307	Q₈	—	2	6	no	—
308	Q₈	—	4	12	no	—
309	Q₈	—	4	3	no	—
310	Q₈	—	118	2	no	—
311	Q₈	—	236	1	no	—
312	Q₈	C₂×#11745669	1	4	yes	—
313	D₄	—	2	2	no	—
314	D₄	—	4	12	no	—
315	D₄	—	4	1	no	—
316	D₄	—	118	6	no	—
317	D₄	—	236	12	no	—
318	D₄	—	236	12	no	—
319	D₄	—	236	3	no	—
320	D₄	C₂×#11745669	1	12	yes	—
321	C₂×C₄	—	4	12	no	—
322	C₂×C₄	—	4	3	no	—
323	C₂×C₄	—	4	12	no	—
324	C₂×C₄	—	236	1	no	—
325	C₂×C₄	—	236	12	no	—
326	C₂×C₄	C₂×#11745669	1	6	yes	—
327	C₂×C₄	C₂×#11745669	1	3	yes	—
328	C₂×C₄	C₂²×C₂²⋊D₅₉	1	6	yes	—
329	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
331	C₈	C₂²×C₂²⋊D₅₉	1	6	yes	—
332	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
333	C₂²	—	2	6	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	944	12	no	—
337	C₂²	—	472	6	no	—
338	C₂²	—	236	6	no	—
339	C₂²	C₂²×#11745669	1	6	yes	—
340	C₂²	C₂²×#11745669	1	1	yes	characteristic
341	C₄	—	4	6	no	—
342	C₄	—	236	2	no	—
343	C₄	C₂²×#11745669	1	6	yes	—
344	C₄	C₂²×#11745669	1	1	yes	characteristic
345	C₄	C₂³×C₂²⋊D₅₉	1	1	yes	characteristic
346	C₂	—	2	6	no	—
347	C₂	—	8	2	no	—
348	C₂	—	472	6	no	—
349	C₂	#21437240	1	2	yes	direct product factor
350	C₂	C₂³×#11745669	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₈×C₅₉	C₂⁵
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂×C₅₉×#578	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	C₂×#578	D₅₉
hypocenter	C₅₉	C₂×#3416
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#578

Complements

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

N	H ≅ G/N	type
1. C₂×#16838207	C₂	semidirect
2. #21437240	C₂	direct (N × H)
3. C₂×#16837843	C₂	semidirect
4. C₂×#16837837	C₂	semidirect
5. C₂×#16837832	C₂	semidirect
6. C₂²×#14273670	C₂	semidirect
7. C₂×#16837819	C₂	semidirect
8. C₂×C₅₉×#578	C₂	semidirect
11. #16838207	C₂²	semidirect
15. #16837843	C₂²	semidirect
16. #16837837	C₂²	semidirect
17. #16837832	C₂²	semidirect
18. C₂×#14273673	C₂²	semidirect
19. C₂×#14273671	C₂²	semidirect
20. C₂×#14273670	C₂²	semidirect
21. C₂×#14273670	C₂²	semidirect
22. C₂×#14273670	C₂²	semidirect
23. C₂×#14273668	C₂²	semidirect
24. #16837819	C₂²	semidirect
25. C₂×#14273667	C₂²	semidirect
26. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
29. C₅₉×#578	C₂²	semidirect
31. C₂×#135×C₅₉	C₂²	semidirect
32. C₂²×Q₁₆×C₅₉	C₂²	semidirect
49. #14273673	C₂³	semidirect
50. #14273671	C₂³	semidirect
51. #14273670	C₂³	semidirect
52. #14273670	C₂³	semidirect
53. #14273668	C₂³	semidirect
54. #14273667	C₂³	semidirect
57. C₂×C₅₉⋊C₁₆	C₂³	semidirect
67. #135×C₅₉	C₂³	semidirect
71. C₂×Q₁₆×C₅₉	C₂³	semidirect
171. C₂×#578	D₅₉	semidirect
181. C₂×C₅₉	#3416	semidirect
187. #578	C₂×D₅₉	semidirect
193. C₂×#135	C₂×D₅₉	semidirect
194. C₂²×Q₁₆	C₂×D₅₉	semidirect
209. C₅₉	C₂×#3416	semidirect
228. #135	C₂²×D₅₉	semidirect
233. C₂×Q₁₆	C₂²×D₅₉	semidirect
349. C₂	#21437240	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×#16838207	2
#21437240	2
C₂×#16837843	2
C₂×#16837837	2
C₂×#16837832	2
C₂²×#14273670	2
C₂×#16837819	2
C₂×C₅₉×#578	2
C₂×#3416	59

Subgroups — C2×#21437240 #29495205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21437240 #29495205