Subgroups — C₂×#13540 #499557

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1946	C₂	1	1	yes	maximal, characteristic
2	C₂×#1849	C₂	1	1	yes	maximal, characteristic
3	#13540	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1389	C₂	1	2	yes	maximal
5	C₂×#1388	C₂	1	2	yes	maximal
6	C₂×#1375	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂²×#108	—	2	1	no	—
9	C₂²×#108	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	2	yes	—
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂×#424	—	2	1	no	—
13	C₂×#423	—	2	1	no	—
14	C₂²×#97	C₂²	1	1	yes	characteristic
15	#1946	—	2	2	no	—
16	#1946	C₂²	1	4	yes	—
17	#1849	—	2	2	no	—
18	#1849	C₂²	1	4	yes	—
19	C₂×#339	—	2	1	no	—
20	#1389	C₂²	1	8	yes	—
21	#1388	C₂²	1	8	yes	—
22	#1375	C₂²	1	4	yes	—
23	C₂²×M₁₆	—	2	2	no	—
24	C₂²×M₁₆	—	2	1	no	—
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂×#117	—	2	1	no	—
28	C₂²×C₄²	D₄	1	1	yes	characteristic
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	4	no	—
39	C₂×#108	—	2	4	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	4	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	—	4	2	no	—
44	C₂×#108	—	4	2	no	—
45	C₂×#108	C₂³	1	4	yes	—
46	C₂×#108	C₂³	1	2	yes	—
47	#427	—	2	1	no	—
48	#427	—	2	1	no	—
49	#427	Q₈	1	2	yes	—
50	#427	C₂³	1	2	yes	—
51	C₂×#107	—	2	1	no	—
52	C₂×#106	Q₈	1	2	yes	—
53	#424	—	2	4	no	—
54	#424	—	4	2	no	—
55	#423	—	2	4	no	—
56	#423	—	4	2	no	—
57	C₂×C₄⋊C₈	—	4	2	no	—
58	C₂×C₄⋊C₈	—	4	2	no	—
59	C₂×#97	—	2	2	no	—
60	C₂×#97	—	2	2	no	—
61	C₂×#97	D₄	1	2	yes	—
62	C₂×#97	C₂³	1	2	yes	—
63	#339	—	2	4	no	—
64	C₂³×C₄	—	2	1	no	—
65	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	2	4	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂×M₁₆	—	4	2	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂×M₁₆	—	4	1	no	—
72	C₂×M₁₆	—	4	2	no	—
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	4	1	no	—
77	C₂×M₁₆	—	4	1	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	4	1	no	—
83	C₂²×C₈	—	4	2	no	—
84	C₂²×C₈	D₈	1	2	yes	—
85	C₂²×C₈	D₈	1	2	yes	—
86	C₂²×C₈	C₂×D₄	1	2	yes	—
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₈	#41	1	1	yes	characteristic
91	C₂²×C₈	#41	1	1	yes	characteristic
92	C₂²×C₈	#41	1	2	yes	—
93	C₂²×C₈	#41	1	2	yes	—
94	C₂²×C₈	#41	1	2	yes	—
95	C₂²×C₈	#41	1	2	yes	—
96	C₂²×C₈	C₂⁴	1	1	yes	characteristic
97	#117	—	2	2	no	—
98	#117	—	2	2	no	—
99	#117	—	4	1	no	—
100	#117	—	4	1	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₂²⋊C₄	—	4	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	4	1	no	—
107	C₂×C₄²	SD₁₆	1	2	yes	—
108	C₂×C₄²	SD₁₆	1	2	yes	—
109	C₂×C₄²	C₂×D₄	1	2	yes	—
110	M₃₂	—	2	4	no	—
111	M₃₂	—	2	4	no	—
112	M₃₂	—	4	2	no	—
113	M₃₂	—	4	2	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	—	4	2	no	—
116	C₂×C₁₆	—	4	2	no	—
117	#108	—	2	4	no	—
118	#108	—	2	8	no	—
119	#108	—	4	2	no	—
120	#108	—	4	4	no	—
121	#108	—	4	8	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#106	—	2	2	no	—
125	#106	C₂×Q₈	1	4	yes	—
126	C₄⋊C₈	—	4	8	no	—
127	C₄⋊C₈	—	4	8	no	—
128	#97	—	2	4	no	—
129	#97	—	2	4	no	—
130	C₂⁴	#137	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	8	2	no	—
136	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄	#123	1	1	yes	characteristic
139	C₂²×C₄	#123	1	1	yes	characteristic
140	C₂²×C₄	#124	1	1	yes	characteristic
141	C₂²×C₄	#124	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
146	C₂²×C₄	#137	1	1	yes	characteristic
147	C₂²×C₄	#137	1	1	yes	characteristic
148	C₂²×C₄	#137	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
150	M₁₆	—	4	2	no	—
151	M₁₆	—	4	4	no	—
152	M₁₆	—	4	4	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	8	1	no	—
155	M₁₆	—	8	1	no	—
156	M₁₆	—	8	2	no	—
157	M₁₆	—	8	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	8	1	no	—
167	C₂×C₈	—	8	1	no	—
168	C₂×C₈	—	8	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₈	#123	1	2	yes	—
173	C₂×C₈	#124	1	2	yes	—
174	C₂×C₈	#125	1	2	yes	—
175	C₂×C₈	#125	1	2	yes	—
176	C₂×C₈	#125	1	2	yes	—
177	C₂×C₈	#125	1	2	yes	—
178	C₂×C₈	#126	1	2	yes	—
179	C₂×C₈	#126	1	2	yes	—
180	C₂×C₈	#126	1	2	yes	—
181	C₂×C₈	#126	1	2	yes	—
182	C₂×C₈	C₂×D₈	1	4	yes	—
183	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₈	C₂×#41	1	2	yes	—
185	C₂×C₈	C₂×#41	1	2	yes	—
186	C₂×C₈	C₂×#41	1	1	yes	characteristic
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	C₂×SD₁₆	1	4	yes	—
197	C₁₆	—	4	4	no	—
198	C₂³	—	4	2	no	—
199	C₂³	#474	1	1	yes	characteristic
200	C₂³	#474	1	1	yes	characteristic
201	C₂³	#483	1	1	yes	characteristic
202	C₂³	#483	1	1	yes	characteristic
203	C₂³	#491	1	1	yes	characteristic
204	C₂³	#491	1	1	yes	characteristic
205	C₂³	C₂×#137	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	#400	1	1	yes	characteristic
213	C₂×C₄	#400	1	1	yes	characteristic
214	C₂×C₄	#400	1	1	yes	characteristic
215	C₂×C₄	#400	1	1	yes	characteristic
216	C₂×C₄	#474	1	1	yes	characteristic
217	C₂×C₄	#474	1	1	yes	characteristic
218	C₂×C₄	#475	1	1	yes	characteristic
219	C₂×C₄	#475	1	1	yes	characteristic
220	C₂×C₄	#475	1	1	yes	characteristic
221	C₂×C₄	#475	1	1	yes	characteristic
222	C₂×C₄	#480	1	1	yes	characteristic
223	C₂×C₄	#480	1	1	yes	characteristic
224	C₂×C₄	#480	1	1	yes	characteristic
225	C₂×C₄	#480	1	1	yes	characteristic
226	C₂×C₄	#483	1	1	yes	characteristic
227	C₂×C₄	#483	1	1	yes	characteristic
228	C₂×C₄	#486	1	1	yes	characteristic
229	C₂×C₄	#486	1	1	yes	characteristic
230	C₂×C₄	#486	1	1	yes	characteristic
231	C₂×C₄	#486	1	1	yes	characteristic
232	C₂×C₄	#491	1	1	yes	characteristic
233	C₂×C₄	#491	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#123	1	1	yes	characteristic
236	C₂×C₄	C₂×#124	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
239	C₂×C₄	C₂×#137	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	8	2	no	—
242	C₈	—	8	4	no	—
243	C₈	#400	1	2	yes	—
244	C₈	#400	1	2	yes	—
245	C₈	C₂×#125	1	2	yes	—
246	C₈	C₂×#126	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	#2099	1	1	yes	characteristic
255	C₂²	#2099	1	1	yes	characteristic
256	C₂²	#2099	1	1	yes	characteristic
257	C₂²	#2099	1	1	yes	characteristic
258	C₂²	C₂×#474	1	1	yes	characteristic
259	C₂²	C₂×#483	1	1	yes	characteristic
260	C₂²	C₂×#491	1	1	yes	characteristic
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	#2099	1	1	yes	characteristic
264	C₄	#2099	1	1	yes	characteristic
265	C₄	#2099	1	1	yes	characteristic
266	C₄	#2099	1	1	yes	characteristic
267	C₄	C₂×#400	1	1	yes	characteristic
268	C₄	C₂×#475	1	1	yes	characteristic
269	C₄	C₂×#480	1	1	yes	characteristic
270	C₄	C₂×#486	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	#13540	1	2	yes	direct product factor
278	C₂	C₂×#2099	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#2099
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #13540	C₂	direct (N × H)
277. C₂	#13540	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×#1946	2
C₂×#1849	2
#13540	2
C₂×#1389	2
C₂×#1388	2
C₂×#1375	2

Subgroups — C2×#13540 #499557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13540 #499557